measuring mortality, fertility, and natural increase

Measuring Mortality, Fertility, and Natural Increase A Self-Teaching Guide to Elementary Measures

James A. Palmore and Robert W. Gardner

M X

z X

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Measuring Mortality, Fertility, and Natural Increase


F I F T H E D I T I O N

]ames A . Palmore and Robert W. Gardner

East-West Center Honolulu

About the Authors

fames A. Palmore was a senior fellow in the East-West Center's Program on Population and professor of sociology and population studies at the University of Hawaii. He directed the university's Population Studies Program from 1976 to 1991 and for 30 years taught graduate-level courses in demography, survey design, and statistical analysis. He was the author of numerous articles on Asian demography and coeditor of several books, including Choosing a Contraceptive: Method Choice in Asia and the United States.

Robert W. Gardner, who lives in Maine, teaches demographic methods at Harvard University's School of Public Health and indexes demographic books (including this one). For 20 years prior to 1992 he was a research associate in the East-West Center's Program on Population and taught courses in the Population Studies Program and the School of Public Health at the University of Hawaii. During 1990-92 he served as assistant director of the Program on Population. His publications include Asian and Pacific Americans in the United States (with Herbert Barringer and Michael Levin) and Migration Decision Making (edited with Gordon F. De Jong).

© 1994 by East-West Center. All rights reserved. Second printing 1996. Printed in the United States of America.

L I B R A R Y O F C O N G R E S S C A T A L O G I N G - I N - P U B L I C A T I O N D A T A

Palmore, James A. Measuring mortality, fertility, and natural increase : a self-

teaching guide to elementary measures / James A. Palmore and Robert W. Gardner. — [Rev. ed.]

p. cm. Includes bibliographical references and index. ISBN 0-86638-165-1 (pbk. : alk. paper) : $15.00 1. Demography. 2. Vital statistics. 3. Mortality—Statistical

methods. 4. Fertility, Human—Statistical methods. I. Title. HB849.4.P34 1996 304.6'01'5195—dc20 94-36934

CIP Published in 1994 by the East-West Center 1777 East-West Road, Honolulu, Hawaii 96848, U.S.A.

Lis t of figures and tables v i i

Preface x i

1. Rates, ratios, percentages, and probabil i t ies 1

2. M o r t a l i t y 9

3. Fer t i l i ty , natural increase, and reproduct ion rates 63

Appendixes

1. No ta t ions and fo rmulas 131

2. Rela t ionship between qx and Mx values 135

3. Answers to selected exercises 139

4. Count r ies w i t h populat ions of fewer than 1 m i l l i o n

(1990 estimates or latest census) 149

References 153

Index 159

LIST OF FIGURES A N D T A B L E S

F I G U R E S 2.1. Age-specif ic death rates: Gua temala , 1985, and

Japan, 1989 12

3.1. Age-specif ic f e r t i l i ty rates: Cos ta R ica , 1984,

Guatemala , 1985, Japan, 1989, and U n i t e d States,

1988 75

T A B L E S 1.1. Rat ios frequent ly used i n demographic work 2

1.2. C a l c u l a t i o n of the number of person-years l i ved

dur ing one year i n a hypothet ica l s m a l l t o w n hav ing

a popula t ion of 700 persons on January 1 and very

erratic demographic behavior 4

2.1. Highest and lowest crude death rates, by region:

recent years 11

2.2. Highes t and lowest age-specific death rates, by sex:

recent years 13

2.3. Age-specif ic and crude death rates for three hypo

thet ical populat ions 16

2.4. Age-specif ic death rates and populat ions for M a i n e

and South Caro l ina : 1930 17

2.5. Age standardization of crude death rates: Japan

(1989) and C h i l e (1989-90) 23

2.6. Formulas for direct age-standardization of the crude

death rate for two hypothet ica l populat ions 25

2.7. Standardized death rates for selected places and

years 26

2.8. Infant mor ta l i ty rates for selected countries, by age

and region: recent years 32

viii List of Figures and Tables

2.9. Highes t and lowest infant mor ta l i ty rates, by region:

latest available data 34

2.10. Comple t e l i fe table for females: U n i t e d States,

1979-81 38

2.11. Abr idged l i f e table for females: U n i t e d States,

1988 48

2.12. Age compos i t ion of the stationary popula t ion and

the actual popula t ion for U . S . wh i t e females:

1988 52

2.13. Crude death rates and life-table death rates for U . S .

wh i t e females: selected years, 1900-88 53

2.14. Survivors to exact age x(ej of 100,000 male infants

[t0): Sri Lanka , 1920-81, and U n i t e d States, 1900-02

to 1988 55

2.15. Examples of h igh and l o w values of l i f e expectancy

at b i r th (e0) for males and females: recent years 57

2.16. L i f e expectancies at selected exact ages for males and

females, by color: U n i t e d States, 1988 57

2.17. L i f e expectancies for wh i t e males and females

at exact ages 0, 40, and 70: U n i t e d States,

1850-1988 58

3.1. Highest and lowest crude b i r th rates, by region:

recent years 67

3.2. D i s t r i b u t i o n of countries by leve l of crude b i r th rate:

1980s 68

3.3. Average crude rates of natural increase, by region:

1985-90 69

3.4. Approx ima te number of years a popula t ion takes to

double, triple, and quadruple i n size, given specif ied

rates of growth (based on the compound interest

fo rmu la of Pn = P0 [l+r\n] 70

List of Figures and Tables ix

3.5. Est imates of midyear populations, by region: selected

years, 1650-1990 71

3.6. Lowest and highest age-specific fe r t i l i ty rates per

1,000 w o m e n : 1980s 76

3.7. Age-specif ic f e r t i l i ty rates per 1,000 w o m e n :

Hutter i tes and a l l U . S . w o m e n , around 1940 77

3.8. D i s t r i b u t i o n of major countries and territories by

level of age-specific fe r t i l i ty rates: recent years 77

3.9. B i r t h rates by l ive b i r th order and percentage change

i n rates: U n i t e d States, selected years,

1942-88 83

3.10. C r u d e b i r th rates and direct ly standardized b i r t h

rates: selected places and dates 86

3.11. Crude b i r th rates and di rect ly standardized b i r th

rates: U n i t e d States, selected years, 1940-88 87

3.12. Genera l f e r t i l i ty rates for selected countries, age-

standardized by the direct method: late 1980s

(standard = Sweden, 1988) 88

3.13. Observed general f e r t i l i ty rates and age-standardized

general f e r t i l i ty rates w i t h Sweden (1988), India

(1990), and the Republ ic of Korea (1990) as standard

populat ions: Egypt, Phi l ipp ines , Sweden, U n i t e d

States, and Japan 89

3.14. C a l c u l a t i o n of total f e r t i l i ty rates for the U n i t e d

States: 1957, 1976, and 1988 93

3.15. C a l c u l a t i o n of the gross reproduct ion rates for Cos t a

R ica : 1984 96

3.16. Es t imated crude b i r th rates and gross reproduct ion

rates for w o r l d regions: 1985-90 98

3.17. D i s t r i b u t i o n of countries by w o r l d region and leve l of

gross reproduction rate: 1985-90 99

x List of Figures and Tables

3.18. C a l c u l a t i o n of the gross and net reproduct ion rates

and the length of a generation for the nonwhi t e

populat ion: U n i t e d States, 1988 100

3.19. Measures of reproduction: selected countries, recent

years 102

3.20. Gross and net reproduction rates: Europe, Great

Depression years, pos t -Wor ld War II, and recent

past 104

3.21. Gross and net reproduction rates, by color: U n i t e d

States, 1905-10 to 1988 105

3.22. Percentage ever marr ied and number of ch i ld ren ever

born for w o m e n of ages 40-44 and 30-34: U n i t e d

States, selected years 1940-90 113

3.23. N u m b e r of ch i ldren ever born per 1,000 w o m e n and

per 1,000 ever-married w o m e n , by age: U n i t e d

States, selected years 1940-90 114

3.24. Average number of chi ldren ever born per w o m a n

among ever-married w o m e n of ages 45-49, by

province and urban/rural residence: Indonesia, 1980

and 1990 115

3.25. B i r t h probabil i t ies w i t h i n successive b i r t h intervals

2, 3, and 4-8, by durat ion of in te rva l and contracep

tive use status: Phi l ipp ines and Republ ic of Korea,

1973-74 122

3.26. Year i n w h i c h any b i r th in terval had to begin, g iven a

woman 's age at the beginning of the in terval and her

age at the t ime of being interviewed, for a survey

tak ing place i n 1990: a l l intervals 126

Rona ld Freedman f i rs t suggested the idea of w r i t i n g a series

of self-teaching guides about elementary demographic mea

sures. F o l l o w i n g up on his idea, fames Pa lmore drafted t w o

short manuals i n the f a l l of 1969: Measuring Mortality: A

Self-Teaching Guide to Elementary Measures and Measuring Fertility and Natural Increase: A Self-Teaching Guide to Elementary Measures. In 1971 revised drafts of the manuals

were publ ished as Papers of the East-West Popula t ion Insti

tute, N o s . 15 and 16. Several more revisions fo l lowed .

Robert Gardner joined this enterprise fo r the f o u r t h

edi t ion, publ ished i n 1983. W i t h that edi t ion we combined

the two manuals in to a single short textbook. T h e Guide, as

i t has come to be called, has gone through seven pr int ings

and been translated in to Chinese , Indonesian, and A l b a n i a n .

We hope y o u w i l l f i n d this f i f t h edi t ion as usefu l . W h i l e

the basic content of this edi t ion remains the same as i n the

four th edi t ion, we have rewri t ten the text to make the ma

terial more accessible, updated the tables to inc lude more

recent data, used more recent research and advanced meth

ods as examples i n the text, and changed the exercises to

reflect the present demographic s i tuat ion.

T h i s v o l u m e may be usefu l for several k inds of reader.

In in t roductory courses on popula t ion issues, usua l ly nei

ther the instructor nor the students want to spend m u c h class

t i m e d iscuss ing such basic demographic measures as the

crude b i r th rate. T h e G u i d e is designed to f ami l i a r i z e gradu

ates, undergraduates, and even advanced high school students

w i t h most measures pf mortal i ty , fer t i l i ty , and natural i n

crease that are l i k e l y to be encountered i n such a course. It

may also be used as an in t roductory text i n courses on demo

graphic methods . S ince mos t demograph ic me thods are

readily mod i f i ed for other k inds of socia l science measure

ment , a th i rd use is i n socia l science courses that do not con-

xii Preface

centrate on popula t ion studies. For example, the ins t ruc tor

of an in t roductory course i n methodology for sociologis ts

might draw on this book for some por t ion of the course.

The Guide is designed for self-teaching. M o s t ins t ruc

tors w i l l f i n d that they need a m a x i m u m of three hours of

class t ime to review the exercises w i t h students and c l a r i fy

any points that are confusing. O u r classroom experience w i t h

these exercises, many of w h i c h appeared i n earlier edi t ions

of the Guide, and comments we have received f r o m other

instructors both i n the U n i t e d States and i n A s i a have en

abled us to incorporate changes that shou ld enhance th is

edition's self-teaching value.

Each chapter explains elementary pr inciples of demo

graphic measurement , and each consists of the f o l l o w i n g

parts: (a) def in i t ions of measures and examples of their usual

values, (b) exercises and questions for the student that em

phasize interpretat ion rather than computa t ion , and (c) ref

erences to other sources that use the measures i n interest ing

or important ways.

For m u c h of the discussion, we borrow heavi ly f r o m

standard references on demographic methods , i n c l u d i n g :

George W. Barclay, Techniques of Population Analysis (New

York : John W i l e y and Sons, Inc., 1958); A . J. Jaffe, Handbook

of Statistical Methods for Demographers (Washington, D . C . :

U . S . Bureau of the Census , U . S . G o v e r n m e n t P r i n t i n g Of

fice, 1951); and H e n r y S. Shryock, Jacob S. Siegel, and Asso

ciates, The Methods and Materials of Demography (Wash

ington, D C : U . S . Gove rnmen t P r in t i ng O f f i c e , 1971), 2 v o l

umes. (A condensed edi t ion of this last w o r k is available i n

one vo lume and is s t i l l i n pr int f r o m A c a d e m i c Press, 1978.)

Other valuable resources have been a manuscr ip t copy of the

" M a n u a l of Demographic Research Techniques ," by D o n a l d

J. Bogue and Eve lyn Kitagawa, and various manuals of the

U n i t e d Na t ions . A d d i t i o n a l ci tat ions are provided at appro

priate places i n the text, and a complete l is t of references is

found at the end of the Guide.

For data i n our tables we have rel ied most heavi ly on

several basic sources, i nc lud ing the U n i t e d N a t i o n s Demo-

Preface xiii

graphic Yearbook series and other U N publ ica t ions , K e y f i t z

and Flieger's World Population Growth and Aging: Demographic Trends in the Late Twentieth Century (1990), and Vital Statistics of the United States. A good s u m m a r y source

of data on countr ies of the w o r l d is the annual World Popula

tion Data Sheet publ ished by the Popula t ion Reference B u

reau.

In preparing a vo lume l i k e this, the authors are a lways

indebted to colleagues and students w h o have pat ient ly read

earlier versions and contr ibuted to the f i n a l c la r i ty of the

product through their comments . We are part icularly indebted

to Rona ld Freedman, w h o not on ly f irst suggested the idea

but also provided several of the exercises used here and made

many valuable comments . We have also benefi ted f r o m c o m

ment s made a long the w a y by R e y n o l d s Far ley, N a t h a n

K e y f i t z , Susan Palmore, M o n i c a Fong, D a v i d Swanson, D a v o r

Jedlicka, M e a d C a i n , J. S. M a c D o n a l d , Robert Retherford ,

Peter Xenos, Sandra Ward, Robert Hearn , Maureen St. M i c h e l ,

A n d r e w Kantner, and G r i f f i t h Feeney. We were very for tu

nate to have Sandra Ward as our editor for this edi t ion. T h a n k s

are also due to C o n n i e Kawamoto , Steven Swapp, Lo i s Bender,

C l i f f o r d Takara, and Russe l l Fu j i t a for their assistance i n pre

paring the Guide for publ ica t ion .

F ina l ly , we are grateful to the m a n y students at the

U n i v e r s i t y of M i c h i g a n and the U n i v e r s i t y of H a w a i i and to

numerous part icipants i n East-West Cente r workshops w h o

discovered errors and tried their best to save us f r o m m a k i n g

s imple matters seem complex. Nevertheless , we must bear

responsibi l i ty for any errors that remain . We w o u l d appreci

ate i t if y o u w o u l d br ing them to our at tention.

C H A P T E R 1

Rates, Ratios, Percentages, and Probabilities

We can measure the occurrence or inc idence of an event

(death, for example) i n many ways. In this chapter we dis

cuss various types of rates, ratios, percentages, and probabi l i

ties, us ing the measurement of mor ta l i ty as an i l lus t ra t ion .

W h e n we as demographers measure an event, we wan t

to be precise about:

(a) the t ime per iod to w h i c h w e are referring,

(b) the group of people to w h i c h we are referring, and

(c) the type of occurrence w e are measuring.

Differences i n the spec i f ic i ty of each of these three fac

tors are responsible for the existence of many different de

mographic measures.

R A T I O S . Y o u are probably already f ami l i a r w i t h the everyday use of

P R O P O R T I O N S , ratios, proportions, and percentages. Examples of everyday

A N D P E R C E N T A G E S ratios are:

(a) " I ' l l give y o u odds of 3 to 1 that Russ ia w i n s the G o l d

M e d a l for gymnast ics at the next S u m m e r O l y m p i c s "

and

(b) "Lee's Supermarket is twice as expensive as Fong's."

General ly, a ratio is a single number that expresses the

relative size of two other numbers. T h e result of d i v i d i n g a

number X by another number Y is the ratio of X to Y; that is,

X — = ratio ofX toY. Y

M a n y ratios are used i n demographic, measurement ,

several of w h i c h are defined i n Table 1.1. For any ratio, we

2 Rates, Ratios, Percentages, and Probabilities

should specify careful ly what type of event or popula t ion is

the referent. For example, the sex ratio, or number of males

per 100 females, might refer to:

(a) the total popula t ion of the U n i t e d States i n 1994,

(b) persons 15-34 years of age i n Ba l i , Indonesia, i n 1987,

or

(c) l ive bir ths occurr ing i n H o n g K o n g i n 1985, 1986, and

1987.

We can also use the sex ratio i n mor t a l i t y analysis . For

example, we might compare the number of male deaths w i t h

the number of female deaths f r o m a certain disease.

A proportion is a special type of ratio i n w h i c h the de

nomina to r includes the numerator. We might , for example,

calculate the proport ion of a l l deaths that occurred to males,

as i n the f o l l o w i n g fo rmu la :

Proportion of deaths that occurred to males

D™ Dm + D'

deaths to males deaths to males plus deaths to females.

Table 1.1. Ratios frequently used in demographic work

Ratio Formula" Definition

Dependency ratio

P + P P

50*15

100 x no. of persons under 15

or over 64 no. of persons 15-64

years old

Sex ratio P/

100 x no. of males in group i no. of females in group i

Population density a.

no. of persons in geographic area i

no. of sq. km. {or miles) of land area in geographic area i

Child-woman ratio

5^0

35* 15

So Pf

30 15

1,000 X no. of children under

five years no. of females 15-49

or 15-44 years old

Note: See Appendix 1 for a discussion of the notation system and formulas used in this volume. a. The symbol ~> stands for infinity. In the formulas throughout this volume it indicates an open-ended age group. For example, _P t f refers to the population 65 and over, or 65t.

Rates, Ratios, Percentages, and Probabilities 3

A percentage is a special type of p ropor t ion , one i n

w h i c h the ratio is m u l t i p l i e d by a constant, 100, so that the

ratio is expressed per 100. If y o u leaf through the tables i n

this vo lume, y o u w i l l see many examples of ratios, propor

tions, and percentages. A l l of these s imple measures are use

f u l to the demographer.

R A T E S General ly , ratios and percentages are usefu l for ana lyz ing the

compos i t ion of a set of events or of a popula t ion . Rates, i n

contrast, are used to study the dynamics of change. A rate

refers to the occurrence of events over a g iven in te rva l i n

t ime. We can define a rate of inc idence i n general terms as

fo l lows :

Rate of incidence •

number of events that occur within. _ ^ a given time interval number of members of the population who were exposed to the risk of the event during the same time interval.

Spec i fy ing the number of persons "exposed to r i s k " i n

the denominator is important . If y o u were s tudying mor ta l

i ty over a one-year period i n country A , y o u should note that

a person w h o died before the year ended was not exposed to

risk for the w h o l e year, nor was a c h i l d w h o was born half

way through the year. People w h o moved to count ry A on ly

one m o n t h before the year ended were not exposed to the

risk of dy ing i n country A for the who le year either.

T h e concept of "person-years l i v e d " is the ideal way to

specify the popula t ion exposed to the risk of an event and

thus the ideal denominator for a demographic rate. It is s i m

p ly the product of the number of persons m u l t i p l i e d by the

number of years, or fract ions of years, that each person l i ved

i n a given place. Table 1.2 presents the ca lcu la t ion pf person-

years l ived for a hypothet ical s m a l l town. N o t e that the popu

la t ion at the beginning and the end of the year, 700, differs

f r o m the number of person-years l ived . T h e example is un

usual because, among other i tems:

(a) no net growth occurred i n the town,

(b) 200 people died on one day (January 15), and


(c) 100 people arrived f r o m elsewhere on one day (Octo

ber 25).

Such occurrences w o u l d be h igh ly unusual , but they

i l lustrate how the number of person-years l ived can be qui te

different f r o m the popula t ion at either the beginning or the

end of a period under study.

T h e ca lcu la t ion of actual person-years l i ved for a real

popula t ion of any large size w o u l d be d i f f i cu l t , if not impos

sible. For this reason, most demographic rates use an approxi-

Table 1.2. Calculation of the number of person-years lived during one year in a hypothetical small town having a population of 700 persons on January 1 and very erratic demographic behavior

Number Number Number of of of days person-years persons Events and dates lived lived

700 Alive on January 1

493 Lived in the town continuously from January 1 to December 31 179,945 493.00

1 Born January 11 354 .97 1 Born January 11, died November 9 302 .83

200 Died January 15 3,000 8.22 1 Born February 21, died April 27 65 .18 1 Born March 6, died March 31 25 .07 2 Died April 8 196 .54

94 Born April 10 24,910 68.25 4 Arrived from outside town April 18 1,032 2.83 1 Died June 1 152 .42 1 Died June 5 156 .43 1 Born June 7 207 .57 1 Died June 22 173 .47 1 Born June 24 190 .52 1 Died June 30 181 .50 1 Left town August 16 228 .62 1 Born August 24 129 .35 1 Born September 13, died November 13 61 .17 1 Born October 1 91 .25 2 Born October 7 170 .46 2 Born October 19 146 .40

100 Arrived from outside town October 25 6,700 18.36

Total person-years lived 598.41

700 Alive on December 31

Source: Modified from Barclay 11958, 39).


m a t i o n of person-years l i ved i n the denominator . We assume

that bir ths, deaths, and movements i n and out of the popula

t i on occur at u n i f o r m intervals , or " smooth ly , " dur ing the

per iod under study. If this assumpt ion is true, then the n u m

ber of people a l ive at the midd le of the year (July 1) w i l l equal

the number of person-years l ived . T h i s popula t ion a l ive at

the midd le of the year is cal led the midyear or central popu

la t ion , and so a death (or birth) rate w i t h the midyear popula

t ion as a denominator is k n o w n as a central rate.

If (as we have assumed) bir ths , deaths, and movemen t s

i n and out of the popula t ion are evenly dis t r ibuted through

out the year:

(a) for every b i r t h at midn igh t on January 1, there is one at

midn igh t on December 30. T h e average number of per

son-years l ived for the two bir ths is:

(b) for every death at midn igh t on January 15, there is one

at midn igh t on December 16. T h e average number of

person-years l ived for the two deaths is:

T h i s is w h y the midyear popula t ion (which is of ten ca lcu

lated by t ak ing the average of the popula t ion at the begin

n ing and the end of the year) is usua l ly a good approx imat ion

of person-years l ived . No te , however, the s igni f icance of the

assumpt ion of evenly dis t r ibuted bir ths , deaths, and move

ments i n and out of the popula t ion . In a popula t ion subject

to condi t ions l i k e the s m a l l t o w n of Table 1.2, the midyear

popula t ion , whether actual ly measured or calcula ted as an

average, is not a good, approximat ion of the number of per

son-years l i v e d .

We can further i l lustrate the errors that might arise f r o m

us ing the midyear popula t ion approximat ion for person-years

l i ved w i t h two more real is t ic examples. In the f irst , a " c o l

lege t o w n " whose m a i n indus t ry is higher educat ion migh t

have h igh percentages of its popula t ion out of t o w n every

January 1st, celebrating N e w Year's D a y at home. Es t imat -


ing person-years l ived i n such a t own by t ak ing the average

of the popula t ion on successive January firsts w o u l d lead to a

substantial underenumerat ion of the true number of person-

years l ived . A second example involves an agr icul tura l v i l

lage w i t h considerable temporary i n -mig ra t ion du r ing the

harvest season at the midd le of the year. In this example the

actual size of the popula t ion at midyear w o u l d be an over

estimate of the number of person-years l ived dur ing the w h o l e

year. A s both examples i l lustrate , i n cer ta in s i tuat ions (espe

c i a l ly s m a l l populations) and on certain occasions, use of the

midyear populat ion, actual or est imated, can lead to inaccu

rate approximat ions of person-years l i ved . In the vast major

i ty of si tuations, however, the midyear popula t ion is an ac

ceptable approximat ion .

To calculate a midyear popula t ion , y o u w o u l d usua l ly

take the popula t ion on January 1 of y e a r X , add it to the popu

la t ion on January 1 of year X + 1, and then d iv ide by 2. For

our s m a l l t o w n i n Table 1.2, the midyear popula t ion is [(700

+ 700J/2] = 700. A l t h o u g h the convent ions used by various

countries for report ing and es t imat ing their popula t ions dif

fer, i t shou ld usua l ly be possible to calculate the midyear

popula t ion by using the average or some other s imp le ap

proach.

E X E R C I S E i Cons t ruc t a s m a l l hypothe t ica l popula t ion , spec i fy ing the

same characteristics and events as are specif ied i n Table 1.2.

Ca lcu la te the midyear populat ion. Ca lcu la t e the number of

person-years l ived . A r e they close to the same value? If not,

w h y not?

A N O T E O F Because demographers come f r o m various academic d i sc i -

C A U T I O N pl ines and for h is tor ica l reasons, the "rates" used by demog

raphers are not always rates as we have described them above.

By convent ion , some ordinary percentage figures are cal led

rates. One example of such usage is the " l i te racy rate," w h i c h

is s i m p l y the percentage of the adul t p o p u l a t i o n that is


l i terate .You must learn h o w to determine whether a rate is

real ly a rate, a s imple percentage, or someth ing else. In each

case, the de f in i t ion of the measure should be clear enough to

a l low readers to decide whether i t is a rate or another type of

measure. M o s t of the rates discussed i n this guide are, for tu

nately, real rates,- the exceptions are the reproduct ion rates

discussed i n Chapter 3, w h i c h are more l i k e probabil i t ies than

rates. 1

P R O B A B I L I T I E S A s we have noted, rates refer to the occurrence of events

over a g iven in te rva l of t ime. T h e denominator of a rate is,

ideally, the number of person-years of exposure and more

c o m m o n l y the average or midper iod popula t ion exposed to

the event i n quest ion. A probability is s i m i l a r to a rate, w i t h

one important difference: the denominator is composed of

a l l those persons i n the given popula t ion at the beginning of

the period of observation. Thus , if 10 people die i n one year

out of a popula t ion that numbered 1,000 at the start of the

year, we say that the probabi l i ty of dy ing for this group dur

ing that year was 10/1000, or 0.01000. N o t e that this is dif

ferent f r o m the death rate for the same period, w h i c h w o u l d

be (if the deaths were evenly distributed) 10/(1/2(1000+990)]

= 10/995 = 0.01005. For populat ions experiencing on ly deaths

(and not migra t ion or births), probabil i t ies of dy ing w i l l al

ways be smal ler than the comparable death rates, because

the numerators w i l l be the same but the denominators w i l l

be larger. We deal w i t h the concept of probabili t ies more w h e n

we reach the discussion of the l i f e table and nqx i n Chapte r 2.

W i t h this brief i n t roduc t ion to concepts used i n the

measurement of any demographic event, we now tu rn to the

measures used i n s tudying morta l i ty .

1. The survival ratios used in connection with the life table are sometimes called survival rates, but not in this Guide.

C H A P T E R 2

Mortality

For h is tor ica l reasons we begin w i t h measures of mor ta l i ty .

Throughout most of human his tory the fate of a popu la t ion—

whether i t grew, stagnated, or fa i led to survive—depended

more on mor ta l i ty than on fe r t i l i ty or migra t ion . U n t i l four

decades ago mor ta l i ty and its cont ro l were the central issue

i n popula t ion po l i cy and of chief demographic interest for

most of the world 's countries. Fer t i l i ty and migra t ion gained

the demographic spotlight on ly recently. Consequently, m u c h

of the earliest w o r k on the deve lopment of demographic

measures concentrated on measures of mor ta l i ty . For ex

ample, w o r k on the l i fe table (discussed later i n this chapter)

began as early as the mid-seventeenth century. Here we start

w i t h s imple r measures, the f irst being the crude death rate.

C R U D E D E A T H

R A T E

T h e crude death rate (CDR) is defined as the number of people

w h o die i n a given year divided by the number of people i n

the popula t ion i n the midd le of that year. Conven t iona l ly ,

we express the rate per 1,000 persons. 1 A s a f o r m u l a , w e have:

CDR = 1,000 number of deaths

midyear population

where D = deaths in the year,

P- midyear population, and

k= 1,000.

P

1. You should be sensitive to the (act that different constants (100, 1,000, and 100,000 are common constants) are used for different demographic measures. For example, crude birth rates and crude death rates are usually expressed per 1,000, but growth rates are expressed as percentages. When calculating a rate, it is always safest to proceed without the use of a constant until you get the final answer, then to use the constant to express the rate per thousand, per hundred (percent), or whatever is the usual constant for that type of rate.

10 Mortality

For example, suppose a very sma l l , hypothe t ica l coun

try had a popula t ion of 550 people on December 31, 1980,

and a popula t ion of 650 on December 31, 1981. T h e m i d -

1980 popula t ion w o u l d be: [(550 + 650)/2] = 600. If 15 deaths

occurred i n this s m a l l country i n 1980, the crude death rate

w o u l d be: (15/600 x 1,000) = 25 per 1,000 or, s imply , 25.

In the 1980s, crude death rates for countries w i t h a popu

la t ion of 1 m i l l i o n or more 2 ranged f r o m just over 2 to more

than 23 per 1,000 per a n n u m (see Table 2.1). In other words,

for each 1,000 persons exposed to the r isk of dy ing i n the

1980s, between 2 and 24 died each year. T h e crude death rate

thus indicates that the r isk of death was more then 10 t imes

higher i n the h igh-mor ta l i ty countries than i n the low-mor

ta l i ty countries.

E X E R C I S E i T h e f o l l o w i n g statements about mor ta l i ty are a l l inadequate

i n some way. In what way is each statement inadequate?

1. Ten people died in 1991.

2. Ten people died i n 1991 out of a popula t ion that n u m

bered 1,000 on 31 December 1991.

3. Ten people died out of 1,000 al ive on 30 June 1991 i n

West Count r idad .

T h e correct answers to the exercises and review quest ions

are found i n Append ix 3.

A G E - S P E C I F I C A S its name impl ies , the crude death rate is a crude measure.

D E A T H R A T E S yVe a l l k n o w that a 95-year-old m a n is more l i k e l y to die

than a 20-year-old w o m a n . Soldiers f igh t ing on the f ront l ines

i n a war are more l i k e l y to die than a student i n a lmost any

class. In other words, different subgroups i n a popula t ion are

exposed to different r isks of dying—because of their occupa

t ion or their age or some other characterist ic.

Because of these differences i n exposure to the risk of

dying, demographers often use specific death rates. A spe-

2. The omitted countries are listed in Appendix 4.

Table 2.1. Highest and lowest crude death rates, by region: recent years

Crude death rates

Year or <Per ^ Region and country period High Low

Africa Sierra Leone Guinea Mauritius Tunisia

America, North Haiti United States Costa Rica Jamaica Panama

America, South Uruguay Argentina Venezuela Chile

Asia (excluding former USSR) Yemen Lads Kuwait United Arab Emirates

Europe (excluding former USSR) Hungary Former German Democratic

Republic Albania Spain

Former USSR Ukraine Azerbaijan

Oceania' Papua New Guinea Australia New Zealand

1985-90 23.4* 1985-90 22.0' 1985-90 6.4* 1985-90 7.3*

1985-90 13.2* 1990 8.6 1989 3.8 1988 5.2 1985-90 5.2*

1989 9.6 1988 8.4 1989 4.4 1989 5.8

1990 21.2 1985-90 16.9' 1987 2.2 1985-90 3.8'

1990 13.4

1989 12.4 1989 5.7 1988 8.2

1989 —10.0— b 12 b 6

1985-90 11.6* 1989 7.4 1989 8.2

Source: United Nations Statistical Office, Demographic Yearbook 1990 (1992, table 4). Notes: Countries for which data are known to be incomplete or of unknown reliability have been omitted. Data for periods before circa 1985 have been omitted. Countries with populations of fewer than 1 million are excluded (see list of such countries in Appendix 4). ' Estimates prepared by the Population Division of the United Nations. a. Only three countries in this region have populations of more than 1 million. b. Data from Population Reference Bureau, 1992 World Population Data Sheet (1992).

12 Mortality

c i f i c death rate refers on ly to a subgroup i n a popula t ion . T h e

most c o m m o n l y used specif ic death rates are age-specific

death rates. We define an age-specific death rate ( A S D R ) i n

the f o l l o w i n g fo rmula :

„M= A S D R for age group x to x + n

number of deaths to persons agex, x + n

midyear population of persons age*, x + n aPx

where nDx = deaths to persons of age groupx to x + n,

nP = midyear population of age group* to x + n, and

*= 1,000.

Figure 2.1 shows two typ ica l patterns of age-specific

death rates, one for an economica l ly developed country, the

other for a less developed country. In both cases the death

rates are highest for the very young and the very o ld . T h i s is

the most c o m m o n pattern for age-specific death rates.

L o o k i n g at the m i n i m u m and m a x i m u m figures for age-

specif ic death rates by sex shown i n Table 2.2, y o u should

Figure 2.1. Age-specific death rates: Guatemala, 1985, and Japan, 1989

1,000 I 1

0.1 1

0-4 5-9 10-14 15-19 20-24 24-29 30-34 35-39 40-44 45-49 50-54 55-59 60-«4 65-«9 70-74 75-79 80* Age group

Source: United Nations Statistical Office, Demographic Yearbook 1990 (1992, table 20). Note: The vertical scale of the graph is logarithmic.

Table 2.2. (continued)

Ages Sex

Highest ASDR Lowest ASDR

Ages Sex Rate" Country and year Rate* Country and year

Female '3.4 Bangladesh, 1986 0.3 Ireland, 1988 2.6 Guatemala, 1985 Hong Kong, 1989

Netherlands, 1989 30-34 Male *5.3 El Salvador, 1986 0.8 Japan, 1989

5.3 Puerto Rico, 1988 Female '3.5 Bangladesh, 1986 0.5 Spain, 1986

3.1 Guatemala, 1985 Italy, 1987 Switzerland, 1987 Ireland, 1988 Japan, 1989

35-39 Male *6.4 El Salvador, 1986 0.9 Kuwait, 1986 6.4 Guatemala, 1985

Kuwait, 1986

Female *4.9 Bangladesh, 1986 0.7 Italy, 1987 4.6 Guatemala, 1985 Japan, 1989

Norway, 1989 40-44 Male '7.1 Algeria, 1982 1.8 Kuwait, 1986

7.0 USSR, 1989 Kuwait, 1986

Female '6.6 Bangladesh, 1986 1.0 Hong Kong, 1989 4.7 Guatemala, 1985

45̂ *9 Male 11.1 Hungary, 1989 3.1 Kuwait, 1986 Japan, 1989

Female '7.3 Bangladesh, 1986 1.6 Japan, 1989 5.8 Guatemala, 1985

50-54 Male 16.6 Hungary, 1989 5.2 Japan, 1989 Female •10.1 Bangladesh, 1986 2.6 Japan, 1989

8.8 Egypt, 1986 55-59 Male 26.3 Egypt, 1986 8.8 Japan, 1989

Female '21.4 Bangladesh, 1986 3.8 Japan, 1989 17.8 Egypt, 1986

60-64 Male 34.1 Hungary, 1989 12.8 Japan, 1989 Female •22.4 Bangladesh, 1986 5.7 Japan, 1989

19.9 Egypt, 1986 65-69 Male 60.7 Egypt, 1986 19.5 Japan, 1989

Female 51.8 Egypt, 1986 9.5 Japan, 1989 70-74 Male 73.6 Egypt, 1986 33.8 Japan, 1989

Female 59.5 Egypt, 1986 17.2 Japan, 1989 75-79 Male •110.9 Korea, Rep. of, 1989 57.3 Japan, 1989

100.7 Czechoslovakia, 1989 Female 75.5 Guatemala, 1985 32.0 Japan, 1989

80-84 Male 156.3 Germany, Dem. Rep., 1988 88.7 Hong Kong, 1989 Female 127.4 Romania, 1989 57.2 Hong Kong, 1989

85+ Male 276.9 Singapore, 1988 117.2 Hong Kong, 1989 Female 256.3 Mexico, 1985 95.7 Hong Kong, 1989

Source: United Nations Statistical Office, Demographic Yearbook 1990 (1992, table 20). Notes: Many of the rates are estimates that vary in reliability. An asterisk (*) indicates that the data on which the highest rate is based are incomplete or of unknown reliability, and the highest reliable estimate is also given. Data are unavailable for the oldest ages in many countries. Countries with populations of under 1 million are excluded; a list of such countries appears in Appendix 4. a. Rates per 1,000 population.

Table 2.2. Highest and lowest age-specific death rates, by sex: recent years


Ages Sex Rate' Country and year Rate* Country and year

<1 Male * 105.1 98.8

Algeria, 1982 Egypt, 1986

4.8 Japan, 1989

Female *95.4 91.3


4.2 Japan, 1989

1-4 Male '12.5 7.6


0.3 Ireland, 1988 Sweden, 1988 Hong Kong, 1989 Norway, 1989

5-9 Male *2.9 Bangladesh, 1989 0.2 Denmark, 1987 Finland, 1987 France, 1987 Switzerland, 1987 Puerto Rico, 1988 Sweden, 1988 Australia, 1989 Hong Kong, 1989 Japan, 1989 Netherlands, 1989 United Kingdom, 1989

Female 2.0 '2.8

Guatemala, 1985 Bangladesh, 1986

0.1 Austria, 1989 Hong Kong, 1989 Japan, 1989 Netherlands, 1989

10-14 Male '1.7 1.3

Bangladesh, 1989 Guatemala, 1985

0.2 Denmark, 1987 Germany, Fed. Rep., 1988 Italy, 1987 Switzerland, 1987 France, 1988 Ireland, 1988 Israel, 1988 Sweden, 1988 Austria, 1989 Hong Kong, 1989 Japan, 1989 Netherlands, 1989 United Kingdom, 1989

Female *1.1 1.1


0.1 Israel, 1988 Japan, 1989

15-19 Male '3.6 1.9

Iran, 1986 Guatemala, 1985

0.4 Hong Kong, 1989

Female '2.3 1.4


0.2 Italy, 1987 Japan, 1989

20-24 Male *5.1 3.3

El Salvador, 1986 Guatemala, 1985

0.6 Hong Kong, 1989

Female '3.1 Bangladesh, 1986 0.3 Italy, 1987 Israel, 1988 Japan, 1989 United Kingdom, 1989

25-29 Male '5.3 4.0

El Salvador, 1986 Guatemala, 1985

0.7 Hong Kong, 1989 Japan, 1989 Netherlands, 1989

Table 2.2. (continued)

Ages Sex


Ages Sex Rate" Country and year Rate* Country and year

Female •3.4 Bangladesh, 1986 0.3 Ireland, 1988 2.6 Guatemala, 1985 Hong Kong, 1989

Netherlands, 1989 30-34 Male *5.3 El Salvador, 1986 0.8 Japan, 1989

5.3 Puerto Rico, 1988 Female "3.5 Bangladesh, 1986 0.5 Spain, 1986

3.1 Guatemala, 1985 Italy, 1987 Switzerland, 1987 Ireland, 1988 Japan, 1989

35-39 Male *6.4 El Salvador, 1986 0.9 Kuwait, 1986 6.4 Guatemala, 1985

Kuwait, 1986

Female *4.9 Bangladesh, 1986 0.7 Italy, 1987 4.6 Guatemala, 1985 Japan, 1989

Norway, 1989 40-44 Male •7.1 Algeria, 1982 1.8 Kuwait, 1986

7.0 USSR, 1989 Female *6.6 Bangladesh, 1986 1.0 Hong Kong, 1989

4.7 Guatemala, 1985 45-49 Male 11.1 Hungary, 1989 3.1 Kuwait, 1986

Japan, 1989 Female '7.3 Bangladesh, 1986 1.6 Japan, 1989

5.8 Guatemala, 1985 50-54 Male 16.6 Hungary, 1989 5.2 Japan, 1989


55-59 Male 26.3 Egypt, 1986 8.8 Japan, 1989 Female '21.4 Bangladesh, 1986 3.8 Japan, 1989

17.8 Egypt, 1986 60-64 Male 34.1 Hungary, 1989 12.8 Japan, 1989


65-69 Male 60.7 Egypt, 1986 19.5 Japan, 1989 Female 51.8 Egypt, 1986 9.5 Japan, 1989

70-74 Male 73.6 Egypt, 1986 33.8 Japan, 1989 Female 59.5 Egypt, 1986 17.2 Japan, 1989

75-79 Male '110.9 Korea, Rep. of, 1989 57.3 Japan, 1989 100.7 Czechoslovakia, 1989

Female 75.5 Guatemala, 1985 32.0 Japan, 1989 80-84 Male 156.3 Germany, Dem. Rep., 1988 88.7 Hong Kong, 1989

Female 127.4 Romania, 1989 57.2 Hong Kong, 1989 85+ Male 276.9 Singapore, 1988 117.2 Hong Kong, 1989

Female 256.3 Mexico, 1985 95.7 Hong Kong, 1989

Source: United Nations Statistical Office, Demographic Yearbook 1990 (1992, table 20). Notes: Many of the rates are estimates that vary in reliability. An asterisk (*) indicates that the data on which the highest rate is based are incomplete or of unknown reliability, and the highest reliable estimate is also given. Data are unavailable for the oldest ages in many countries. Countries with populations of under 1 million are excluded; a list of such countries appears in Appendix 4. a. Rates per 1,000 population.

Mortal i ty 15

note that data needed to calculate age-specific rates are not

available for m a n y h igh-mor ta l i ty countries. We therefore

present the range as i t appears f r o m est imated rates reported

i n the U n i t e d Na t ions 2990 Demographic Yearbook. H ighe r

A S D R s may exist . 3 T h e lowest rates shown are generally for

European countries. It is l i k e l y that these lower figures are

really the lowest, s ince countries w i t h l o w death rates usu

a l ly also have better systems for co l lec t ing demographic data

on mor ta l i ty than do countries w i t h h igh death rates.

T H E E F F E C T O F

A G E C O M P O S I T I O N

O N T H E C R U D E

D E A T H R A T E

T h e crude death rate is a weighted s u m of age-specific death

rates. Take the f o l l o w i n g s imple calculat ions:

Number of persons Number of Death rate

in midyear deaths in year z Ages population in year z (per 1,000)

0-34 35+ Total, all ages

2,000 1,000 3,000

40 80

120

20 80 40

T h e crude death rate is 40 for this hypothet ica l popula t ion .

It is a weighted s u m of two age-specific rates, 20 and 80. T h e

weights are the proportions of the total popula t ion i n each

age group i n the midyear populat ion. Tha t is,

CDR = 2,000 3,000

40 80 3 3

x20 1,000

120 3,000

= 40

x80 x2oJ + ^ x80

We can express this basic re la t ionship i n the f o l l o w i n g for

m u l a :

CDR:

where nPs = midyear population in age group x to x + n,

P = total midyear population,

„MX = age - specific death rate per 1,000 for age group x to x + n, and

L = the sum of the quantity in brackets for all age groups.

3. In many cases the U N estimates are based on civil registers known to be incomplete or of unknown reliability. The actual ASDRs in such cases are likely to be even higher.

16 Mortal i ty

T h e fact that the crude death rate is a f u n c t i o n of bo th

the age-specific death rates and the age d i s t r ibu t ion is dem

onstrated by the calculat ions for three hypothet ica l popula

tions presented i n Table 2.3. Count r ies A and B have the same

age-specific death rates, but country A's crude death rate is

54 percent higher than country B's. Why? Because count ry A

has a considerably larger proport ion of its popula t ion i n the

youngest age group, w h i c h is subject to higher death rates.

Count r i es B and C have the same crude death rates,

but their age-specific death rates are quite different. C o u n t r y

C has a m u c h larger proport ion of its popula t ion i n the old

est age group (where we might expect to f i n d a higher death

rate), but i n this age group its age-specific death rate is on ly

half of that for countries A and B. Thus , whereas count ry C

has an older popula t ion than either country A or B, i ts crude

death rate is not higher. T h i s example, w h i c h is designed to

demonstrate the relat ionship between the age d i s t r ibu t ion

and age-specific rates, does not necessarily represent realis

t ic figures for actual countries.

T w o populat ions may have the same crude death rate

Table 2.3. Age-specific and crude death rates for three hypothetical populations

Country

Measure A B C

Number of persons in midyear population for age group:

0-4 1,500 500 500 5-39 4,000 5,000 4,000 40+ 500 500 1,500

Number of deaths in age group: 0-4 120 40 50 5-39 40 50 20 40+ 40 40 60

Age-specific death rate (per 1,000) for age group:

0-4 80 80 100 5-39 10 10 5 40+ 80 80 40

Crude death rate (per 1,000) 33.3 21.7 21.7

Mortality 17

even though one has higher death rates than the other i n

every age group. T h i s result w o u l d occur, for example, i f the

popula t ion w i t h the higher age-specific rates were concen

trated i n the age groups between 5 and 45, so that more of i ts

people were subject to l o w death rates. It is even possible for

one populat ion to have a crude death rate that is lower than

another's a l though its death rates are higher at every age.

T h i s paradox is i l lust ra ted i n Table 2.4, w h i c h compares the

death rates for M a i n e and South C a r o l i n a i n 1930, and i t w i l l

be demonstra ted again w h e n we discuss s tandardiza t ion .

South C a r o l i n a had higher A S D R s than M a i n e for every age

group except one. Nevertheless, M a i n e had a higher crude

death rate because a larger proport ion of its popula t ion was

i n the age groups 55 and over, w h i c h experience higher mor

ta l i ty than most younger groups.

We have i l lustrated the idea that a crude death rate can

be subdivided, or decomposed, in to two elements: (1) age-

specif ic death rates and (2) the age d is t r ibut ion , w h i c h deter-

Table 2.4. Age-specific death rates and populations for Maine and South Carolina: 1930

Maine South Carolina

Percentage Percentage ASDR distribution ASDR distribution

(per 1,000 of popu (per 1,000 of popuAges population) Population lation population) Population lation

0-4 20.56 75,037 9.4 23.92 205,076 11.8 5-9 0.86 79,727 10.0 1.85 240,750 13.9 10-14 1.40 74,061 9.3 1.84 222,808 12.8 15-19 2.23 68,683 8.6 4.26 211,345 12.2 20-24 3.70 60,575 7.6 6.45 166,354 9.6 25-34 3.91 105,723 13.3 8.71 219,327 12.6 35-44 5.45 101,192 12.7 12.42 191,349 11.0 45-54 10.85 90,346 11.3 19.94 143,509 8.3 55-64 20.36 72,478 9.1 33.13 80,491 4.6 65-74 .52.19 46,614 5.8 61.47 40,441 2.3 75+ 136.45 22,396 2.8 61.47 16,723 1.0

All ages 796,832 99.9 1,738,173 100.1

Crude death rate (per 1,000) 13.9 12.9

Notes: Deaths and populations of unknown ages are excluded. Percentages do not sum exactly to 100.0 because of rounding.

18 Mortality

mines the proport ions of the popula t ion to w h i c h the age-

specif ic rates apply. We can decompose a crude death rate

in to rates that apply to any set of characterist ics that m igh t

be h e l p f u l i n an analysis, and we can divide, or dis t r ibute ,

the popu la t ion according to those character is t ics . For ex

ample, i t is possible to calculate sex-specific death rates and

the sex d is t r ibu t ion . It is also possible to calculate age-sex-

specif ic death rates (e.g., the death rate for m e n or w o m e n of

ages 20-24) and the d is t r ibu t ion of the popula t ion by age and

sex. Since mor ta l i ty rates do vary s ign i f ican t ly by age and by

sex, and because data on the popula t ion d i s t r ibu t ion and on

deaths are usual ly available by age and sex, i t is c o m m o n for

such age-sex-specific death rates to be calculated. In fact, the

approach and logic are quite general. A l s o calculated are death

rates specif ic to age, sex, and occupat ion s imul taneous ly , al

though data for these are less c o m m o n l y available. Obvious ly ,

the characteristics for w h i c h i t is usefu l to decompose a death

rate (or any other rate) are usual ly those that migh t make a

difference i n the death rate. There w o u l d be l i t t l e po in t i n

ca lcu la t ing death rates spec i f ic to eye color, for example ,

unless eye color had some bearing on morta l i ty .

F I R S T S E T O F 1. In two countries, A and B, the age-specific death rates per

M U L T I P L E - C H O I C E i ; 0 0 0 are as fo l lows :

Ages Country A Country B

0-4 70 70 5-24 5 5 25-44 10 10 45-64 30 30 65+ 80 80

W h i c h of the f o l l o w i n g is true?

(a) T h e crude death rate is higher i n country A than i n

country B.

(b) T h e crude death rate is higher i n country B than i n coun

try A .

(c) T h e crude death rates are equal i n the two countries.

(d) T h e crude death rate i n country A may be higher, lower,

or the same as i n country B.

Mortality 19

2. T h e crude death rates per 1,000 i n countr ies A and B are as

fo l lows for specif ic areas of the two countries:

Areas Country A Country B

Metropolitan areas 15 14 Small towns 17 15 Rural areas 30 29

T h e crude death rate for the whole country is:

(a) def in i te ly less i n A than i n B.

(b) def in i te ly less i n B than i n A .

(c) probably higher i n A than i n B, but the reverse is pos

sible.

(d) probably higher i n B than i n A , but the reverse is pos

sible.

S T A N D A R D I Z A T I O N We have seen that the age compos i t ion of the popula t ion has

a pronounced effect on the crude death rate. O the r aspects of

popula t ion compos i t ion may also affect the death rate. Ex

amples of other variables that of ten in f luence death rates are:

(a) urban or rura l residence, perhaps because of unequal

ava i lab i l i ty of heal th care fac i l i t ies , l i v i n g standards, or

infrastructures;

(b) different occupat ional composi t ions (miners or soldiers

are more subject to risk than are judges or most profes

s ional workers);

(c) different i ncome composi t ions (the wea l thy can afford

better medica l care);

(d) sex (women almost universa l ly have lower death rates

than m e n at most ages); and

(e) mar i ta l status (the marr ied usual ly have lower mor ta l

i ty than the single, w idowed , or divorced).

Since we are interested here i n measur ing mor t a l i t y i tself

rather than age or occupat ional compos i t ion , h o w do we re

move, or control for, the effects of these other variables (which

are said to " c o n f o u n d " the comparison)?

We cou ld s i m p l y look at the detailed schedule of age-

spec i f i c or occupa t ion-spec i f i c or age-occupat ion-spec i f ic

death rates for two countries and compare them. But w o u l d

20 Mortality

i t not be usefu l to have one single measure, such as the crude

death rate, that has somehow taken in to account the effect

of any extraneous variable bel ieved to in f luence the crude

death rate? To obtain this single measure, demographers usu

al ly use a technique k n o w n as standardization.*

Look again at the age compos i t ion of countr ies A and B

i n Table 2.3. If they both had the same age compos i t ion , i t is

obvious that their crude death rates w o u l d be the same—

because they have ident ica l age-specific rates. In standard

iza t ion the procedure is to apply the same age compos i t i on

(or occupat ion compos i t ion or whatever) to different sets of

specif ic rates and observe what the crude rate w o u l d then be.

T h e age compos i t ion used for the standardization is cal led

the standard populat ion. T h e rates used are those of the ac

tual populat ions being studied.

Age standardization is used to answer the question, H o w

w o u l d the crude death rates of two populat ions compare i f

they had exact ly the same age d i s t r ibu t ion (the "s tandard"

we select) but each retained i ts o w n d is t inc t ive age-specific

death rates? In this way we " h o l d constant," or cont ro l for,

the effect of the age d is t r ibut ion , so that any variat ions i n

the total death rates must result f r o m real differences i n age-

specif ic mor ta l i ty rates between the two populat ions.

T h i s same procedure can be applied to any rate c o m

parison that we can separate in to two parts, (1) the effect of

differences i n the d is t r ibut ion of the characterist ic and (2)

the effect of differences i n the characteris t ic-specif ic rates.

For example, we could ask: H o w w o u l d popula t ions A and B

compare on the death rate i f they had the same (standard)

d i s t r ibu t ion by mar i ta l status and different mar i t a l status-

specif ic death rates?

T h e s tandardizat ion technique also applies to m a n y

fields besides demography and to measures other than rates,

such as ratios or percentages. In a study compar ing the per

centages of people vo t ing for a certain po l i t i c a l party i n c i t ies

4. We discuss only the technique of direct standardization in this Guide. For an instructive discussion of indirect standardization, see Barclay 11958, 164-66).

Mortality 21

A and B, we might ask whether the difference results f r o m

differences between the two cities i n the d is t r ibut ion of people

by age and income. We can apply the age and income-spe

c i f i c percentages vo t ing for the party i n each c i ty to a stan

dard age and income d i s t r ibu t ion to ascertain whether the

difference between the cit ies s t i l l persists or is mod i f i ed .

To i l lus t ra te s tandardizat ion, we again begin w i t h a

s imple , hypothet ica l example. T w o countries, A and B, have

the f o l l o w i n g age-specific death rates and age composi t ions :

Country A Country B

Midyear Death rate Midyear Death rate Ages population per 1,000 population per 1,000

0-44 1,000 25 4,000 30 45+ 4,000 40 1,000 45

T h e crude death rates are 37 per 1,000 for count ry A and 33

per 1,000 for country B. Suppose both countries had the age

compos i t ion of count ry A . T h e n the crude death rates w o u l d

be 37 for country A and 42 for count ry B. (Try to dupl icate

this result.) In this case we w o u l d say that count ry A was the

"standard p o p u l a t i o n " and that 42 was the "s tandardized

crude death rate" for count ry B. We can also standardize the

crude death rates us ing count ry B as the standard popula

t ion . In this case the standardized rates are 28 for count ry A

and 33 for country B. N o t e that the choice of standard affects

the absolute values of the standardized death rates. T h e f o l

l o w i n g table summarizes the calculat ions:

Rate Country A Country B

Unstandardized crude death rate per 1,000 37 33 Standardized crude death rate per 1,000

with country A as the standard 37 42 with country B as the standard 28 33

C o u n t r y A has a higher crude death rate than country B. W h e n

we standardize on the age d i s t r ibu t ion of either count ry A or

country B, however, country B has a higher death rate be

cause the age-specific death rates for count ry B are higher

than those for country A i n every age group.

A s an example us ing actual data, we present the ca lcu

lations for an age standardization of crude death rates for C h i l e

22 Mortality

i n 1989-90 and Japan i n 1989 (Table 2.5). T h e crude death

rates are 5.73 per 1,000 for C h i l e and 6.40 per 1,000 for Japan.

Standardized on the age d is t r ibut ion of C h i l e , the Japan death

rate w o u l d be on ly 3.56. Standardized on the age d is t r ibu t ion

of Japan, the C h i l e death rate w o u l d be 9.67. Hence, a l though

the unstandardized crude death rate for Japan is higher than

Ch i l e ' s , the standardized rates (wi th ei ther count ry as the

standard population) show that C h i l e has higher death rates.

T h e age composi t ions of the two countr ies i n 1989-90 were

conspicuous ly different, C h i l e hav ing a m u c h younger popu

la t ion.

Dea th rates at different ages i n the same popula t ion

tend to be h ighly correlated. C o u n t r y A , w i t h l o w death rates

at one age, is l i k e l y to have relat ively l o w death rates at a l l

other ages. C o u n t r y B, w i t h higher death rates at one age,

w i l l usual ly also have higher death rates at a l l ages (the A S D R

curves, as shown i n Figure 2.1, w i l l not cross except perhaps

at the oldest ages). W h e n this is true, i t means that any stan

dard popula t ion selected w i l l produce the same results: the

standardized rate for count ry B w i l l exceed the standardized

rate for country A . T h e reason is that we are m u l t i p l y i n g the

same set of numbers (the standard) by higher numbers for

count ry B than for count ry A for every age. Ord inar i ly , stan

dardizat ion under these circumstances w i l l at least c l a r i fy

the direction of the difference. It w i l l show that count ry B

has higher mor ta l i ty than country A . O f course, the choice

of the standard populat ion, even i n this n o r m a l case, cou ld

affect the amount of the mor ta l i ty difference between coun

tries A and B. Suppose the mor ta l i ty difference between coun

tries A and B is especial ly large for ages 40-49. In that case

the amount of the difference i n the standardized rates w i l l

depend on the proport ion of the standard popula t ion that is

i n the age group 40-49.

Occas iona l ly the s i tua t ion is unclear. It may be that

populat ion A has higher death rates than populat ion B at some

ages but not at others. In such a case, not on ly the amount of

the difference but also the direction after standardization w i l l

depend on the standard age d is t r ibu t ion selected. In si tua-

Table 2.5. Age standardization of crude death rates for Japan (1989) and Chile (1989-90)

Chilean Chilean Japanese Japanese deaths with deaths with deaths with deaths with

own age Japan's age own age Chile's age distribution distribution distribution distribution

Age-specific death rate (per 1,000)

Age-specif: rate (per

ic death 1,000)

(for population

of 1,000)'

(for population

of 1,000]

(for population

of 1,000)

(for population

of 1,000)

(for population

of 1,000)'

(for population

of 1,000]

(for population

of 1,000)

(for population

of 1,000) Chile Japan Chile Japan (5) (6) (7) (8)

Ages (1) (2) (3) (4) (l)x(3) (l)x(4) (2) x (4) (2) x (3)

<1 17.1 4.5 23 10 .39 .17 .05 .10 1-4 0.8 0.4 89 45 .07 .04 .02 .04 5-9 0.3 0.2 103 62 .03 .02 .01 .02 10-14 0.3 0.1 91 72 .03 .02 .01 .01 15-19 0.7 0.4 94 81 .07 .06 .03 .04 20-24 1.1 0.5 94 72 .10 .08 .04 .05 25-29 1.2 0.6 93 65 .11 .08 .04 .06 30-34 1.5 0.6 80 64 .12 .10 .04 .05 35-39 2.0 1.0 68 77 .14 .15 .08 .07 40-44 3.1 1.5 56 82 .17 .25 .12 .08 45-49 4.5 2.3 48 75 .22 .34 .17 .11 50-54 7.1 3.9 38 65 .27 .46 .25 .15 55-59 11.0 6.3 34 62 .37 .68 .39 .21 60-64 15.9 9.1 29 53 .46 .84 .48 .26 65-69 23.3 13.7 22 39 .51 .91 .53 .30 70-74 38.0 24.1 16 29 .61 1.10 .70 .39 75-79 62.8 42.1 11 24 .69 1.51 1.01 .46 80+ 124.5 105.5 11 23 1.37 2.86 2.43 1.16 All ages 1,000 1,000 5.73b 9.67' 6.40d 3.56=

Source: United Nations Statistical Office, Demographic Yearbook 1990 (1992, tables 7,19). a. Age distribution for Chile is for 1990; deaths are for 1989. b. Total is the Chilean crude death rate. c. Total is the Chilean death rate standardized on Japan's age distribution. d. Total is the Japanese crude death rate. e. Total is the Japanese death rate standardized on Chile's age distribution.

24 Mortality

tions like this the process of standardization depends on the

arbitrary choice of a standard and the results are probably

misleading and not very worthwhile. Here is a simple hypo

thetical case:

Country A Country B

Midyear Death rate Midyear Death rate Ages population per 1,000 population per 1,000

0-44 1,000 35 4,000 25 45+ 4,000 50 1,000 75

The crude death rates and the standardized rates are summa

rized in the following table:


Unstandardized crude death rate per 1,000 47 35 Standardized crude death rate per 1,000

with country A as the standard 47 65 with country B as the standard 38 35

In this case, country A has a higher crude death rate than

country B and also has a higher standardized death rate if we

standardize on country B's age distribution. If we standard

ize on country A's age distribution, however, country B has a

higher standardized rate. The choice of the standard popula

tion thus reverses the direction of our answer! In cases like

this, techniques other than standardization, such as the life

table, are often used to summarize the underlying mortality

situation. We will shortly examine life-table functions that

might be used.

Fortunately, cases like the one just presented are not

common. For this reason, and because standardization is rela

tively easy to use, it is widely used by demographers. The

formulas for the age standardization of death rates are given

in Table 2.6. A comparison of the formulas for populations A

and B in the last two rows of the table shows what standard

ization does—namely, it uses the age composition of the stan

dard population as the weights in obtaining the weighted sum

of age-specific rates that cumulate to form the standardized

crude death rate.

Developing countries often have low crude death rates

(see Table 2.7). Their populations are very young—that is,

Mortality 25

they have large proport ions of people i n the youngest age

groups—as a result of their recent h igh b i r t h rates. Age stan

dardizat ion us ing the age compos i t ion of the U n i t e d States

i n 1980 shows that most developing countr ies w o u l d have

m u c h higher crude death rates i f they had the U . S . age corn-

Table 2.6. Formulas for direct age-standardization of the crude death rate for two hypotheti

cal populations

Formula for:

Item Population A Population B

Number of people in age group x,x+n nP!

Total population J C

PB=lnPxB

X

Deaths in age group x,x+n

Deaths in total population X J C

Death rate in age group x,x+n

DA

MA = " * " * pA

n x

nB

MB = - * " " pB

n x

Crude death rate

a y DA

J^A Ami " X

X

y DB

D B Z j n^x

PB~lAB

X

Death rate standardized on age distribution of Population A l n p A

X

lAA

X

Death rate standardized on age distribution of Population B

XU B )Ur) y PB

/ > n x X

X U ' ) U ' ) y PB

JC

Table 2.7. Standardized death rates for selected places and years

Standardized death rate using as the standard:

Crude United death States, Mexico,

Country and year rate 1980 1980

Mauritius, 1970 7.78 16.48 7.80 Mauritius, 1980 7.21 16.26 6.71 South Africa (white), 1970 10.50 15.59 8.54 South Africa (white), 1985 7.90 12.38 5.58 Chile, 1970 8.78 12.98 7.93 Chile, 1980 6.67 11.27 5.09 Mexico, 1970 9.49 14.40 8.56 Mexico, 1983 5.54 11.33 5.44 Venezuela, 1970 6.46 12.14 6.35 Venezuela, 1985 4.56 9.52 4.32 Canada, 1970 7.32 9.39 3.99 Canada, 1980 7.13 8.15 3.29 Canada, 1985 7.15 7.55 2.92 United States, 1970 9.43 10.37 4.54 United States, 1980 8.77 8.77 3.65 United States, 1985 8.74 8.27 3.35 Taiwan, 1970 4.90 12.69 5.27 Taiwan, 1980 4.76 10.20 4.11 Taiwan, 1985 4.80 9.40 3.64 Hong Kong, 1970 5.24 10.32 4.46 Hong Kong, 1980 4.98 7.99 3.18 Japan, 1970 6.83 10.71 4.15 Japan, 1980 6.19 7.95 2.93 Peninsular Malaysia, 1970 6.99 12.64 6.64 Peninsular Malaysia, 1980 5.55 11.54 5.20 Peninsular Malaysia, 1985 5.27 11.08 4.69 Singapore, 1970 5.16 12.61 5.23 Singapore, 1980 5.18 10.99 4.23 Singapore, 1985 5.22 10.37 3.82 Austria, 1970 13.23 12.15 4.99 Austria, 1980 12.32 10.32 4.06 Austria, 1985 11.85 9.42 3.57 Belgium, 1970 12.30 11.39 4.58 Belgium, 1980 11.54 9.83 3.80 Belgium, 1985 11.22 9.05 3.44 Bulgaria, 1970 9.08 10.83 4.51 Bulgaria, 1980 11.05 11.51 4.49 Bulgaria, 1985 12.00 12.00 4.54 Denmark, 1970 9.79 9.42 3.71 Denmark, 1980 10.92 9.05 3.44 Denmark, 1985 11.42 8.82 3.35 France, 1970 10.63 10.09 4.10 France, 1980 10.15 8.80 3.51

Table 2 .7. (continued)

Standardized death rate using as the standard:

Crude United death States, Mexico,

Country and year rate 1980 1980

France, 1985 10.01 8.17 3.19

Fed. Rep. Germany, 1970 12.12 11.80 4.80 Fed. Rep. Germany, 1980 11.60 9.79 3.80 Fed. Rep. Germany, 1985 11.54 8.87 3.31

Greece, 1970 8.76 9.04 4.09 Greece, 1980 9.80 9.04 3.56 Greece, 1985 10.20 8.80 3.34

Hungary, 1970 11.63 12.50 5.28 Hungary, 1980 13.57 12.89 5.19 Hungary, 1985 11.86 12.68 5.15

Italy, 1970 9.68 10.27 4.38 Italy, 1980 9.83 9.42 3.61 Italy, 1983 9.93 9.08 3.39

Netherlands, 1970 8.41 9.51 3.66 Netherlands, 1980 8.08 8.05 3.04 Netherlands, 1985 8.47 7.84 2.89

Norway, 1970 9.99 9.15 3.54 Norway, 1980 10.12 8.13 3.05 Norway, 1985 10.67 7.92 2.99

Poland, 1970 8.20 11.98 5.11 Poland, 1980 9.84 11.63 4.79 Poland, 1985 10.25 11.77 4.67

Spain, 1970 8.42 9.97 4.25 Spain, 1980 7.71 8.50 3.23 Spain, 1983 7.93 7.90 3.02

Sweden, 1970 9.95 8.74 3.34 Sweden, 1980 11.05 8.19 3.01 Sweden, 1985 11.26 7.64 2.77

England and Wales, 1970 11.76 10.72 4.20 England and Wales, 1985 11.83 9.05 3.32

Australia, 1971 8.47 10:81 4.38 Australia, 1980 7.40 8.68 3.41 Australia, 1985 7.53 8.23 3.17

Fiji, 1975 6.89 13.92 7.13 Fiji, 1980 6.40 12.39 6.37

New Zealand, 1970 8.78 13.92 4.93 New Zealand, 1980 8.52 10.18 3.96 New Zealand, 1985 8.38 9.30 3.59

Former USSR, 1979 10.05 12.41 5.83 Former USSR, 1987 9.85 11.16 5.12

Source: Keyfitz and Flieger (1990, 294-583].

28 Mortality

pos i t ion . For example, M a u r i t i u s had a crude death rate of

7.2 i n 1980. Standardized on the age composi t ion of the U n i t e d

States, the rate w o u l d be above 16. S imi l a r results are evi

dent f r o m standardizing the rates for M e x i c o , Singapore, and

M a l a y s i a .

Y o u should make some addi t ional comparisons us ing

Table 2.7 to acquire an i n tu i t i ve feel ing for the effects of age

standardization. N o t i c e , for example, that the amount of the

differences i n rates is affected by the standard populat ion used.

There is a difference of 1.01 points between the crude death

rates of Japan and Singapore i n 1980, Japan's rate being higher.

W h e n the rates are standardized on the age compos i t i on of

the U n i t e d States, the difference is 3.04, but in the other

direction, Singapore's rate being higher. W h e n they are stan

dardized on the age d i s t r ibu t ion of M e x i c o , the difference is

1.30 points, w i t h Singapore's rate higher.

A s a second example, the trend i n the C D R s can be

different f r o m the trend i n the standardized rates. L o o k at

the data for Denmark . T h e crude death rate rose f r o m 1970

to 1985, w h i l e the standardized rates f e l l . Age-spec i f ic mor

ta l i ty rates were actual ly f a l l i ng i n D e n m a r k dur ing that pe

r iod, but this improvement i n mor ta l i ty was not reflected i n

the C D R because Denmark ' s popula t ion was also aging dur

ing the same t ime period.

S E C O N D S E T O F 1. In countries A and B the age-specific death rates per 1,000

M U L T I P L E - C H O I C E are as fo l lows :

Q U E S T I O N S Ages Country A Country B

0-4 40 29 5-24 20 19 25-54 25 22 55+ 60 58

If the crude death rates for the two countr ies are standard

ized on the same age d is t r ibut ion , w h i c h of the f o l l o w i n g is

true?

(a) T h e standardized death rate is higher i n count ry A than

i n country B.

Mortal i ty 29

(b) T h e standardized death rate is higher i n count ry B than

i n country A .

(c) T h e standardized death rates i n the two countries are

equal.

(d) T h e s tandardized death rates i n c o u n t r y A m a y be

higher, lower, or equal to those of count ry B.

2. T h e range of values for nat ional crude death rates i n the

w o r l d today is about:

(a) 10 to 80.

(b) 2 to 25.

(c) 10 to 120.

(d) 2 to 150.

3. Dea th rates are standardized:

(a) to e l imina te the di f ferent ia l in f luence of one or more

variables.

(b) to obtain an estimate of the ideal rates.

(c) to determine the future rates that may be expected.

(d) to obtain a correct statement of the actual or experi

enced rates.

(e) to correct for underregistration of the phenomenon i n

quest ion.

4. A high sex ratio:

(a) indicates a h igh proport ion of males i n the popula t ion .

(b) indicates a l o w proport ion of males i n the popula t ion .

(c) indicates a h igh proport ion of infants i n the popula

t ion .

(d) measures the extent of mor ta l i ty to males i n the popu

la t ion .

(e) is i m m o r a l .

E X E R C I S E 2 In 1988 c i ty A had a crude death rate of 15 per 1,000 and c i ty

B a crude death rate of 9. In the same year the crude death

rate of the U n i t e d States was about 9.

T h e age-specific death rates of the two ci t ies are stan

dardized on the age d i s t r ibu t ion of the U n i t e d States as a

who le i n 1988. Cons ider each of the f o l l o w i n g s ix possible

30 Mortal i ty

results and indicate what interpretat ion y o u w o u l d give them

in the absence of any othei information:

Death rates per 1,000 standardized on the age distribution of the United States

Results City A City B

Crude death rate 15 9 Standardized rates:

Case 1 15 9 Case 2 15 15 Case 3 9 9 Case 4 9 15 Case 5 12 10 Case 6 7 10

E X E R C I S E 3 Y o u are given the f o l l o w i n g data for countries A and B:

Country A Country B

Midyear Death rate Midyear Death rate Areas population per 1,000 population per 1,000

Metropolitan areas 500 20 6,000 25

Small towns 1,500 35 1,500 40 Rural areas 8,000 40 2,500 45

Calcu la t e the crude death rates for each country. A l s o ca lcu

late the area-standardized death rates, us ing (1) count ry A as

the standard popula t ion and (2) count ry B as the standard

popula t ion . Compare the answers and interpret them.

T H E I N F A N T E s t i m a t i n g the number of person-years l i v e d for c h i l d r e n

M O R T A L I T Y R A T E under age 1 is usual ly d i f f i cu l t because the requisi te statis

t ics are not col lected or not publ ished even i f col lected. Fur

thermore, for the reasons given i n Chapter 1, the midyear

popula t ion is usual ly a poor estimate of the number of per

son-years l i ved i n the age group under 1. Hence , demogra

phers use a special method for ca lcula t ing mor t a l i t y for c h i l

dren under 1 year of age. T h e y ca l l ch i ldren under age 1 " i n

fants" and calculate the infant mortality rate* ( IMR) accord

ing to the f o l l o w i n g fo rmula :

5. Barclay (1958| calls this rate the infant death rate to distinguish it from another

Mortal i ty 31

Bz

where D„ = number of deaths to children under 1 year of age in year z,

B' = number of live births in year z, and

k = 1,000.

T h e infant mor ta l i ty rate is thus closer to being a probabi l i ty

than a rate, since the denominator is persons (infants) ex

posed to death beginning at a certain t ime (birth), rather than

the number of person-years l i ved by infants . 6

A s we ment ioned i n Chapter 1, there is a special pat

tern of mor ta l i ty dur ing the f irst year of l i f e . T h i s is i l l u s

trated by the data for selected countries and years i n Table

2 . 8 / Deaths are not evenly dis tr ibuted throughout the f i rs t

year of l i fe . Instead, a h igh proport ion of in fan t mor t a l i t y

occurs i n the f irst m o n t h of l i fe . Furthermore, a h igh propor

t i on of the deaths i n the f irst m o n t h of l i f e occurs dur ing the

first week, and a h igh proport ion of the deaths i n the f i rs t

week of l i f e occurs dur ing the very f irst day. M o r t a l i t y of

ch i ld ren under 28 days of age is generally a lmost as h igh as.

or even higher than mor ta l i ty i n the next f ive months c o m

bined; mor ta l i ty rates for the second half of the f irst year are

always less than half and usual ly less than one-third of those

for the f irst s ix months .

Figures f r o m countries w i t h good data usual ly show that

the lower the infant mor ta l i ty rate, the higher is the propor

t ion of deaths that occur i n the f irst mon th , the f irst week,

and even the f irst hour of l i f e . T h i s is so because the causes

type of rate applied to infants in life tables (See Barclay 1958,47 ff., 106 ff., and 138 ff.]. We prefer the present usage, however, to maintain consistency with tables in the United Nations Demographic Yearbook series and other common reference materials. 6. When constructing life tables, demographers often use the IMR as the value for ,</„, the probability of dying between birth and the first birthday. (See the next section of this.chapter.) 7. Data on the number of days and weeks within infancy are not available for many developing countries. The examples shown in Table 2.8 are not necessarily based on good data and hence do not necessarily exhibit the "ideal" patterns described in the text. In particular, the low values for the death rate in the first day of life for Egypt, Pakistan, and Albania are suspect, the actual values are probably higher.

32 Mortality

of very early infant deaths tend to be congenital m a l f o r m a

t ion, b i r th injuries, prematuri ty, and other causes that are

not easily prevented by modern medica l and heal th measures.

Causes of later infant deaths (such as infec t ious diseases or

poor nutr i t ion) are more susceptible to prevent ion or treat

ment. Hence, whenever pub l i c heal th and i n d i v i d u a l med i

cal care improve, late infant deaths d i m i n i s h faster than early

deaths, and a higher proport ion of a l l infant deaths are early

deaths.

Three addi t ional problems i n measuring infant mor ta l

i ty are caused by the f o l l o w i n g facts: (a) there are seasonal

f luctuat ions i n the number of births; (b) many babies are bo rn

Table 2.8. Infant mortality rates for selected countries, by age and region: recent years

Age Age Age Region, country, Age 1-6 7-28 29̂ 364 All ages and year <1 day days days days <1 year

Africa Algeria, 1980' "26.1 (27.4) "69.3 (72.7) * 101.9 (100.1) Egypt, 1987 1.5 (3.0) 5.4(10.9) 6.4(13.0) 36.1(73.1) 49.4(100.0)

Americas Guatemala, 1988 3.4 (7.3) 5.3(11.4) 6.9(14.8) 31.0(66.5) 46.6(100.0) Canada, 1988 2.6(36.6) 1.2(16.9) 0.7 (9.9) 2.6(36.6) 7.2(100.0)

Asia (excluding former USSR) Pakistan, 1988" 0.1 (0.1) 36.6(34.0) 22.8(21.2) 48.2(44.8)107.7(100.1) Japan, 1989 1.0(21.7) 0.9(19.6) 0.7(15.2) 2.0(43.5) 4.6(100.0)

Europe (excluding former USSR) Albania, 1989 0.9 (2.9) 3.5(11.4) 3.1(10.1) 23.4(76.0) 30.8(99.9) Netherlands, 1989 1.8(26.5) 2.0(29.4) 0.8(11.8) 2.2(32.4) 6.8(100.1)

Oceania Australia, 1989 2.4(30.0) 1.4(17.5) 0.9(11.3) 3.3(41.3) 8.0(100.1)

Former USSR 7.8(33.9) 2.3(10.0) 12.9(56.1) 23.0(100.0)

Source: United Nations Statistical Office, Demographic Yearbook 1990 (1992, table 16). Notes: Rates are the number of deaths of infants per 1,000 live births. Figures in parentheses are percentages of the total. The rates for specific ages are based on the same denominator (1,000 live births) as is the total. Consequently, the sum of the rates for the specific ages equals the total infant mortality rate shown in the last column. Totals may not equal the sum of constituent rates or percentages because of rounding. Rates are shown only for countries having at least 1,000 infant deaths in a given year and with populations of 1 million or more. Data from registers that are incomplete or of unknown completeness are indicated with an asterisk ('). Ranges may not encompass the actual worldwide range because few countries have the requisite data available.

a. Excludes live-born infants who died before their births were registered. Not included in the calculations of percentages are 6.6 infant deaths/1,000 births of unknown age. b. Based on Pakistan's Population Growth Survey.

Mortal i ty 33

and die i n the same calendar year and are omi t t ed i n counts

of the popula t ion under age 1 at both the beginning and the

end of the year,- and (c) i n most censuses and surveys, in fan ts

are of ten underenumerated more than older persons, appar

ent ly because many parents do not t h i n k of infants as per

sons when asked " H o w m a n y persons l i ve here?"

Infant mor t a l i t y rates dur ing the 1980s ranged f r o m 4.5

per 1,000 bir ths (in Japan) to 172 (in Afghanistan) , as s h o w n

i n Table 2.9. I M R s were m u c h higher i n the past. Rates as

h igh as 200 were recorded for B e l g i u m i n 1900, France i n the

per iod 1851-1903, and Sweden i n the per iod 1778-1832

(Keyf i tz and Flieger 1968, 24-39). T h i s means, roughly, that

for every f ive infants born i n Be lg ium, France, or Sweden

dur ing those years, one died before its f irst birthday.

T h e infant mor ta l i ty rate calculated i n the s imp le way

described above is reliable on ly when the number of b i r ths

does not change rapidly f r o m one calendar year to the next.

W h e n there are rapid yearly changes i n the number of bir ths,

adjusted rates of various k inds are needed and can be ca lcu

lated; but we delay our d iscuss ion of these u n t i l A p p e n d i x 2.

F I R S T S E T O F De te rmine whether each of the f o l l o w i n g statements is true

T R U E / F A L S E or false:

Q U E S T I O N S I Infant mor ta l i ty rates are generally higher i n the less

developed countries than they are i n the more devel

oped countries.

2. A c c o r d i n g to available data, crude death rates i n the

less developed countries are always higher than those

i n the more developed countries.

3. T h e more developed countries probably never had i n

fant mor ta l i ty rates as h igh as those n o w recorded i n

many less developed countries.

4. T h e midyear popula t ion is always a good est imate of

the person-years l ived i n a given year.

5. O n the average, age-specific death rates are h igh o n l y

for persons over age 65.

Table 2.9. Highest and lowest infant mortality rates, by region: latest available data

Infant mortality rate (per 1,000 live births)

Region and country Year High Low

Africa Sierra Leone 1985-90 "154.0 Malawi 1985-90 "150.0 Mauritius 1985-90 "23.0 Tunisia 1985-90 "52.0

Americas Haiti 1985-90 "97.0 Bolivia 1985-90 "110.0 Canada 1988 7.2 UnitedStates 1990 t9.1

Asia (excluding former USSR) Afghanistan 1985-90 "172.0 Cambodia 1985-90 "130.0 Japan 1990 4.5 Hong Kong 1990 t6.1

Europe (excluding former USSR) Romania 1990 '26.9 Yugoslavia 1990 20.2 Sweden 1990 t5.6 Finland 1989 5.8

Oceania Papua New Guinea 1985-90 "59.0 Australia 1989 8.0 New Zealand 1989 10.2

Former USSR' 1989 23.0 Turkmenistan 1990-91 93 Belarus 1990-91 20

Sources: Except for the former USSR, all data are from United Nations Statistical Office, Demographic Yearbook 1990 (1992, table 15). Data for the former Soviet republics are from Population Reference Bureau (1992). Note: Rates are shown only for countries having at least 100 infant deaths in the specified year and a population of 1 million or more. Data from registers that are incomplete or of unknown completeness are not included. * Estimate prepared by Population Division of the United Nations. t Provisional figure. a. The U N Demographic Yearbook reports a figure of 39 deaths per 1,000 births for the former Soviet republics.

Mortality 35

T H E L I F E T A B L E Rates and ratios provide us w i t h a use fu l set of measures for

answering questions about morta l i ty . There are s t i l l m a n y

questions, however, that we cannot answer w i t h these mea

sures alone. To cite a few examples:

1. O u t of 100 persons i n country A w h o were 20 years o ld

i n 1968, how many are l i k e l y to l ive to age 50?

2. Immediate ly after b i r th , h o w m a n y years cou ld a c h i l d

born i n 1950 i n country B expect to live?

3. A m o n g young m e n and w o m e n entering the labor force

at ages 20-24, what proport ion can be expected to be

al ive at age 67 (when, for example, i n the U n i t e d States

they are ent i t led to col lect socia l securi ty benefits)? Of

those w h o do begin to col lect benefits at age 67, h o w

many can be expected to survive for one year, two years,

three years, etc.?

4. Is there a measure that can be used to compare the

mor ta l i ty of many countries so that differences i n their

age dis t r ibut ions w i l l not be d is tor t ing factors and so

that an arbitrary choice of a standard popula t ion for an

age standardization w i l l not be necessary?

Quest ions of this type have immense prac t ica l impor

tance. For example , pro jec t ions of the fu tu re p o p u l a t i o n

needed to determine how many schools or hospi tals are re

quired depend on estimates of how long people survive . In

addi t ion, l i fe insurance companies need accurate answers to

questions about average l i f e expectancy, for w i t h o u t t h e m

they w o u l d not be able to construct the actuarial tables on

w h i c h they base the p remiums customers mus t pay. Such

questions as these can best be answered by life tables, a l

though the answers are s t i l l approximate.

Cons t ruc t ing a l i f e table can be a complex process. Here,

we emphasize interpretat ion rather than computa t ion , be

g inn ing w i t h a descript ion of the l i f e table and af terward dis

cussing elementary applicat ions i n demographic analysis .

Instead of the more usual no t ion of a popula t ion , sup

pose we were to define a populat ion to be everyone born i n a

country dur ing a part icular year, say 1879. A s demographers,

we ca l l this group the "1879 b i r t h cohor t" for that country.

36 Mortality

N o w suppose we had the death rates for the 1879 b i r t h co

hort as i t passed through each age, u n t i l every member of the

1879 cohort had died (presumably a l l w o u l d have died by

now). In this s i tua t ion , we cou ld easi ly answer quest ions

about the surv iva l of members of the cohort f r o m one age to

the next, s ince we w o u l d k n o w their entire m o r t a l i t y his

tory. From data of this type, we could construct what is k n o w n

as a longitudinal, or generation, life table, w h i c h refers to

one b i r th cohort as i t ages. A generation l i f e table can be

constructed only after a l l or a lmost a l l of the members of the

b i r th cohort have died. For this reason, and because the re

quired data are not often available, generation l i fe tables have

l i m i t e d practical u t i l i t y and are not c o m m o n l y used, a l though

his tor ical studies of morta l i ty benefit greatly f r o m such tables.

Let us return to our more usual de f in i t i on of a popula

t ion . Suppose we have a set of age-specific death rates that

represent the incidence of mor t a l i t y i n each age group for a

cross-section of the popula t ion over a short per iod of t ime (a

year, 1990, for example). We assume that the age-specific

mor ta l i ty experiences dur ing 1990 represent the death expe

rience of a w h o l e generation of persons. T h a t is, we assume

that a cohort of persons w i l l pass through l i f e exper iencing

at each age the age-specific death rates for 1990. M a k i n g this

assumption, we can determine what the number of surv i

vors at any given age w o u l d be out of an i n i t i a l group of births,

according to the given mor ta l i ty schedule. T h e l i fe table, then,

becomes a model of what w o u l d happen to a hypothe t ica l

b i r th cohort if the age-specific death rates for a given period

were to remain constant and were to apply throughout the

experience of an entire generation. M o r t a l i t y analyses based

on l i f e tables are n o r m a l l y based on the assumpt ion that a

single mor ta l i ty schedule applies to a hypothetical group of

persons u n t i l a l l the persons have died. In other words, the

more c o m m o n type of l i fe table is calculated for a synthetic,

or hypothetical, cohort of bir ths .

T h i s more c o m m o n type of l i f e table is cal led a period,

cross-sectional, current, or time-specific life table. It answers

the question, Wha t w o u l d be the mor ta l i ty h is tory and aver-

Mortality 37

age l i f e expectancy of a cohort of people subject throughout

their l i f e h i s tory to the age-specific death rates of a par t icular

year or period of years? T h e per iod l i f e table is a ma themat i

ca l mode l of the l i f e h is tory of a hypothet ica l cohort . It is a

mode l because we must make s i m p l i f y i n g assumptions to

construct the table and because i t refers to a hypothe t ica l

rather than a real b i r th cohort.

T h e l i f e table begins w i t h the b i r t h , dur ing one year, of

a hypothe t ica l cohort of persons. T h e number of bir ths is

usual ly set arbi t rar i ly at 100,000. 8 T h i s s tar t ing number of

bir ths is cal led the radix of the l i f e table. T h e l ife-table record

cont inues u n t i l a l l the members of the cohor t have died,

deaths at each age occurr ing i n accordance w i t h a mor t a l i t y

schedule that is f ixed i n advance and does not change. N o

factors other than mor ta l i ty operate to reduce the size of the

start ing cohort; that is, the hypothet ica l cohort is " c lo sed"

to migra t ion of any k i n d . A t each age, except for the f irst f ew

years of l i fe , the deaths are assumed to be evenly dis t r ibuted

throughout the year. Hence , half of the deaths to persons

between the ages 15 and 16, say, w o u l d occur by the t i m e the

average person i n the cohort has reached the age of 1 5 V i . M o s t

l i f e tables refer to on ly one sex, p r i m a r i l y because the death

rates for males and females differ substantial ly. L i f e tables

can be specif ic to any subgroup w i t h i n a popula t ion , provided

the necessary A S D R s are available.

T h e convent iona l l i fe table consists of seven co lumns ,

s ix of w h i c h present what are cal led the life-table functions.

A brief descr ipt ion of each c o l u m n fo l lows . To i l lustrate , we

use a l i fe table for females i n the U n i t e d States for the years

1979-81 (Table 2.10).

C O L U M N t: E X A C T A C E : x

Each of the l ife-table func t ions refers to a specif ic age or age

in terval . T h e first c o l u m n of the l i f e table specifies the age to

8. Other starting numbers are found in the literature, the most common being 1

and 10,000.

Table 2 .10 . Complete life table for females: United States, 1979-81

Probability Number Expectation of dying of deaths Number of Total of life between between years lived number (average

exact exact between of years number of age x and Number of age x and exact age x lived years

Exact age exact survivors at exact and exact after exact remaining) in years age x+1 exact age x age x+1 age x+1 age x at exact age x

X <?, Cx dx K T, e ,

(1) (2) (3) (4) (5) (6) (7)

0 .01120 100,000 1,120 99,085 7,762,496 77.62 1 .00086 98,880 84 98,838 7,663,411 77.50 2 .00056 98,796 56 98,768 7,564,573 76.57 3 .00042 98,740 41 98,720 7,465,805 75.61 4 .00033 98,699 33 98,682 7,367,085 74.64

5 .00031 98,666 30 98,651 7,268,403 73.67 6 .00027 98,636 27 98,623 7,169,752 72.69 7 .00024 98,609 24 98,596 7,071,129 71.71 8 .00022 98,585 22 98,575 6,972,533 70.73 9 .00019 98,563 19 98,553 6,873,958 69.74

10 .00018 98,544 17 98,536 6,775,405 68.75 11 .00018 98,527 18 98,518 6,676,869 67.77 12 .00020 98,509 20 98,499 6,578,351 66.78 13 .00026 98,489 25 98,477 6,479,852 65.79 14 .00033 98,464 32 98,448 6,381,375 64.81

15 .00040 98,432 40 98,411 6,282,927 63.83 16 .00047 98,392 46 98,369 6,184,516 62.86 17 .00052 98,346 52 98,320 6,086,147 61.89 18 .00055 98,294 54 98,267 5,987,827 60.92 19 .00057 98,240 56 98,212 5,889,560 59.95

20 .00058 98,184 57 98,156 5,791,348 58.98 21 .00060 98,127 59 98,097 5,693,192 58.02 22 .00062 98,068 61 98,037 5,595,095 57.05 23 .00063 98,007 61 97,977 5,497,058 56.09 24 .00064 97,946 63 97,914 5,399,081 55.12

25 .00065 97,883 63 97,851 5,301,167 54.16 26 .00066 97,820 65 97,788 5,203,316 53.19 27 .00067 97,755 66 97,722 5,105,528 52.23 28 .00070 97,689 68 97,655 5,007,806 51.26 29 .00072 97,621 70 97,586 4,910,151 50.30

30 .00075 97,551 74 97,514 4,812,565 49.33 31 .00079 97,477 77 97,439 4,715,051 48.37 32 .00083 97,400 81 97,360 4,617,612 47.41 33 .00089 97,319 86 97,276 4,520,252 46.45 34 .00096 97,233 93 97,186 4,422,976 45.49

35 .00104 97,140 101 97,890 4,325,790 44.53 36 .00114 97,039 111 96,984 4,228,701 43.58 37 .00125 96,928 121 96,868 4,131,717 42.63 38 .00127 96,807 132 96,741 4,034,849 41.68 39 .00149 96,675 144 96,603 3,938,108 40.74

Table 2 .10 . (continued)




X Q, d X K T

X e »

(1) (2) (3) (4) (5) (6) (7)

40 .00163 96,531 157 96,452 3,841,505 39.80 41 .00180 96,374 174 96,287 3,745,053 38.86 42 .00199 96,200 191 96,104 3,648,766 37.93 43 .00218 96,009 210 95,904 3,552,662 37.00 44 .00239 95,799 229 95,684 3,456,758 36.08

45 .00262 95,570 250 95,445 3,361,074 35.17 46 .00286 95,320 273 95,184 3,265,629 34.26 47 .00315 95,047 299 94,897 3,170,445 33.36 48 .00347 94,748 329 95,584 3,075,548 32.46 49 .00381 94,419 359 94,239 2,980,964 31.57

50 .00416 94,060 391 93,864 2,886,725 30.69 51 .00452 93,669 424 93,457 2,792,861 29.82 52 .00490 93,245 457 93,017 2,699,404 28.95 53 .00532 92,788 494 92,541 2,606,387 28.09 54 .00578 92,294 534 92,028 2,513,846 27.24

55 .00627 91,760 575 91,472 2,421,818 26.39 56' .00678 91,185 618 90,876 2,330,346 25.56 57 .00733 90,567 664 90,235 2,239,470 24.73 58 .00796 89,903 716 89,545 2,149,235 23.91 59 .00867 89,187 773 88,800 2,059,690 23.09

60 .00947 88,414 837 87,996 1,970,890 22.29 61 .01035 87,577 907 87,123 1,882,894 21.50 62 .01129 86,670 979 86,181 1,795,771 20.72 63 .01226 85,691 1,050 85,166 1,709,590 19.95 64 .01325 84,641 1,121 84,081 1,624,424 19.19

65 .01427 83,520 1,192 82,923 1,540,343 18.44 66 .01538 82,328 1,267 81,695 1,457,420 17.70 67 .01664 81,061 1,349 80,387 1,375,725 16.97 68 .01811 79,712 1,443 78,990 1,295,338 16.25 69 .01980 78,269 1,549 77,495 1,216,348 15.54

70 .02169 76,720 1,665 75,887 1,138,853 14.84 71 .02375 75,055' 1,782 74,164 1,062,966 14.16 72 .02600 73,273 1,905 72,321 988,802 13.49 73 .02842 71,368 2,028 70,354 916,481 12.84 74 .03106 69,340 2,154 68,263 846,127 12.20

75 .03388 67,186 2,276 66,048 777,864 11.58 76 .03704 64,910 2,404 63,707 711,816 10.97 77 .04073 62,506 2,546 61,233 648,109 10.37 78 ,04515 59,960 2,707 58,607 586,876 9.79 79 .05033 57,253 2,881 55,812 528,269 9.23

40 Mortality

Table 2 .10 . (continued)




X <7, tx dx I X

T, e, (1] (2) 13] (4) (5) (6) (7)

80 .05622 54,372 3,057 52,844 472,457 8.69 81 .06269 51,315 3,217 49,706 419,613 8.18 82 .06973 48,098 3,354 46,422 369,907 7.69 83 .07722 44,744 3,455 43,106 323,485 7.23 84 .08519 41,289 3,517 39,531 280,469 6.79

85 .09409 37,772 3,554 35,995 240,938 6.38 86 .10405 34,218 3,561 32,437 204,943 5.99 87 .11420 30,657 3,501 28,907 172,506 5.63 88 .12427 27,156 3,374 25,469 143,599 5.29 89 .13471 23,782 3,204 22,180 118,130 4.97

90 .14661 20,578 3,017 19,069 95,950 4.66 91 .16024 17,561 2,814 16,154 76,881 4.38 92 .17460 14,747 2,575 13,459 60,727 4.12 93 .18904 12,172 2,301 11,022 47,268 3.88 94 .20348 9,871 2,029 8,867 36,246 3.67

95 .21823 7,862 1,715 7,004 27,379 3.48 96 .23221 6,147 1,428 5,433 20,375 3.31 97 .24560 4,719 1,159 4,140 14,962 3.17 98 .25834 3,560 919 3,101 10,802 3.03 99 .27040 2,641 714 2,283 7,701 2.92

100 .28176 1,927 543 1,655 5,418 2.81 101 .29242 1,384 405 1,182 3,763 2.72 102 .30237 979 296 831 2,581 2.64 103 .31163 683 213 577 1,750 2.56 104 .32023 470 150 394 1,173 2.50

105 .32817 320 105 268 779 2.44 106 .33550 215 72 178 511 2.38 107 .34224 143 49 119 333 2.33 108 .34843 94 33 77 214 2.28 109 .35411 61 22 50 137 2.24

110+ 1.00000 39 39 87 87 2.23

Source: Modified from United States, National Center for Health Statistics (1985, table 3).

w h i c h the later co lumns of the table refer. In the l i f e table

the word "age" is used very precisely, and the precis ion is

emphasized by the addi t ion of the mod i f i e r "exact ." W h e n

we say that a person is exact age 0, we mean that he or she

Mortality 41

was just born. W h e n he or she is exact age 5, he or she has

l i ved exact ly f ive f u l l years. In contrast, when we say i n ev

eryday conversat ion that someone " is 5 years o l d , " we mean

that the person is between exact age 5 and exact age 6—that

is, the person completed 5 years of l i f e on his or her last b i r th

day.

T h e letter x is used to represent exact age. Some of the

l ife-table func t ions refer to the exact agex and others refer to

the age in te rva l between exact age x and exact age x + 1.

C O L U M N 2 : P R O B A B I L I T Y O F D Y I N G B E T W E E N E X A C T A G E x

A N D E X A C T A G E X + 1 : fl.

T h e second c o l u m n of the l i f e table [qx] represents the prob

ab i l i ty of dy ing between exact age x and exact age x + 1. T h i s

c o l u m n summar izes the l ife-table mor ta l i ty rates, w h i c h are

probabil i t ies and thus different f r o m the age-specific death

rates discussed earlier i n this chapter. T h e qx f u n c t i o n is the

numer i ca l answer to the quest ion, A m o n g persons w h o reach

exact age x, what proport ion w i l l die before their next b i r th

day—that is, w i t h i n one year? T h e t ^ values usual ly are some

what lower than the age-specific death rates [MJ we discussed

earlier, but the rates are closely paral lel . T h e technica l ques

t ion of how to derive a set of qx values f r o m a set of age-

speci f ic death rates need not concern us at this po in t . 9 For

now, just remember that the qx values are a set of m o r t a l i t y

probabil i t ies for the cohort as i t begins each successive year

of l i fe .

C O L U M N 3 : S U R V I V O R S A T E X A C T A G E x: t w

T h e th i rd c o l u m n of the l i fe table [lx] represents the number

of people w h o have survived f r o m b i r t h to exact age x . T h e

i n i t i a l cohort, the radix, is 100,000 i n Table 2.10 [lQ = 100,000).

9. Appendix 2 provides a brief introduction to the relationship between qt values

and Mx values.

42 Mortality

In the f irst year of l i fe , the probabi l i ty of dy ing is .01120 (the

value of q0). Consequen t ly , 1,120 persons of the o r ig ina l

100,000 die i n the first year of l i fe (see c o l u m n 4) and on ly

98,880 persons reach exact age 1 (ix = 98,880).

T h e number of survivors to any age [lx) is equal to the

product of i x , and the value of the mor ta l i ty probabi l i ty for

the preceding age in terval [qx_x], subtracted f r o m the number

w h o surv ived to the beginning of the preceding age in te rva l

| / ), In a fo rmula :

To i l lustrate, we calculate the value of £ x for exact age 19 for

the l i f e table i n Table 2.10:

= 98,294 - [(.00055)(98,294)]

= 9 8 , 2 9 4 - 5 4

= 98,240.

If y o u f i n d these formulas confus ing at f irst , y o u shou ld bear

i n m i n d that they are s i m p l y algebraic statements of the fact

that the number of survivors at any exact age consists of those

a l ive one year earlier m i n u s those w h o died dur ing the inter

vening year.

T h e meaning of the l x c o l u m n m a y be clearer i f w e re

fer to i ts possible use by an insurance company. In Table 2.10,

note that 96,531 people reach age 40 and that 157 die dur ing

their 40th year. Suppose that the insurance company desires

to provide $1,000 i n t e rm insurance for one year's coverage

for each of the 96,531 people reaching age 40. Since 157 of

the 96,531 are expected to die before the i r 41st bir thday,

$157,000 must be available to be paid out i n benefits. T h e

p r e m i u m for the insurance is to be paid by 96,531 people;

therefore each must pay:

$157,000

plus any charges for adminis t ra t ion or profi ts for the c o m

pany.

Mortality 43

C O L U M N 4 : N U M B E R O F D E A T H S B E T W E E N E X A C T A G E x

A N D E X A C T A G E x + 1: d t

T h e four th c o l u m n represents the number of deaths to the

cohort between exact age x and exact age x + 1. S y m b o l i z e d

as dx, it is equal to the number su rv iv ing to exact age x [tx)

m u l t i p l i e d by the probabi l i ty of dy ing between exact age x

and exact age x + 1:

dMtM T h e number of cohort deaths \dx) is also equal to the differ

ence between the number su rv iv ing to exact age x and the

number su rv iv ing to exact age x + 1 ; that is:

In Table 2.10, the number of deaths i n the f irst year of

l i fe is 1,120, w h i c h is the product of 100,000 and .01120. T h e

number of deaths at age 84 is 3,517, w h i c h is the product of

41,289 and .08519.

C O L U M N 5 : Y E A R S L I V E D B E T W E E N E X A C T A G E x A N D

E X A C T A G E x + 1 : LK

T h e f i f t h c o l u m n of the l i f e table \Lx) represents the number

of person-years l ived by the cohort dur ing an age in te rva l .

A l t h o u g h a precise de terminat ion of Lx values is not u sua l ly

possible, we can approximate the values by assuming that

deaths are evenly distributed throughout the in terval between

exact age x and exact age x + 1 except dur ing the f irst f ew

years of l i fe . M a k i n g this assumption, we can est imate the

value of Lx by averaging the number of survivors at the be

g inn ing of the age in terval [ix] and the number of surv ivors at

the end of the in te rva l (^ , ) . In other words, i t is usua l ly as

sumed that:

* - 2 '

T h i s approximat ion uses the same logic we used earlier w h e n

we exp la ined h o w the m i d y e a r p o p u l a t i o n is used as an

44 Morta l i ty

approximation of the number of person-years lived for calcu

lating death rates.

For the first few years of life, the average of l x and i x t l is

not a reasonable approximation of Lx because deaths are not

evenly distributed throughout the year. Instead, they are con

centrated at the earlier part of the year, as documented in

our earlier discussion of the infant mortality rate. For this

reason, values of Lx for the first few years should be closer to

l x t l than to lx. In the absence of good data for estimating the

relative weighting of l x and £ x t l , it is often assumed that:

L0=.3eo+J£l and

Ll=Aei+.6£2.

For L 2 and for ages greater than 2, the .5{tx + £xtl) approximation is used. The formulas above are approximations based on empirical observations. When data are available on the mortality of children by the number of months or days since birth, more refined estimates of L 0 and L, are possible. We do not describe them here, but more sophisticated techniques are often used for calculating the Lx values for the first few years of l i fe . 1 0

It may help you to look at these observations another way. A l l the persons who survive the year (that is, live from £x to t x t l ) live for one year. Therefore the minimum number of years lived is equal to t x t V For example, all of the 96,374 people who live from age 40 to age 41 in Table 2.10 contribute one year of life,- hence we begin with a minimum of 96,374 41-year-olds. In addition, the persons who die during that year (157 persons during their 40th year) live for some part of the year. If all of them were to die one second after their 40th birthday, then we could ignore the addition. On the other hand, if all the deaths occurred one second before the 41st birthday, we could assume that all the decedents lived a fu l l year. Our assumption is that deaths are likely to be more or

10. More sophisticated techniques were, in fact, used to construct the life table shown in Table 2.10, which is why the values given forL 0 and I, in Table 2.10 are slightly different from what the above formulas would give. More sophisticated methods also modify the .5 assumption at other ages.

Mortal i ty 45

less evenly spaced throughout the year. If that is so, each decedent wi l l have lived for an average of one-half year; hence we add one-half of the deaths in the 40th year (78) to the total number alive at age 41 to obtain the total number of years lived between birthdays 40 and 41. The resulting number is 96,452.

The logic of the life table permits us to make any other reasonable assumptions about the distribution of deaths during the year. Since we know that most infant deaths occur early in the first year of life, data on that first year are used to obtain an Lx figure that assumes much less than half a year of life for the infant decedents. When we approach the oldest ages of the age distribution, there may be similar effects.

C O L U M N 6 : T O T A L Y E A R S L IVED A F T E R E X A C T A G E x: T t

The sixth column of the life table gives the number of person-years lived after exact age x. We have already considered the number of years lived during the 40th year of life using the Lx column, which gives these figures for each year of life. The Tx figure at age 0 is the sum of all Lx entries—that is, how many years w i l l be lived in the first, second, third, etc., years of life when all are added together. The Tx figure for any other age (e.g., exact age 40) is the sum of the years lived for that age (L40) and all later ages by those survivors st i l l alive at the beginning of the age in question.

The entries in the sixth column thus show the number of person-years that the cohort wi l l live after reaching exact age x. It is the sum of the values of Lx for exact age x and all ages greater than x that are presented in the life table. In a formula:

i=x

where L. entry i in the Lx column, and

means "take the sum of the Lx column starting with entry x and add entries x + 1, x + 2, etc., until you have added the last entry (°°)."

46 Mortal i ty

C O L U M N 7: E X P E C T A T I O N O F L I F E , OR A V E R A G E N U M B E R O F

Y E A R S L IVED A F T E R E X A C T A G E x: e .

The seventh and last column in the life table is the one most commonly used. It answers the question, If all the persons reaching any exact age could share equally the total number of years that all wi l l live from that age onward, how many years would each live on the average? After having calculated Tx (the total number of person-years lived after exact age x), and £x (the number of persons who survived to attain age x), it is easy to determine how long the average person in the life table lives after exact age x. We simply divide the entries in the Tx column by the entries in the £ column:

In the life table for U.S. females for 1979-81 (Table 2.10), females of exact age 35 had an expectation of l iving 44.53 more years on the average. That is, their expected time of death, on the average, was at exact age 79.53. Females of exact age 0, on the other hand, had an expectation of life of 77.62 years. Expressed informally, this means that women in the hypothetical cohort who survive the hazards of the first 35 years exhibit an increase in the average age to which they w i l l live over the age expected at their birth. The increase is, however, modest: 1.91 years (79.53 - 77.62).

T H E A B R I D G E D LIFE T A B L E

The life table we have just described is known as a complete life table because it presents the life-table functions for single years of age. There are also life tables that present the functions for groups of ages. They may refer, for example, to the probability of dying between exact age 5 and exact age 10 and present all the values in the table only for intervals of 5 years. In these abridged life tables, the first year of life and the ages from 1 to 4 are usually presented separately. Later ages are usually presented in 5- or 10-year intervals. Although the

Mortal i ty 47

calculation of the abridged table is slightly different from that of the complete table," the interpretation of the values of the life-table functions is the same as for the complete table. Only the time interval must be adjusted in discussions of values taken from an abridged table. A small number placed below and to the left of the letter for the life-table function (e.g., 5 I 1 0 , 4 d , , or nqJ indicates the length of the interval. A n abridged life table for U.S. females in 1988 is presented in Table 2.11.

T H I R D S E T O F 1. Life expectancy at birth for females in the United States in M U L T I P L E - C H O I C E 1988 was:

Q U E S T I O N S ( a , about 35 years.

(b) about 55 years. (c) about 78 years. (d) about 100 years.

What is the life expectancy at birth for females in your own country? What is the life expectancy at birth for males in your own country?

2. The difference between a generation life table and a period life table is that: (a) the radix is different. (b) one refers to a true birth cohort and the other does not. (c) one uses a different method for calculating q0 than the

other. (d) none of the above.

3. Life-table mortality rates (qx) are usually: (a) about the same as age-specific death rates [Mx). (b) higher than age-specific death rates (MJ. (c) exactly the same values as age-specific death rates (MJ. (d) lower than age-specific death rates (Mx).

4. If country A has a higher life expectancy and also a higher crude death rate than country B, it is likely that:

11. For example, for calculating ^L^, the formula is (5/2) \ tM + ?liS).

Table 2.11. Abridged life table for females: United States, 1988


exact exact between of years number of Size of age x and Number of age x and exact age x lived years

Exact age interval exact survivors at exact and exact after exact remaining) in years in year age x+1 exact age x age x+1 age x+1 age x at exact age x

X n 1, d. K T, e x (1) (2) (3) (4) (5) (6) (7) (8)

0 1 .0089 100,000 890 99,243 7,831,495 78.3 1 4 .0018 99,110 175 396,021 7,732,252 78.0 5 5 .0010 98,935 101 494,400 7,336,231 74.2 10 5 .0010 98,834 100 493,954 6,841,831 69.2

15 5 .0024 98,734 240 493,108 6,347,877 64.3 20 5 .0028 98,494 272 491,802 5,854,769 59.4 25 5 .0033 98,222 321 490,324 5,362,967 54.6 30 5 .0041 97,901 405 488,539 4,872,643 49.8

35 5 .0058 97,496 567 486,163 4,384,104 45.0 40 5 .0084 96,929 817 482,754 3,897,941 40.2 45 5 .0135 96,112 1,293 477,562 3,415,187 35.5 50 5 .0219 94,819 2,077 469,225 2,937,625 31.0

55 5 .0347 92,742 3,217 456,141 2,468,400 26.6 60 5 .0537 89,525 4,810 436,300 2,012,259 22.5 65 5 .0793 84,715 6,716 407,664 1,575,959 18.8 70 5 .1210 77,999 9,435 367,619 1,168,295 15.0

75 5 .1843 68,564 12,640 312,711 800,676 11.7 80 5 .2981 55,924 16,671 239,106 487,965 8.7 85 1.0000 39,253 39,253 248,859 248,859 6.3

Source: United States, National Center for Health Statistics (1991, table 6-1).

Mortal i ty 49

(a) A's population is younger than that of B. (b) A's population is older than that of B. (c) A's population has a higher infant mortality rate. (d) none of the above is likely.

S E C O N D S E T O F Determine whether each of the following statements is true T R U E / F A L S E or false: Q U E S T I O N S 1 Survival ratios from age 0 to age 1 are higher than other

one-year survival ratios. 2. Life expectancy in the United States is greater for males

than for females.

3. A period life table is a hypothetical model because mortality rates actually change from one time period to the next.

4. In a country where mortality rates have remained relatively constant for many years, a generation life table and a period life table would be almost identical.

A N A L Y S I S

A P P L I C A T I O N S The life-table functions provide useful tools for analyzing the O F T H E L I F E T A B L E effects of mortality alone because migration is explicitly ex-T O M O R T A L I T Y eluded and fertility is held constant. The uses of the life table

are many and varied, but we concentrate on only three here: (1) uses of the stationary population concept, (2) survival ratios, and (3) comparisons of life expectancy in two or more populations at various ages.

T H E S T A T I O N A R Y P O P U L A T I O N C O N C E P T

The numbers in the Lx column are similar to the midyear population in each age group for a hypothetical Or model population that demographers call the stationary population. The nature of this model population may be understood as resulting from the following process. Suppose that 100,000 persons are born each year and are subject to the mortality rates shown in Table 2.10. After 40 years the population would consist of all the age groups shown in the Lx column up

50 Mortal i ty

through age 40. The persons shown at age 2 would be the survivors of the 100,000 babies born two years before, the persons in the 40th year would be the survivors of the 100,000 babies born 40 years previously, etc. After about 110 years, the whole population structure shown in theL x column would have been created. From that time on—ad infinitum—the 100,000 entering the population at birth would be exactly balanced by the 100,000 dying at all ages: this is the characteristic that leads to the label "stationary." The size of this total population would be T v and the Lx column would give the age distribution of the stationary population.

The stationary population has many of the characteristics of a real population. For example, it has a crude birth rate,12 called the life-table birth rate, which is defined as follows:

b = k^-

where t0 = the radix, usually 100,000, T 0 = the first entry of the Tx column, and Jc = 1,000.

Note again that T 0 is the total size of the stationary population, since it is the sum of all the values in the Lx column.

The stationary population also has a crude death rate, called the life-table death rate. The life-table death rate is equal to:

d = k^-T

or, alternatively, to the reciprocal of e0 multiplied by a constant:

The life-table death rate is the same as the life-table birth rate, of course, since everyone in the hypothetical cohort dies at some age. This hypothetical population is "stationary" because the number of births and the number of deaths are

12. Fertility measures are discussed in Chapter 3. The crude birth rate is the same as the crude death rate except that the numerator for the crude birth rate is the number of live births in a given year.

Mortality 51

equal and therefore the population is neither growing nor declining in size.

Earlier, we described the life table as the life history of a cohort of persons born (i.e., reaching exact age zero) during a single year who move through a series of mortality rates until all of them have died. Alternatively, we can think of the life table as describing what happens each year in a hypothetical stationary population. In talking about a stationary population, demographers look at it the following way:

The stationary population is a model without immigration or

emigration in which the same age-specific probabilities of death

apply continuously and in which there are the same number of

births and deaths each year (Greville 1946, 21).

Another characteristic of a stationary population is that the number of persons living in each age group never changes. The figures in the Lx column, as we have said, specify the age composition of the stationary population, and this age composition never changes.

The stationary population concept has limited descriptive value because the life-table model is very different from what happens in a real population. It is useful for analytic purposes, however, because it summarizes what the age structure of a population would be if it were subject to the fixed mortality and birth conditions in the life table. A comparison of the age composition of females in the United States in 1988 with that of the female stationary population for the same time period shows, for example, that the stationary population is older than the actual population (Table 2.12). This reflects two facts: (1) mortality conditions for American females have improved over time, 1 3 and (2) crude birth rates have actually been higher than crude death rates, resulting, in the absence of migration, in a growing population

13. Declining mortality does not automatically make a population older or younger; rather, the effect depends on the age pattern of the mortality changes. Historically, a decline in mortality has usually been especially important at the youngest ages, resulting in a younger population by reinforcing the effect of high fertility. Because infant and childhood mortality is now low in most countries, future mortality declines are likely to be concentrated at the older ages and will result in an older population (all other factors being equal).

52 Mortal i ty

and a young age distribution. A similar result would obtain even if only the latter condition were true—that is, birth rates had been consistently higher than death rates.

Perhaps the most frequent use of the stationary popu

lation concept is for comparing death rates in a stationary

population with those of an actual population. Table 2.13

presents life-table death rates for white females in the United

States for various periods between 1900 and 1988 and corn-

Table 2.12. Age composition of the stationary population and the actual population for U.S.

white females: 1988

Difference

Compositon of stationary population

Composition of actual population

Estimated actual Percentage

between actual and stationary

populations' percentage

between actual and stationary

populations' percentage

Percentage population distribution distributions distribution on 1 July 1988 of actual (col. 4 minus

Ages A o f A (in thousands) population col. 2)

(i) (2) (3) (4) (5)

<1 99,363 1.3 1,517 1.4 +0.1 1-4 396,632 5.0 5,732 5.4 +0.4

5-9 495,226 6.3 7,063 6.7 +0.4 10-14 494,822 6.3 6,489 6.1 -0.2 15-19 494,000 6.3 7,216 6.8 +0.5 20-24 492,755 6.2 7,866 7.4 + 1.2 25-29 491,466 6.2 9,067 8.6 +2.4 30-34 489,978 6.2 9,074 8.6 +2.4

35-39 488,011 6.2 8,112 7.7 + 1.5 40-44 485,110 6.1 7,025 6.6 +0.5 45-49 480,509 6.1 5,697 5.6 -0.5 50-54 472,928 6.0 4,903 4.6 -1.4

55-59 460,750 5.8 4,933 4.7 -1.1 60-64 441,931 5.6 5,132 4.8 -0.8 65-69 414,254 5.2 4,829 4.6 -0.6 70-74 374,756 4.7 4,038 3.8 -0.9 75-79 319,782 4.1 3,227 3.0 -1.1 80-84 245,202 3.1 2,162 2.0 -1.1

85+ 253,240 3.2 1,940 1.8 -1.4

Al l ages 7,890,715 99.9 106,023 100.2 0.3

Sources: United States, Bureau of the Census (1990, table 1, p. 411; United States, National Center for Health Statistics (1991, table 6-1). Note: Totals may not equal sums because of rounding.

Mortal i ty 53

Table 2.13. Crude death rates and life table death rates for U.S. white females: selected years, 1900-88

Difference (life table

Crude death Life table rate minus Years rate death rate crude rate)

1900-02 15.4 19.6 +4.2 1909-11 13.2 18.6 +5.4 1919-21 11.5 17.1 +5.6 1929-31 9.9 16.0 +6.0 1939-41 9.1 14.9 +5.8 1949-51 8.0 13.9 +5.9 1959-61 7.9 13.5 +5.6 1969-71 8.1 13.2 +5.1 1979-81 7.9 12.8 +4.9 1988 8.6 12.7 +4.1

Source: United States, National Center for Health Statistics (1991, tables 1-2, 6-4).

parable figures for the crude death rates. The life-table death rates are consistently higher than the crude death rates because the age composition of the actual population has been consistently younger than the age composition of the stationary population.

S U R V I V A L R A T I O S

The life table is particularly valuable for making population projections or for making estimates of the population by age between census years. If we assume that the mortality conditions of a particular life table w i l l continue in the future, we can determine what proportion of people in a given age group w i l l survive from that particular age group to another. For most age groups in low-mortality societies such as Japan or the United States, this is a safe assumption. Death rates at most ages are so low and so stable that changes are not l ikely to be great. Even a considerable percentage change in death rates that are very low w i l l make little difference in survival ratios. That is why population projections for countries like Japan or the United States are not likely to be seriously in error as a result of assumptions about future mortality

54 Mortal i ty

rates.14 For example, since 93 percent of the women in the 1960 life table were surviving to age 45, projections for women less than 45 years of age could not be much affected by further reductions in mortality.

The Lx column specifies the midyear population of the stationary population in the age interval x to x + 1. If we want to determine the proportion of persons surviving from age group x to the later age group x + n, we simply determine:

forward survival ratio from age* to age* + n - ^£tn-.

On the other hand, if we want to know how many persons would have been alive n years in the past, we can determine:

L reverse survival ratio from age x + n to agex =

L x+n

To illustrate forward survival simply, suppose the life table of U.S. females (Table 2.10) is the latest life table available. Government officials want to know how many females wi l l be age 6 (i.e., between exact age 6 and exact age 7) in 1996 if roughly 2 mill ion females were of age 0 (not having reached their first birthday) in 1990. This information is needed, let us say, to determine how many girls w i l l enter the first grade of elementary school in 1996. Using Table 2.10, we calculate:

^ = ̂ 6 2 3 = 0.9953. La 99,085

Using the life table, then, we can estimate that 99.53 percent of the girls at age 0 in 1990 w i l l survive to age 6 in 1996. Multiplying 0.9953 by the number of girls at age 0 in 1990 (2,000,000) gives us the projected number of females who wi l l be age 6 in 1996, barring immigration and emigration of young children: 1,990,600.

We introduced another type of survival ratio earlier, one

14. Experience has shown that such projections may nevertheless be far off the mark, a result of faulty assumptions about fertility.

Mortality 55

based on the £x column of the life table. These survival ratios vary dramatically from country to country and have improved rapidly in the less developed countries. Compare, for example, the survival ratios for males in Sri Lanka (formerly Ceylon) in 1920, 1946, 1954, 1967, and 1981 with those for white males in the United States from 1900 to 1988 (Table 2.14). Whereas only 67 percent of the Sri Lankans born in 1920 would have survived to exact age 5 according to that life table, the figure was almost 96 percent by 1981. The recent ratios for the United States were even more favorable to survival. Since these survival ratios depend only on mortality (and exclude the effects of migration), they show that the mortality conditions in the two countries during this century were strikingly different and that the Sri Lankan survival ratios improved significantly in the years between 1920 and 1981, as did those of the United States between 1900-02 and 1988.

Table 2.14. Survivors to exact age x {(J of 100,000 male infants (£0): Sri Lanka, 1920-81, and United States, 1900-02 to 1988

Country and year

Exact age Country and year 0 5 20 50 65

Sri Lanka 1920 100,000 67,167 56,681 34,458 19,174 1946 100,000 75,448 70,089 51,963 33,245 1954 100,000 86,948 84,332 76,085 62,541 1967 100,000 92,472 90,584 81,651 66,679 1981 100,000 95,825 94,276 84,819 69,053

United States 1900-02 100,000 80,548 75,984 56,736 38,736 1909-11 100,000 82,718 78,792 60,118 40,264 1919-21 100,000 88,505 84,440 67,553 49,218 1929-31 100,000 91,294 88,220 71,518 50,154 1939-41 100,000 •93,624 91,617 78,254 55,776 1949-51 100,000 96,077 94,695 84,158 61,566 1959-61 100,000 96,643 95,491 86,199 64,177 1969-71 100,000 97,395 96,126 86,070 64,318 1979-81 100,000 98,333 97,316 89,007 70,646 1988 100,000 98,676 97,758 89,886 73,517

Sources: United Nations Statistical Office Demographic Yearbook 1957 (1957, table 26|, Demographic Yearbook 1974 (1975, table 35), Demographic Yearbook 1985 (1987, table 36); United States, National Center for Health Statistics (1991, table 6-4). Note: For the United States until 1919-21, the <r figures include only the states that voluntarily joined the federal death registration system.

56 Mortality

C O M P A R I S O N S O F LIFE E X P E C T A N C I E S IN D I F F E R E N T

P O P U L A T I O N S

The ex column of the life table is particularly useful. In comparing the mortality of two countries, we have seen that crude death rates and even standardized death rates have some weaknesses. Since the life expectancy figures in the life table are derived from a model that excludes migration and holds fertility constant, the values of the ex function are often used to compare the mortality of different countries or the same country at several points in time.

The values of c0, life expectancy at birth, are used especially often. Although the most accurate comparison of mortality in two countries would involve a detailed analysis of all the qx values or all the ex values, the life expectancy at birth is a good summary measure. It has some hazards, because the value of e0 is disproportionately affected by the infant mortality rate, but infant mortality rates are usually highly correlated with death rates at other ages. Further, the value of e0 has an immediately appealing interpretation: e0

for a given year of birth measures how long members of cohort born that year can expect to live on the average if mortality conditions remain the same in the future as they were during the year of birth.

In the late 1970s and the 1980s, values of life expectancy at birth in countries for which data were available ranged from 38.1 years to 75.9 years for males (Table 2.15). For females, who usually live longer than males, the values ranged from 41.2 years to 81.8 years.

Other illustrations of the use of life expectancy figures are shown in Tables 2.16 and 2.17. From these tables, we can make the following statements:

1. Males, regardless of color, had lower life expectancies than females at all ages in the United States in 1988 (Table 2.16).

2. Within each sex, nonwhites had lower life expectancies than whites at all ages in the United States in 1988 (Table 2.16).

Mortal i ty 57

3. Whether measured by the absolute number of years gained or proportionate gain, life expectancy at birth increased substantially between 1850 and 1988 for both white males and white females. Life expectancy at age 40, however, increased only moderately. Life expectancy at age 70 increased even less (Table 2.17).

You may find other interesting comparisons.

Table 2.15. Examples of high and low values of life expectancy at birth |e0) for males and females: recent years

Males Females

Year or Year or Country period eo Country period

High e0s High e0s Japan 1989 75.91 Japan 1989 81.77 Hong Kong 1989 74.25 Switzerland 1987-89 80.70 Sweden 1988 74.15 France 1988 80.46 Switzerland 1987-89 73.90 Netherlands 1988-89 80.23 Israel 1988 73.87 Hong Kong 1989 80.05

Low e0s Low e0s Malawi 1977 38.12 Malawi 1977 41.16 Sierra Leone 1985-90 39.40* Afghanistan 1985-90 42.00* Afghanistan 1985-90 41.00* Sierra Leone 1985-90 42.60* Guinea 1985-90 42.00* Guinea 1985-90 43.00* Ethiopia 1985-90 42.40* Ethiopia 1985-90 45.60*

Source: United Nations Statistical Office, Demographic Yearbook 1990 (1992, tables 4 , 1 1 ) . Note: Countries with populations under 1 million are excluded. * Estimate prepared by the United Nations Population Division.

Table 2.16. Life expectancies at selected exact ages for males and females, by color: United States, 1988

g e x Life expectancy at exact age

and color 0 10 20 30 40 50 60 70

Males White 72.3 63.2 53.6 44.4 35.2 26.3 18.4 11.8 Nonwhite 67.1 58.6 49.1 40.3 32.0 24.1 17.2 11.6

Females White 78.9 69.7 59.9 50.2 40.6 31.2 22.6 15.0 Nonwhite 75.1 66.4 56.6 47.1 37.9 29.2 21.3 14.5

Source: United States, National Center for Health Statistics (1991, table 6-1).

58 Mortal i ty

Table 2.17. Life expectancies for white males and females at exact ages 0, 40, and 70: United

States, 1850-1988

White males, by age White females, by age

Year 0 40 70 0 40 70

1850 38.3 27.9 10.2 40.5 29.8 11.3

1890 42.5 27.4 9.4 44.5 28.8 10.2

1900-02 48.2 27.7 9.0 51.1 29.2 9.6

1901-10 49.3 27.6 8.9 52.5 29.3 9.5

1919-21 56.3 29.9 9.5 58.5 30.9 9.9

1920-29 57.8 29.4 9.2 60.6 31.0 10.2

1930-39 60.6 29.6 9.3 64.5 32.2 10.2

1939-41 62.8 30.0 9.4 67.3 33.3 10.5

1949-51 66.3 31.2 10.1 72.0 35.6 11.7

1959-61 67.6 31.7 10.3 74.2 37.1 12.4

1969-71 67.9 31.9 10.4 75.5 38.1 13.4

1979-81 70.8 34.0 11.4 78.2 40.2 14.9

1988 72.3 35.2 11.8 78.9 40.6 15.0

Sources: 1850-1929: Dublin, Lotka, and Spiegelman (1949, table 12); 1939-61: Grove and Hetzel (1968, 308), 1969-71: United States, National Center for Health Statistics (1978b, table 5-11; 1979-81: United States, National Center for Health Statistics (1985, tables 5, 6); 1988: United States, National Center for Health Statistics (1991, table 6-1). Note: 1850, 1890: coverage is restricted to Massachusetts; 1900-29: coverage is restricted to death registration states; 1929-51: coverage is restricted to continental United States.

F O U R T H S E T O F 1. The stationary population is a model that: M U L T I P L E - C H O I C E (a) excludes migration. Q U E S T I O N S ( b ) holds fertility constant.

(c) has fixed mortality rates. |d) is not very good as a descriptive model and is mainly

useful for analytic purposes. (e) Only (a), (b), and (c) are true. (f) Answers (a), (b), (c), and (d) are all true.

2. If the death rate of a stationary life-table population is 10, this implies a life expectancy of about: (a) 10 years. (b) 30 years. (c) 50 years.

Mortal i ty 59

(d) 65 years. (e) 100 years.

3. In a country with a high life expectancy, the fact that the actual death rate is lower than the death rate of the stationary population means that:

(a) the actual population is growing through natural increase.

(b) the country has a younger actual population than the stationary population.

(c) neither of the above is true. (d) both (a) and (b) are true.

4. Survival ratios may be used for: (a) making projections of the future population. (b) comparing the mortality of several countries or the same

country at different points in time.

(c) estimating the effect of different levels of qx on future population sizes.

(d) only (a) and (b). (e) Answers (a), (b), and (c) are all true.

T H I R D S E T O F Determine whether each of the following statements is true T R U E / F A L S E or false:

Q U E S T I O N S i The life-table death rate for females in the United States is higher than the crude death rate.

2. In a life table the life-table death rate is twice the life-table birth rate.

3. Life expectancy at age 70 has not increased very much in the United States in the past 138 years.

4. If you know the life expectancy at birth for a life table prepared for the year of your birth, you know how long you are going to live.

5. Standardized rates are almost always better measures of mortality than crude rates.

6. q0 is usually larger than ql0 and ql0 is usually larger than q

60 Mortal i ty

A D D I T I O N A L For further reading on the materials in this chapter, we rec-R E A D I N G ommend George W. Barclay (1958), Techniques of popula

tion analysis-, A . J. Jaffe (1951), Handbook of statistical methods for demographers-, Warren S. Thompson and David T. Lewis (1965), Population problems, 5th ed. ; and L. I. Dublin, A . J. Lotka, and M . Spiegelman (1949), Length of life, rev. ed. For a complete discussion of standardization and the closely related technique of decomposition, see Prithwis Das Gupta (1993), Standardization and decomposition of rates: A user's manual.

For data on mortality for many nations and for many time periods, we found the following sources by Nathan Keyfitz and Wilhelm Flieger of particular value: (1968), World population: An analysis of vital data-, (1990), World population growth and aging: Demographic trends in the late twentieth century-, and (1971), Population: Facts and methods of demography. In addition, we recommend Samuel H . Preston, Nathan Keyfitz, and Robert Schoen (1972), Causes of death: Life tables for national populations-, United Nations Statistical Office, Department of Economic and Social Affairs, United Nations demographic yearbook, published annually (various issues), and United Nations (1993), World population prospects: The 1992 revision.

Illustrations of the wide utility of the measures discussed in this chapter can be found by referring to studies of mortality in such sources as Richard A. Easterlin (1980), Population and economic change in developing countries; Samuel H . Preston (1978), The effects of infant and child mortality on fecundity-, Preston (1976), Mortality patterns in national populations; Preston and Michael R. Haines (1991), Fatal years: Child mortality in late nineteenth-century America; and Tai-Hun K i m (1990), Mortality transition in Korea: 1960-1980. We also recommend two United Nations publications: (1986a), Consequences of mortality trends and differentials; and (1986b), Determinants of mortality change and differentials in developing countries.

Complete bibliographic information about these works is given in the list of references at the end of the Guide.

Mortality 61

More advanced discussion of the material in the Guide can be found in such books as the following: Donald J. Bogue et al., eds. (1993), Readings in population research methodology, 8 vols.; Mortimer Spiegelman (1968), Introduction to demography, rev. ed. ; Nathan Keyfitz (1968), Introduction to the mathematics of population-, Hugh H . Wolfenden (1954), Population statistics and their compilation, rev. ed. ; Henry S. Shryock, Jacob S. Siegel, and Associates (1971), The methods and materials of demography, 2.vols.,- Roland Pressat (1972), Demographic analysis: Methods, results, applications.

Fertility, Natural Increase, and Reproduction Rates

At the beginning of the last chapter we noted the importance of mortality as a determinant of population growth for most of human history. In more recent history, fertility and fertility control have become dominant in population policy and demographic interest. To illustrate the facts that prompted the shift in attention, in 1985-90 there were annually about 86 mil l ion more births than deaths in the world, and the ratio of births to deaths was more than two to one (United Nations Population Division 1992, Demographic Yearbook 1990, table 1). The increases in population due to these "natural" processes of birth and death, known as natural increase,. led many concerned nations in the 1960s and 1970s to adopt national programs for fertility control—just as in the past they had emphasized death control through campaigns against smallpox, the plague, malaria, tuberculosis, polio, and other diseases.

In this chapter we examine the most common measures of fertility and natural increase. In studying these measures, you w i l l note that most of them are rates, and consequently we follow some of the same procedures we use in measuring mortality. For example, we talk about crude rates, specific rates, and standardized rates.

Demographers distinguish between fertility and fecundity. Fertility refers to actual reproductive performance, whereas fecundity refers to the physiological capacity of a woman, man, or couple to reproduce (United Nations Population Branch 1958, 38; IUSSP 1982, 73 and 78). The definitions are reversed in the Romance languages, sometimes causing confusion at international conferences. In Spanish, for

64 Fertility, Natural Increase, and Reproduction Rates

example, the word for fertility is fecundidad and the word for fecundity is fertilidad. (See Petersen and Petersen 1985 for a multilingual glossary of common demographic terms in English, French, Spanish, Italian, German, Japanese, Ch i nese, and Russian.)

The measurement of fertility poses special problems. We discuss these first before describing particular rates.

F E R T I L I T Y

S P E C I A L Fertility measures always relate the number of live births to P R O B L E M S I N a specific population base and time reference period. Unfor-M E A S U R I N C tunately, it is difficult to establish accurate statistical records

on live births because many infants die in the first few hours or days of infancy. A definition that describes a live birth accurately is complex and difficult to establish; and once one is established, it is difficult to be certain that the definition is actually used by local registration authorities or by respondents answering questions in a sample survey. Nevertheless, the following definition of a live birth has international approval:

A live birth is the complete expulsion or extraction from its

mother of a product of conception, irrespective of the duration

of pregnancy, which, after such separation, breathes or shows

any other evidence of life, such as beating of the heart, pulsation

of the umbilical cord, or definite movement of voluntary muscles,

whether or not the umbilical cord has been cut or the placenta

is attached; each product of such a birth is considered live born

(United Nations Statistical Office 1955, 6).

It is unlikely that this definition is observed everywhere in all cases. When a child dies before the birth is registered, neither the birth nor the death may be registered, only one or the other may be registered, or the birth may be registered as a stillbirth. The registration system is thus prone to error. Similarly, when a survey respondent is asked to report in retrospect on live births, such short-lived children are particularly apt to be omitted.

Fertility measurement also presents special problems not encountered with mortality measurement because a

Fertility, Natural Increase, and Reproduction Rates 65

woman can die only once but she may have no births or more than one birth. This allows us to consider two approaches to fertility measurement: the cumulative fertility approach and the vital rates (or yearly-birth-rates) approach. In using the cumulative fertility approach, we measure the average number of children ever born to women up to some specified age. In using the vital rates approach, we measure the number of live births in a given year as related to the population exposed to the "risk" of giving birth in that year.

The vital rates in fertility measurement are similar to the mortality rates discussed in the second chapter, but there are important differences. The population exposed to the risk of childbearing is not ordinarily decreased by having a birth. Dying, on the other hand, completely removes a person from the population exposed to the risk of dying. Moreover, plural births (e.g., twins or triplets) are possible even though infrequent.

Fertility measurement is also complicated by the fact that fertility involves two parents, whereas death involves only one person. The fact that a couple is the "base" is problematic when we want to consider specific rates because we have to decide whose characteristics to use, the father's or the mother's.

There are also minimum and maximum ages at which men and women are physiologically capable of reproduction. Yet another problem is that not every woman is truly exposed to the risk of childbearing, for the reason that not every woman in the population is paired with a member of the opposite sex. In addition, through widowhood, divorce, separation, and the like, individuals may enter or leave a couple unit at various times in their lives.

Because of these special problems in measuring fertility, no one measurement system comparable to the life table has become dominant in fertility studies. Instead, a wide variety of rates and ratios are currently used, each of which has advantages and limitations in particular analytic situations.


T H E C R U D E The crude birth rate (CBR} is defined as the number of births B I R T H R A T E in a given year divided by the number of people in the popu

lation in the middle of that year. Again, as for the C D R , the ideal denominator is the number of person-years lived, which is often impossible to calculate for a real population. The rate is usually expressed per 1,000 persons. In a formula, we have:

CBR = 1,000 number of births

midyear population P

During the late 1980s the range of crude birth rates for major countries of the world ranged from 10 to more than 56 births per 1,000 per annum (Table 3.1). The highest recorded rates were found in Africa and Asia; the lowest, in Europe and Japan. Eighty-three percent of the more developed nations had rates under 15 whereas 73 percent of the less developed nations had rates of over 30 (Table 3.2). Although the crude birth rate is not a refined measure of fertility, most other fertility measures also show this pattern of higher rates in the developing world.

T H E C R U D E R A T E As one might imagine from the recurrent concern about the O F N A T U R A L "population explosion," typical values of the crude birth rate I N C R E A S E a r e h i g n e r t h a n typical values of the crude death rate. The

crude rate of natural increase measures this gap, as in the following formula:

/ , n v „ , _ J number of births - number of deaths CRNI = 1,000 midyear population

crude birth rate - crude death rate.

In the 1985-90 period the population of the world had a crude rate of natural increase of around 17 per 1,000 (Table 3.3). The nations with the highest rates (20 to 33) were those in the developing regions: almost all of Africa, most of Latin America except for the temperate region, most of Asia ex-

Table 3.1. Highest and lowest crude birth rates, by region: recent years

Region and country Year or period

Crude birth rates (per 1,000)

High Low

Africa Malawi 1985-90 Uganda 1985-90 Mauritius 1985-90 Tunisia 1989

Central and North America Nicaragua 1985-90 Honduras 1985-90 Canada 1988 United States 1990

South America Bolivia 1985-90 Paraguay 1985-90 Uruguay 1989 Argentina 1988

Asia (excluding former USSR) Yemen 1990 Afghanistan 1985-90 Japan 1990 Hong Kong 1990

Europe (excluding former USSR) Albania Ireland Italy Greece

Former USSR Tajikistan Ukraine

Oceania' Papua New Guinea Australia New Zealand

1989 1990 1989 1989

1989

1985-90 1989 1989

56.3* 52.2*

41.8* 39.8*

42.8* 34.8*

51.2T 49.3*

24.6 15.lt

38

34.2*

17.7

18.6* 25.2

14.5 16.7t

18.0 20.7

9.9t 11.7t

9.7 10.1

13

14.9 17.5

Source: United Nations Statistical Office, Demographic Yearbook 1990 (1992, table 9|. ' Estimate prepared by the United Nations Population Division, t Provisional figure. a. Only three countries have populations of more than 1 million. b. Data from Population Reference Bureau, 1992 World Population Data Sheet |1992|. Note: Countries with populations of less than one million are excluded.

http://15.lt

Table 3.2. Distribution of countries by level of crude birth rate: 1980s

Crude birth rate (per 1,000 population)

World total

Less developed regions

More developed regions Africa

North America

Latin America

Asia (excluding-

former USSR) Europe

Former USSR Oceania

All countries 129 100 29 43 2 23 34 23 1 3

Less than 10.0 2 0 2 0 0 0 1 1 0 10.0-14.9 24 2 22 0 1 0 2 20 0 0 15.0-19.9 9 5 4 1 1 3 1 1 1 1 20.0-24.9 12 11 1 0 0 4 7 1 0 1 25.0-29.9 9 9 0 0 0 7 2 0 0 0 30.0^4.9 13 13 0 2 0 3 7 0 0 0 35.0^9.9 12 12 0 4 0 4 4 0 0 1 40.0-44.9 16 16 0 8 0 1 7 0 0 0 45.0^19.9 24 24 0 21 0 1 2 0 0 0 50.0-54.9 8 8 0 7 0 0 1 0 0 0 55+ 0 0 0 0 0 0 0 0 0 0

Source: United Nations Statistical Office, Demographic Yearbook 1990(1992, table 4). Notes: Countries with populations under 1 million are excluded. More developed regions, as defined by the United Nations Statistical Office. World Urbanization Prospects, 1990 (1990) include North America, Japan, Europe, Australia, New Zealand, and the former USSR. Less developed regions include all regions of Africa, Latin America, and Asia except fapan and all regions of Oceania except Australia and New Zealand.


cept East Asia, and all of Oceania except Australia and New Zealand. Europe, the former USSR, North America, Australia, and New Zealand had the lowest rates (2 to 7). East Asia and the Caribbean had intermediate rates. Of course, if continued, any positive rate of natural increase in the absence of net out-migration would lead to very large populations over time. Using the compound interest formula and compounding annually, even a yearly natural increase rate of only 5 per 1,000 would quadruple a population in less than 300 years. At the high natural increase rate of 30 per 1,000 (or 3 percent), which is found in many developing countries, a popu-

Table 3.3. Average crude rates of natural increase, by region: 1985-90

Crude rate of Region natural increase

Africa, total 30 West Africa 31 East Africa 33 North Africa 27 Central Africa 29 Southern Africa 24

North America 6

Latin America, total 22 Caribbean 17 Central America 25 South America 20

Asia, total (excluding former USSR) 19 East Asia 13 South Asia 23 Southeast Asia 21 West Asia 27

Europe (excluding former USSR) 2

Former USSR 7

Oceania, total 11 Australia and New Zealand 7 Melanesia 23 Micronesia 20 Polynesia 29

Al l regions 17

Source: United Nations Demographic Yearbook 1990 (1992, table 1.) Note: Many of these rates are estimates and vary in reliability.


lation doubles in only 24 years, triples in 38 years, and quadruples in 47 years (Table 3.4).

The natural increase rates of recent decades are very high compared with those of previous historical periods. Using the data for all regions shown in Table 3.5, we have estimated the crude rates of natural increase for periods from 1650 to 1990 to be:

Annual crude rate of natural increase

Years (per 1,000)

1650-1750 3.7 1750-1850 4.7 1850-1900 5.4 1900-50 8.4 1950-60 18.3 1960-70 20.2 1970-80 18.5 1980-90 17.4

Table 3.4. Approximate number of years a population takes to double, triple, and quadruple in size, given specified rates of growth (based on the compound interest formula of Pn = P0[l+r]")

Approximate number of years (n) that a population takes to

Rate (%) of growth Double Triple Quadruple per annum (r) in size in size in size

0.50 139 220 278 0.75 93 14 188 1.00 70 111 139 1.25 56 88 112 1.50 47 74 93 1.75 40 63 80 2.00 35 55 70 2.25 31 49 62 2.50 28 45 56 2.75 26 40 51 3.00 24 38 47 3.25 22 34 43 3.50 21 32 41 3.75 19 30 38 4.00 18 28 35

Source: Modified from Marty and Neebe (1966, 1-8). Note: The percentage growth rate used here is equivalent to one-tenth the growth rate expressed per 1,000. For example, a growth rate of 0.5 percent is the same as a growth rate of 5 per 1,000.

Table 3.5. Estimates of midyear populations, by region: selected years, 1650-1990

Population and region 1650 1750 1850 1900 1950 1960 1970 1980 1990

Millions of persons Africa 100 106 111 133 222 279 362 477 642 North America 1 2 26 82 166 199 226 252 276 Central and South America' 12 16 38 74 166 218 286 363 448 Asia 330 498 801 925 1,377 1,668 2,102 2,583 3,113 Europe* 100 167 284 430 572 639 704 750 787 Europe' — — — — 393 425 460 484 498 Former USSR* — — — — 180 214 243 266 289

Al l regions 545 791 1,262 1,650 2,516 3,020 3,698 4,448 5,292

Percentage distribution Africa 18.3 13.4 8.8 8.1 8.8 9.2 9.8 10.7 12.1 North America 0.2 0.3 2.1 5.0 6.6 6.6 6.1 5.7 5.2 Central and South America" 2.2 2.0 3.0 4.5 6.6 7.2 7.7 8.2 8.5 Asia 60.6 63.0 63.5 56.1 54.7 55.2 56.8 58.1 58.8 Europeb 18.3 21.0 22.5 26.1 22.7 21.2 19.0 16.9 14.9 Europe' — — — — 15.6 14.1 12.4 10.9 9.4 Former USSR d — — — — 7.2 7.1 6.6 6.0 5.5

Al l regions 100.0 100.0 100.0 99.8 100.0 99.9 99.9 100.1 100.0

Sources: 1650: Carr-Saunders [1936, 42); 1750-1900: Durand (1968, 109|; 1950-90: United Nations Statistical Office, Demographic Yearbook 1990(1992, table 1, p. 141). a Includes Caribbean beginning in 1950. b. Includes former USSR, including Asian portions. c. Excludes former USSR, including Asian portions. d. Included in Europe totals through 1900 Note: Columns may not total exactly 100 0 percent because of rounding.


We calculated these rates by presuming natural increase to be constant during each decade and by using the exponential growth formula:

where: Pt = population at time t, P0 = population at time 0,

i = the growth rate, t = the number of years, and e = base of natural logarithms (e = 2.71828282 . . . ) .

For example, the growth rate for 1650-1750 can be calculated by making Pt = 791, P0 = 545, and t = 100. Hence:

791 = < J I O O r

545

Solving the equation yields a growth rate of approximately 3.7 per 1,000.

From these estimates it is clear that the rate of growth has been much higher since 1950 than it ever was previously, although the rate has declined somewhat in the past two decades. In the two decades between 1950 and 1970, death rates declined to low or moderate levels for much of the world's population, yet birth rates remained high. It is this fact that led to concern about the "population explosion" and to such dramatic (and admittedly unlikely) projections as the following:

V .

Projection of the post-World War II rate of increase gives a population of one person per square foot of the land surface of the earth in less than 800 years. It gives a population of 50 billion (the highest estimate of the population-carrying capacity of the globe ever calculated by a responsible scholar) in less than 200 years (Hauser 1960, 7).

Because of projections like this, the crude rate of natural increase has been an important and recurrent measure in recent demographic literature. Even though natural increase rates have declined somewhat in the last two decades, the rate of 17.4 from 1980 to 1990 would sti l l lead to a doubling of the world's population in slightly more than 40 years.


T H E G E N E R A L In the mortality chapter, we discussed the rationale for us-F E R T I L I T Y R A T E ing age-specific death rates or death rates specific for other

characteristics. Fertility is also highly variable across subgroups of a population, and it is common to calculate age-specific, age-marital-status-specific, and other specific fertility rates.

The frequency of childbirth varies significantly with the age of the parents, and the age at which maximum fertility occurs may be different for males and females. Furthermore, fertility is higher among couples who have established some type of regular cohabitation (such as legal marriage or common-law marriage) than among persons not in such unions (single persons, for example). Conventionally, specific fertility rates are calculated for female parents and not male parents, and henceforth we w i l l discuss specific birth rates for females only. Male parallels could be developed in each case.

It is rare for a child to be born to a woman less than 15 years old or more than 50 years old. For this reason, one way to refine the measurement of fertility somewhat is by using the midyear population of women in the childbearing years for the denominator of the rate instead of the total midyear population of both sexes. The rate so constructed is called the age-delimited or general fertility rate (GFR). It is defined as the number of births in a given year divided by the midyear population of women in the age groups 15-44 or 15-49, although the ages 10-49 are sometimes used. In a formula:

GFR = 1,000

= k B Pf

number of births in a given year midyear population of women

of ages 15 - 44 or 15 - 49

or A: B

3 5 ' 1 5

The purpose of the GFR is to restrict the denominator to potential mothers, but it is still not restrictive enough. Rates within five-year age groups may be different for two


populations and yet the two populations may have the same general fertility rate if the age composition of women in the childbearing years differs for the two. In this sense the GFR is subject to the same kind of crudeness as the crude birth rate, although it is a distinct improvement in precision.

In the recent past, general fertility rates for various countries have been in the range of the low 40s to the high 100s per thousand women of reproductive age. Data for recent years from table 11 of the United Nations Demographic Yearbook 1990 show that the highest values of the GFR were 188.1 for Guatemala and 163.9 for Egypt. The lowest values were 39.7 for Japan and 41.7 for Italy. As is true of the crude birth rate, the highest rates were found in developing countries and the lowest rates were usually found in Europe.

A G E - S P E C I F I C Within the age range of 15-49 years, there are marked differ-F E R T I L I T Y R A T E S ences in the fertility of women of various ages. For this rea

son it is customary to calculate fertility rates for each age or age group, as in the following formula:

nFx = Age-specific fertility rate for age group*, x + n - 1 000 n u m ' 5 e r °^ births to women in age group x, x + n

midyear population of women in age group*, x + n

- L- " B ' P>

n x

where „BX = births to women of the age group x, x + n, nPj = midyear population of women in the group*, x + n, and

k = 1,000.

In most analyses, five-year age groups are used to calculate the age-specific rates. Typically, the age-specific rates are low or moderate in the 15-19 age group, highest in the 20s, and then decline to moderate levels for women in their 30s. Rates after age 39 are usually low. Rates for the mid- to late-1980s for the United States, Japan, Guatemala, and Costa Rica are portrayed graphically in Figure 3.1 to illustrate the


typical, mountain-shaped patterns of age-specific fertility. Described more formally, the typical distribution is truncated, skewed to the left, and leptokurtic (i.e., having a high, narrow concentration about the mean) relative to a normal distribution.

Although the patterns of age-specific rates are reasonably similar for various populations, the absolute levels of the age-specific rates vary considerably. Table 3.6, which presents recent data on the lowest and highest national age-specific rates by age group, shows that very high age-specific rates have been recorded among selected groups of women.

An example often cited to illustrate very high fertility is the schedule of age-specific rates for the ethnic Hutterites of North America, an Anabaptist religious sect living in small colonies in the United States and Canada. In the book Man's Capacity to Reproduce: The Demography of a Unique Population, Eaton and Mayer (1954) report the age-specific fertil-

Figure 3.1. Age-specific fertility rates: Costa Rica, 1984, Guatemala, 1985, Japan, 1989, and United States, 1988

15-19 20-24 25-29 30-34 35^39 40-44

Age group Source: United Nations Statistical Office, Demographic Yearbook 1990 11992, table 11).


ity rates for Hutterite women in the 1936-40 period. Table 3.7 compares the Hutterite rates with the rates for all U.S. women in 1940. At all ages except for ages 15-19, the Hutterite rates were dramatically higher than the rates for all U.S. women. The reason the Hutterite rates were lower at ages 15-19 is that the Hutterites married at later ages than the average for U.S. women. These figures mean that during the peak fertility years, roughly 46 percent of the Hutterite women gave birth each year (46.2 percent for women of ages 30-34 and 46.6 percent for women of 25-29). Even as late as ages 35-39, 43 percent gave birth each year.

The Hutterite rates and the ranges found in Table 3.6 are examples of the extremes in age-specific rates. Most of the rates in any age group are much closer together. In recent years 93 percent of the countries for which data were available had age-specific rates below 100 at ages 15-19 (Table

Table 3.6. Lowest and highest age-specific fertility rates per 1,000 women: 1980s

Lowest Highest

Women's ages Rate Country, year Rate Country, year

15-19" 3.5 Japan, 1989 137.7* Honduras, 1981 125.5 Guatemala, 1985

20-24 47.0 Japan, 1989 307.4* Honduras, 1981 273.5 Guatemala, 1985

25-29 84.7 Bulgaria, 1989 333.2 Egypt, 1986 30-34 30.9 Bulgaria, 1989 289.5 Egypt, 1986 35-39 9.8 Bulgaria, 1989 183.0 Guatemala, 1985 40-44 1.4 Former German 122.7* Dominican Rep., 1980

Democratic Rep., 1988 81.5 Guatemala, 1985 45-49b 0.1 Japan, Hungary, 212.2* Dominican Rep., 1980

Czechoslovakia, 1989 43.0 Guatemala, 1985 O.lt Norway, Bulgaria,

Denmark, 1989

Source: United Nations Statistical Office, Demographic Yearbook 1990 (1992, table 11). Notes: Countries with populations of less than 1 million are excluded. Of the countries with populations greater than 1 million, coverage is more complete for countries of the more developed world. For example, in Africa, data are available for only Botswana, Egypt, Mauritius, and Tunisia, whereas virtually every European nation is included. * Rate has been calculated from live births recorded in civil registers that are incomplete or of unknown completeness. Such figures are included in the "highest" column only when the rate is higher than any rate from a more reliable source. t Based on 30 or fewer live births. a. Computed using all women under age 20. b. Computed using all women of ages 45 and above.


Table 3.7. Age-specific fertility rates per 1,000 women: Hutterites and all U.S. women, around 1940

Women's ages Hutterite women,

1936-̂ 0 U.S. women,

1940-

15-19 259 136 25-29 466 123 30-34 462 83 35-39 431 46 40̂ 14 203 16 45-49 48 2

Sources: Hutterite women: Eaton and Mayer 11954, table 11); U.S. women: United States, National Center for Health Statistics 11978a, table 1-6). a. U.S. rates have been corrected for underregistration of births.

3.8). At ages 20-24, 80 percent of the age-specific rates were in the range of 50-199. Rates in the remaining age groups showed similar patterns of concentration in a narrower range than the ranges given in Table 3.6.

Table 3.8. Distribution of major countries and territories by level of age-specific fertility rates: recent years

Women's ages

Level of age-specific fertility rates (per 1,000 women) Women's ages 0-49 50-99 100-149 150-199 200-249 250-299 300+ All levels

Number of countries 15-19 37 20 4 0 0 0 0 61 20-24 3 18 17 14 4 2 1 61 25-29 0 6 37 9 4 5 0 61 30-34 5 33 15 4 4 0 0 61 35-39 38 13 6 4 0 0 0 61 40-44 54 6 1 0 0 0 0 61 45-49 61 0 0 0 0 0 0 61

Percentage distribution 15-19 61 33 7 0 0 0 0 101 20-24 5 30 28 23 7 3 2 98 25-29 0 10 61 15 7 8 0 101 30-34 8 54 25 7 7 0 0 101 35-39 62 21 10 7 0 0 0 100 40-44 89 10 2 0 0 0 0 101 45̂ 19 100 0 0 0 0 0 0 100

Source: United Nations Statistical Office, Demographic Yearbook 1990(1992, table 11). Notes: Percentages may not add exactly to 100 percent because of rounding. Countries with populations under 1 million are excluded. Of the countries with populations greater than 1 million, coverage is more complete for more developed than for less developed countries. For example, in Africa, data are available for only Botswana, Egypt, Mauritius, and TUnisia, whereas virtually every European nation is included. To achieve greater coverage, countries with incomplete registration are included; therefore, actual age-specific rates are somewhat higher than those summarized in this table.


F I R S T S E T O F Determine whether each of the following statements is true T R U E / F A L S E or false: Q U E S T I O N S • As a result of postwar progress, only about one-third of

the world's population lives in countries with high rates of natural increase; the other two-thirds has attained relatively low rates of natural increase resulting from low birth rates and low death rates.

2. The majority of countries in the 1980s had crude birth rates above 30 per thousand per annum.

3. The recent crude rates of natural increase for the population of the world were probably never attained in the period between 1650 and 1950.

4. It is unlikely that a population would have a crude birth rate of 40 and a crude death rate of 15 during the same time period.

5. The lowest birth rates recorded in the late 1980s were mostly for European nations.

F I R S T S E T O F 1. In two hypothetical countries, A and B, the age-specific M U L T I P L E - C H O I C E fertility rates for females are as follows: Q U E S T I O N S Ages Country A Country B

15-24 80 80 25-34 250 250 35-44 100 100

(a) Country A has a higher general fertility rate than country B.

(b) Country B has a higher general fertility rate than country A.

(c) Country A has the same general fertility rate as country B.

(d) Country A has the same crude birth rate as country B. (e) The general fertility rate for country A may be the same,

higher, or lower than the general fertility rate for country B.

2. The crude birth rate in the United States is now approximately: (a) 10 per thousand.


(b) 15 per thousand. (c) 25 per thousand. (d) 35 per thousand. What is the crude birth rate in your own country?

3. Characterize as closely as possible the population of the United States, Canada, and the former USSR. (a) crude birth rate of 21-44, crude death rate of 20-30. (b) crude birth rate of 15-20, crude death rate of 5-10. (c) crude birth rate of 20-30, crude death rate of 15-25. (d) crude birth rate of 10-25, crude death rate of 5-15. (e) crude birth rate of 10-16, crude death rate of 5-30.

4. Characterize as closely as possible the populations of the European nations. (Select from the same answer categories as for question 3.)

5. A crude rate of natural increase of 30 per thousand leads to a doubling of the population in approximately: (a) 15 years. (b) 25 years. (c) 50 years. (d) 75 years. (e) 100 years.

6. Typically, age-specific fertility rates for women: (a) are highest at ages 15-24 and lower thereafter. (b) are highest at ages 20-29 and lower at ages 15-19 and

ages over 30. (c) are highest at ages 25-34 and lowest at ages 15-24 and

ages over 35. (d) are fairly constant throughout the childbearing years.

B I R T H R A T E S We often need to study birth rates specific for characteristics S P E C I F I C F O R other than age. Two important characteristics are marital C H A R A C T E R I S T I C S s t a t u s a n d live birth order.

Al l societies have forms of culturally sanctioned reproductive units resulting from a religious marriage, legal marriage, consensual union, common-law marriage, "living in"

O T H E R T H A N A G E


arrangement, or other union.1 For convenience, we call all of these institutionalized arrangements "marriage" in the present discussion. Although marriage is a nearly universal phenomenon, significant variation exists among norms about the proper age to marry, about divorce, and about the remarriage of widows. The norms may change over time and the possibility of adhering to them may be affected by the age and sex composition of a population. For example, in populations where there is a shortage of eligible males or females, persons of the opposite sex who wish to marry may find themselves caught in a marriage squeeze. A marriage squeeze is usually a "by-product of the combination of different-sized cohorts and the fact that women do not usually marry men their own age" (Weeks 1992, 320). The squeeze refers to an imbalance between the number of males and females in the age groups that usually marry one another: either there are too many males or too many females relative to the other sex in the appropriate age groups. (For a more detailed discussion of this phenomenon, see Schoen 1983 and Weeks 1992, 320-21.)

Since the proportions married may vary, and since birth rates generally are much higher for the married than for the total population of women, demographers often construct fertility rates specific for marital status as well as for age, so that one has age-marital-status-specific fertility rates. Five articles illustrate the use of fertility rates specific for marital status and age. Freedman and Adlakha (1968), Hirschman (1986), Ogawa and Retherford (1993), Peng (1993), and Retherford and Rele (1989) examine respectively the factors responsible for declining fertility rates in Hong Kong in the 1960s, Peninsular Malaysia from 1970 to 1980, Japan from 1950 to 1990, China from the mid-1960s to the early 1980s,

1. For an informative discussion of the complexity of marital unions in relation to fertility in the U.S. context, we recommend a series of papers by Ronald Rindfuss and colleagues (Rindfuss and Parnell 1989, Rindfuss and Jones 1991, and Pagnini and Rindfuss 1993). Chapters 4-6 in a book by Stycos and Back (1964) published three decades ago, are still recommended reading on this topic for a look at these relationships in the Caribbean.


and the countries of South Asia from 1960-64 to 1980-84. To quote Freedman and Adlakha (p. 181):

An important question about such declines in crude birth rates is whether they result from real declines in the fertility of married women or from changes in the number of women of childbearing age or in the proportion who are married in the productive childbearing years. Are married women having fewer children or are there simply fewer married women in the important childbearing years?

Clearly, we cannot review the detailed results of all five studies here. Instead, we present a few examples of the results showing how changes in the marital status and age compositions of populations can affect fertility:

1. Changes in the marital-status composition of the populations were among the important causes of the fertility declines in all of the countries studied during the time periods mentioned above. For example, all of the fertility decline in Japan between 1980 and 1990 and roughly half of the decline between 1970 and 1980 were due to changes in the marital-status composition of the population. Between 1960-64 and 1980-84, the percentages of the fertility declines due to changes in the marital-status composition were: 51 percent in Bangladesh, 28 percent in India, and 39 percent in Sri Lanka. In Pakistan from 1960-64 to 1980-84, fertility remained relatively constant, but it would have increased were it not for changes in the marital-status compositon. In China, changes in the marital-status composition accounted for about one-fifth of the declines in rural fertility between the mid-1960s and the early 1980s. In urban areas of China during the same time period, changes in the maritial-status composition accounted for as much as 40 percent of the decline (in Shanghai) and as little as 6 percent (in Jiangsu and Hebei provinces).

2. Changes in the age composition were important in Hong Kong's fertility decline in the 1961-65 period but not in 1965-66. In Peninsular Malaysia, changes in the age composition would have led to increased fertility among the Malays between 1970 and 1980 were it not for the opposite


effect of changes in their marital-status composition. Hence, the crude birth rate for Malays declined only about 1 percentage point in that decade.

Fertility rates specific for live birth order are also useful. The probability of having an additional child is affected by how many children a woman has already borne. This is true because she and her husband may use contraception after a certain number of births and because the physiological capacity to bear children is affected by previous childbirths as well as by age and other factors. One may calculate the birth order-specific fertility rate as follows:

Birth-order-specific fertility rate = 1,000

ff

number of births of order i mi lidyear population of women

of ages 15-44 or 15-49

P' 30^15

or k ff

where B' =

30^15 o r =

k =

Pl

35 15

births of order i,

midyear population of women between ages 15-44 or 15-49, and

1,000.

(Notice that the sum of the birth-order-specific fertility rates is the GFR.) It is often useful to make the rates specific for smaller age groups, and we may therefore wish to calculate age-specific, birth-order-specific fertility rates.

To illustrate the use of rates specific for live birth order, consider the data for the United States summarized in Table 3.9. The general fertility rate in the United States was lower in 1988 (67.2) than in 1942 (91.5). During the intervening 46 years, the rate had both increased and decreased from the 1942 level, reaching a high value of 122.9 in 1957, thereafter declining and then staying fairly steady until 1988. Table 3.9 also shows data for two of the intervening years, 1960 and 1975.

Table 3.9. Birth rates by live birth order and percentage change in rates: United States, selected years, 1942-88

Live births per 1,000 women 15-44 years old Percentage change"

1942 1960 1975 1988 1942-60 1960-75 1975-88 1942-75 1942-88 Live birth order U) 12) 13) 14) 15) (6) 17) 18) 19)

First birth 37.5 31.1 28.1 27.6 -17.1 -9.6 -1.8 -25.1 -26.4 Second birth 22.9 29.2 20.9 22.0 27.5 -28.4 5.3 -8.7 -3.9 Third birth 11.9 22.8 9.4 10.9 91.6 -58.8 16.0 -21.0 -8.4 Fourth birth 6.6 14.6 3.9 4.1 121.2 -73.3 5.1 -40.9 -37.9 Fifth birth 4.1 8.3 1.7 1.5 102.4 -79.5 -11.8 -58.5 -63.4 Sixth and seventh births 4.6 7.6 1.3 0.9 65.2 -82.9 -30.8 -71.7 -80.4 Eighth and higher births 3.9 4.3 0.7 0.3 10.3 -83.7 -57.1 -82.1 -92.3 All births (GFR) 91.5 118.0 66.0 67.2 29.0 -44.1 1.8 -27.9 -26.6

Source: United States, National Center for Health Statistics (1969, table l-8; 1990, table 1-8). a. Column 5 - ((column 2 - column l|/column 1]/ x 100;

Column 6 - ((column 3 - column 2)/column 2)/ x 100; Column 7 - Ifcolumn 4 - column 3)/column 3)/ x 100, Column 8 - |(column 3 - column 1 (/column 1 ]/ x 100; Column 9 - [(column 4 - column l)/column 1]/ x 100.


Rates for all birth orders but the first were higher in 1960 than in 1942. The 1960 birth rate for the first birth order was low because so many women had already had their first children in the "baby boom" period of the 1950s. Most of the difference in the general fertility rates of 1960 and 1942 (118.0 - 91.5 = 26.5) resulted from higher rates for second, third, and fourth births (25.2 of the 26.5 difference). Much of the 1960-75 fall (118.0 - 66.7 = 51.3) was due to declines in the rates for the third and fourth birth orders (total decline of 24.1), but the rates fell for all orders during that period. The GFRs for 1975 and 1988 were almost the same; but rates for second, third, and fourth births rose and for other birth orders declined between the two years.

Fertility rates specific for age, for marital status, or for live birth order are only three examples of many specific rates that may be useful in a particular fertility analysis. Demographers may also be interested in the variation in fertility rates by mothers' parity (i.e., the number of children already borne), educational attainment, income, size of place of residence, ethnic group, occupation, contraceptive use, and other social and economic variables. The method of computing rates specific to other characteristics is similar to that examined for age, marital status, and live birth order; all you need are the data.

S T A N D A R D I Z E D Because we are interested in measuring fertility itself, we B I R T H R A T E S often want to control for, or eliminate the effect of, other

variables. To do this we may look at a detailed schedule of specific fertility rates (by age, marital status, or any other characteristic) and compare two populations. Alternatively, we may want a single measure that corrects for the effects of the extraneous variables. One such measure would be a standardized fertility rate, corresponding to the standardized mortality rates discussed in Chapter 2.

The most common standardized fertility measure in use is the age-sex adjusted birth rate, which is the crude birth


rate standardized for age and sex composition. Because the procedure used in calculating this rate is similar to that used for standardizing the crude death rate, we do not repeat that discussion here. The actual values of age-sex adjusted birth rates, however, are interesting.

Table 3.10 presents the crude birth rates and standardized birth rates for 37 countries and dates, based on calculations by Keyfitz and Flieger (1990). The standard populations used are those of the United States and Mexico in 1980. A l though many statements can be made about the rates summarized there, we focus on two observations:

1. The rank order of countries from highest to lowest birth rate is not affected much by the standardization—that is, the correlation between the two rank orderings is high— but a few countries do change in rank depending on the standard used. For example, Peninsular Malaysia has the third highest unstandardized rate. Standardized on the 1980 U.S. age distribution, it retains this ranking. Standardized on the 1980 Mexican age distribution, however, Peninsular Malaysia has a rank of 5.

2. Even though the rank order of countries is not critically affected by standardization, the amounts of the differences do change substantially. For example, the crude birth rates are 26.40 for Mauritius in 1980 and 20.96 for China in 1981, a difference of 5.44. Standardized on the age distribution of Mexico in 1980, the rates are 21.88 for Mauritius and 19.48 for China, a difference of 2.40. Hence, a large proportion of the difference in crude rates between Mauritius and China is due to differences in their age distributions.2

Data for the United States between 1940 and 1988 provide an additional illustration of the use of age-sex adjusted

2. We can quantify how much of the difference in crude rates is due to differences in the age distribution and how much is due to differences in the ASFRs using a technique called decompositon. For a discussion of decomposition and standardization techniques, see Das Gupta (1993). For a description of how to decompose the total fertility rate, discussed in the next section of this chapter, see Retherford and Rele (1989, 744-45|.

Table 3.10. Crude birth rates and directly standardized birth rates: selected places and dates

Crude birth rates standardized using as standard: Crude hirth United States, 1980 Mexico, 1980

Place, Year rate Rate Rank Rate Rank

Guatemala, 1985 41.04 48.05 1 42.59 1 Mexico, 1983 34.96 37.46 2 33.30 2 Peninsular Malaysia, 1985 31.34 31.34 3 26.62 5 Panama, 1980 28.65 30.50 4 28.03 3 Venezuela, 1985 29.01 29.59 5 26.64 4 Fiji, 1980 29.67 28.89 6 25.79 6 Mauritius, 1980 26.40 24.31 7 21.88 7 China, 1981 20.96 22.68 8 19.48 9 Former USSR, 1987 19.92 21.83 9 19.80 8 Chile, 1980 21.13 20.86 10 19.01 10 Poland, 1985 18.21 20.18 11 18.25 11 Ireland, 1986 17.35 19.64 12 16.55 13 Former Yugoslavia, 1985 15.85 17.81 13 16.30 14 Bulgaria, 1985 13.28 17.70 14 17.08 12 Hong Kong, 1980 16.98 17.16 15 14.65 16 New Zealand, 1985 15.80 16.50 16 14.63 18 Australia, 1985 15.67 16.34 17 14.26 20 Hungary, 1985 12.23 16.20 18 15.17 15 United States, 1985 15.75 15.86 19 14.54 19 German Dem. Rep., 1985 13.68 15.64 20 14.64 17 France, 1985 13.93 15.57 21 13.63 21 England and Wales, 1985 13.15 15.21 22 13.47 22 Spain, 1983 12.71 14.92 23 12.97 25 Japan, 1980 13.51 14.87 24 12.68 26 Sweden, 1985 11.79 14.54 25 12.48 28 Greece, 1985 11.74 14.53 26 13.24 23 Portugal, 1985 12.85 14.46 27 12.98 24 Canada, 1985 14.82 14.23 28.5 12.51 27 Norway, 1985 12.29 14.23 28.5 12.47 29 Finland, 1985 12.81 13.74 30 11.84 30 Singapore, 1985 16.61 13.49 31 11.49 32 Belgium, 1984 11.73 13.25 32 11.68 31 Austria, 1985 11.57 12.77 33 11.48 33 Netherlands, 1985 12.29 12.72 34 10.80 35 Italy, 1983 10.59 12.60 35 11.02 34 Denmark, 1985 10.51 12.29 36 10.64 36 Germany, Fed. Rep., 1985 9.61 10.87 37 9.32 37

Source: Keyfitz and Flieger (1990, 294 ff ).


birth rates (Table 3.11). The crude birth rate for the United States increased from 19.4 in 1940 to a peak of 25.0 in 1955, declined to 14.6 in 1975, then rose to 15.9 in 1980 and again in 1988. The highest crude birth rates were recorded in the baby boom period of the 1950s, right after World War II. A l though the crude rates in that period were high, they were not nearly as high as they would have been if the age and sex composition of the 1940s had still held in the 1950s. For example, if the age and sex composition of the 1955 population had been the same as the 1940 composition, the birth rate for 1955 would have been 30.4 instead of the observed value of 25.0. In fact, the standardized birth rate for every year after 1940 except the early 1980s is higher than the crude rate when the 1940 composition is used as the standard population. The age and sex structure of the United States since 1940 has generally been less favorable to high crude birth rates than was the 1940 age and sex structure. The situation in the 1980s was due to the baby boomers of the 1950s passing through the prime childbearing ages.

We can standardize more refined fertility measures— the general fertility rate, for example, or even age-specific

Table 3.11. Crude birth rates and directly standardized birth rates: United States, selected years, 1940-88

Age-sex adjusted birth rate Crude using 1940 U.S. age and birth sex distribution rate as standard population

1940 19.4 19.4 1945 20.4 20.9 1950 24.1 26.3 1955 25.0 30.4 1960 23.7 31.2 1965 19.4 24.8 1970 18.4 21.3 1975 14.6 15.3 1980 15.9 15.8 1985 15.8 15.8 1988 15.9 16.5

Source: United States, National Center for Health Statistics (1990, tables 1-2, 1-3, pp. 1,4|. Note: For years prior to 1960, data are adjusted for underregistration.

Table 3.12. General fertility rates for selected countries, age-standardized by the direct method: late 1980s (standard - Sweden, 1988)

Standard million Age-specific fertility rates, by country* Expected number of births

for females 15-49, Philip United Philip United Sweden, 1988 Egypt pines Sweden States Japan Egypt pines Sweden States Japan

Ages (1) (2) (3) (4) (5) (6) (l)x|2) U)x(3) (l)x(4) U)x(5) U)x(6)

15-19 135,064 20.1 42.5 11.4 54.8 3.5 2,715 5,740 1,540 7,402 473 20-24 149,706 203.0 162.7 90.7 111.2 47.0 30,390 24,357 13,578 16,647 7,036 25-29 136,531 333.2 180.4 146.6 113.2 144.8 45,492 24,630 20,015 15,455 19,770 30-34 138,687 289.5 135.7 100.9 73.7 90.9 40,150 18,820 13,944 10,221 12,607 35^9 145,961 182.6 96.2 36.8 27.9 19.5 26,652 14,041 5,371 4,072 2,486 40-44 163,270 69.8 40.5 6.0 4.8 2.4 11,396 6,612 980 784 392 45-49 130,781 32.7 8.9 0.3 0.2 0.1 4,277 1,164 39 26 13

All ages 1,000,000 161,072 95,364 55,467 54,607 42,777

Observed GFRs (per 1,000) 163.4 106.8 55.5 60.2 39.7

Standardized GFRs (per 1,000) 161.1 95.4 55.5 54.6 42.7

Sources: United Nations Statistical Office, Demographic Yearbook 1990(1992, tables 7, 11). Note: Philippine data are from the civil register, which is of unknown completeness, a. Data for all countries are for 1988 except data for Egypt, which are for 1986.


fertility rates. To illustrate, we have age-standardized recent general fertility rates for Sweden, Egypt, the Philippines, the United States, and Japan, using three standard populations: Sweden in 1988, India in 1990, and the Republic of Korea in 1990. Table 3.12 shows the calculations using the Swedish population as standard. The comparable calculations for the India standard and the Korea standard are not shown, but the observed rates and the standardized values are summarized in Table 3.13. The rank order of the general fertility rates is similar for the actual rates and each set of standardized rates: Egypt always has the highest rate and Japan the lowest, with the Philippines second highest, and the United States and Sweden very close. The amounts of the differences, however, are affected by the standards. To take just one example, the ratio of the actual GFR for Egypt to that of Japan is 4.1 (163.4/ 39.7). Standardized on the age distribution of India, however, the ratio falls to 3.5 (165.0/47.0). Notice that Japan's rate is more affected by the standardization (it increases from 39.7 to 47.0 than is Egypt's rate (which increases only from 163.4 to 165.0). What this means is that a smaller proportion of Japan's women were concentrated in the peak fertility years than was true in Egypt or India. In other words, Japan's female age distribution was less favorable to high general fertility rates than India's or Egypt's.

Table 3.13. Observed general fertility rates and age-standardized general fertility rates with Sweden (1988), India (1990), and the Republic of Korea (1990) as standard populations: Egypt, Philippines, Sweden, United States, and Japan

Age-standardized GFR using as standard population:

Sweden, India, Korea, Country GFR 1988 1990 1990

Egypt 163.4 161.1 165.0 175.5 Philippines 106.8 95.4 102.3 105.7 Sweden 55.5 55.5 60.9 64.6 United States 60.2 54.6 64.0 64.6 Japan 39.7 42.7 47.0 51.1

Sources: GFRs: United Nations Statistical Office, Demographic Yearbook 1990(1992, table 11). Age-standardized GFRs use standard age distribution of the countries listed and the ASFRs from Table 3.12 in this chapter.


S E C O N D S E T O F 1. In two hypothetical countries, A and B, the age-specific M U L T I P L E - C H O I C E fertility rates per 1,000 are as follows: Q U E S T I O N S

Ages Country A Country B 15-24 50 50 25-34 100 100 35-44 60 60

In country A, 60 percent of the population is female and 30 percent of the females are between the ages of 15 and 44. In country B, 50 percent of the population is female and 35 percent of the females are between the ages of 15 and 44. Assume that births occur only to women between the ages of 15 and 44. (a) The crude birth rate is higher in country A than in coun

try B. (b) The crude birth rate is lower in country A than in coun

try B. (c) The crude birth rate is equal in the two countries. (d) Any of the above may be true.

2. Using the data in question 1, it is possible to say with certainty that: (a) The general fertility rate is higher in country A than in

country B. (b) The general fertility rate is higher in country B than in

country A. (c) The general fertility rate is equal in the two countries. (d) Any of the above may be true.

3. Using the data in question 1, it is possible to say with certainty that: (a) The age-sex adjusted birth rate is higher in country A

than in country B. (b) The age-sex adjusted birth rate is higher in country B

than in country A. (c) The age-sex adjusted birth rates in the two countries

are equal. (d) Any of the above may be true.


4. As compared with less developed nations, the age structures of the more developed nations tend to be unusually favorable to: (a) high crude birth rates and high crude death rates. (b) low crude birth rates and high crude death rates. (c) low crude birth rates and low crude death rates. (d) high crude birth rates and low crude death rates. (e) none of the above.

S E C O N D S E T O F Determine whether each of the following statements is true T R U E / F A L S E or false. Q U E S T I O N S i . The standardization of crude birth rates makes rela

tively little difference in the rank order of countries for values of the birth rate, but it does affect the sizes of rates relative to one another.

2. It would be possible to construct an age-standardized rate of natural increase.

3. Fertility rates specific for live birth order can be constructed only as period rates and not as cohort rates.

4. It is possible to standardize means, percentages, proportions, and ratios as well as rates.

T H E T O T A L By now you know that the standardization technique is quite F E R T I L I T Y R A T E general, and we could apply it to many refined uses—such as

the computation of age- and marital-status-specific fertility rates standardized on the educational composition of a standard population. Instead, we now turn to a discussion of the total fertility rate (TFR), which is a standardized rate whose values are particularly useful in interpreting the cumulative fertility implied by a given set of age-specific fertility rates.

The total fertility rate is defined as the sum of the age-specific fertility rates for women, when age is given in single years. We would usually perform the following calculation to get the total fertility rate:


TFR = [sum of the age- specific fertility rates] x 1,000

= £ ( F > 1,000 j :

where TFR = total fertility rate, X means one should add up the age- specific rates, and X

Fx = the age- specific rates for the age group*, * +1.

The total fertility rate is a standardized measure because the age-specific fertility rate at each age is multiplied by a standard population, usually of 1,000 persons, as above. In other words, the total fertility rate assumes a "rectangular" age distribution for the standard population with the same number of persons at each year of age, namely l,000.3In practice, it is usual to sum rates for five-year age groups and to assume that the age-specific rates for each single year are accurately summarized by the average rate for the five-year age group. The formula then becomes

TFR = 5 £ ,̂ (1,000) (see Table 3.14).

The TFR is only one type of standardized rate, but its use has been particularly widespread because it has a useful interpretation. The total fertility rate summarizes a hypothetical fertility history analogous to the hypothetical mortality history of a cross-sectional life table. It estimates the total number of live births 1,000 women would have if they all lived through their entire reproductive period and were subject to a given set of age-specific fertility rates. In other words, the total fertility rate reports the average number of live births among 1,000 women exposed throughout their childbearing years to the schedule of age-specific fertility rates being used, assuming no women died during the childbearing years.

Actually, age-specific fertility rates change from year

3. The total fertility rate may be expressed either per woman or per 1,000 women. In other words, the constant k is either 1 or 1,000. In this Guide, we express the rate per 1,000 women. If sources for the tables give the TFR (or GRR or NRR) per woman, we have converted the constant to 1,000 women.


Table 3.14. Calculation of total fertility rates for the United States: 1957, 1976, and 1988

Age-specific fertility rates

Women's ages

per 1,000 women for:

Women's ages 1957 1976 1988

10-14 1.0 1.2 1.3 15-19 96.3 52.8 53.6 20-24 260.6 110.3 111.5 25-29 199.4 106.2 113.4 30-34 118.9 53.6 73.7 35-39 59.9 19.0 27.9 40-44 16.3 4.3 4.8 45-49 1.1 0.2 0.2 Sum 753.5 347.6 386.4

Sumx5 - T F R (per 1,000 women) 3,767.5 1,738.0 1,932.0

Source: United States, National Center for Health Statistics (1990, table 1-6, p. 7|.

to year, and it is not likely that the age-specific rates for a specific calendar year would remain the same throughout the reproductive years of a woman. Just like the measures from a cross-sectional life table, the total fertility rate reflects what would happen to a hypothetical or "synthetic" cohort of women. The rate can be interpreted to reflect completed family size only when we assume that the age-specific fertility rates for women 20-24 years old now will still be the same when women 15-19 become 20-24 in five years, and when we also make similar assumptions for the other age groups.

In the late 1980s the United Nations Statistical Office (1992, table A-12) estimated total fertility rates as high as 8,490 (Rwanda) and as low as 1,330 (Italy). Higher total fertility rates are found in the less developed areas than in the more developed areas, just as are higher crude birth rates and higher general fertility rates.

In fact, all the common measures of fertility we have discussed thus far are usually highly correlated with one another. Using the fertility measures for 50 nations with reliable data for the 1955-60 period, Bogue and Palmore (1964) reported the following correlations: (a) .992 between the crude


birth rate and the general fertility rate, (b) .980 between the crude birth rate and the general fertility rate standardized on the estimated age composition of the world, and (c) .982 between the crude birth rate and the total fertility rate. These three correlation coefficients summarize only a few of the relationships, but coefficients above .979 were found between all the fertility rates we have presented thus far except for the various age-specific rates. The age pattern of fertility is more variable within a specific overall fertility level, but even the lowest correlations were still quite high and can be illustrated by the following two values: the lowest correlation between the total fertility rate and an age-specific rate was .711 and the lowest correlation between the standardized GFR and an age-specific rate was .689. Even the correlations between age-specific rates without being controlled for the overall fertility level were .425 or greater.

Since the various measures of fertility are so highly correlated, you may well ask why there are so many of them. Why don't demographers use just one? There are several reasons:

1. The data necessary for calculating any given measure may not be available. For example, for a certain country you may be able to compute only the crude birth rate because data on the age and sex distribution or on live births by age of mother are not available.

2. We cannot be certain that the high correlations of the 1955-60 period have always obtained in the past, and they may not obtain in the future. Rapid changes in fertility are occurring in some countries, and the age distribution depends on fertility. Hence, we may get different results in the future. The articles we referred to earlier when we discussed the effects of the age and marital status compositions of populations on their fertility rates (Freedman and Adlakha 1968; Hirschman 1986; Ogawa and Retherford 1993; Peng 1993; Retherford and Rele 1989) illustrate the types of changes that can occur and how the different measures help us understand more comprehensively what has been happening.

3. The values of different measures are highly corre-


lated, but the values for specific countries may be deviant. It may not always be wise to assume that because country A has a higher crude birth rate than country B, the total fertility rate in country A is also higher than in country B. Further, even if the direction of the difference in two rates is the same with different measures, the amount of the difference between the fertility rates of two populations may be different, depending on which measure is used.

4. Finally, an important reason for having a variety of measures is that each measure answers a somewhat different question about the fertility level.

To cite an example of the last point using the rate we discussed most recently, we can interpret the total fertility rate in a way that is not possible with either the crude birth rate or the general fertility rate. Whereas the total fertility rate summarizes the fertility data for the same group of women as the general fertility rate, the TFR takes into account the distribution of births within the childbearing years and uses the same standard population in every calculation. It is this feature that allows us to interpret the TFR as the completed family size for a hypothetical cohort of women.

G R O S S A N D N E T Other measures give us yet additional information about the R E P R O D U C T I O N reproductive behavior of a population. One meaningful ques-R A T E S tion, for example, is whether a given set of fertility rates

implies that the population will grow, exactly replace itself, or decline. In a way, this is more a question about natural increase (or "reproduction") than about fertility itself. The gross and net reproduction rates are often used to provide partial answers to this type of question.

The gross reproduction rate (GRR) is a standardized rate similar to the total fertility rate except that it is the sum of age-specific rates that include only female live births in the numerators." The formula for the calculation is as follows:

4. No firm standard has been established for expressing the gross reproduciton rate per woman or per 1,000 women. In this Guide we express it per 1,000 women


'5 times the sum of five- year age-specific^ GRR = 5 F x 1,000

^fertility rates including only female births J

= 5X5^/0.000). x

Since the number of female live births by age of mother may not be known, the proportion of all births that are female is often used as a constant multiplier for the age-specific rates to obtain the data required for the gross reproduction rate. An example of the calculation of the gross reproduction rate for Costa Rica in 1984 using this method is given in Table 3.15.

Table 3.15. Calculation of the gross reproduction rate for Costa Rica: 1984

Women's ASFRs ASFRs x proportion of ages (per 1,000 women) births female (0.4916)

15-19 96.0 47.2 20-24 192.1 94.4 25-29 181.7 89.3 30-34 131.0 64.4 35-39 76.8 37.8 40-44 27.0 13.3 45-49 3.1 1.5

Sum 347.9

Sum x 5 - GRR (per 1,000 women) 1,739.5

Source: United Nations Demographic Yearbook 1990 (1992, table 11, p. 332). Note: As is often done, the few births to women less than 15 and more than 49 years old were attributed to women of ages 15-19 and 45-49, respectively.

Note that in the above formula, we multiply the sum of the ASFRs by 5 because we are dealing with five-year rates; each woman in the hypothetical cohort of ages 20-24 will experience 5 ^ 0 for five years. This amounts to the same thing as summing the single-year ASFRs, as we did when calculating the TFR.

Like the total fertility rate, the gross reproduction rate, when expressed per 1,000, can be interpreted as the number

to maintain consistency with the age-specific rates, the crude rate, and the general fertility rate. It is probably somewhat more common to express the rate per woman.


of daughters expected to be born alive to a hypothetical cohort of 1,000 women if no women died during the childbearing years and if the same schedule of age-specific rates applied throughout the childbearing years. The advantage of using only female births in the calculations is that the GRR then measures the extent to which a hypothetical cohort of women will replace itself, provided no women die in the childbearing years.

In 1985 the gross reproduction rate for the world's total population was estimated to be 1,680 per 1,000 women (Keyfitz and Flieger 1990, 65). Values were as high as 4,100 (Rwanda) and as low as 630 (Federal Republic of Germany). For the 1985-90 period the average GRR for the less developed regions was around 1,900, ranging between 1,121 and 3,343, while the average for the more developed regions was about 936 (see Table 3.16). Whereas 97 percent of the more developed regions had gross reproduction rates of 1,299 or less, 82 percent of the less developed regions had GRRs of 1,600 or more and 49 percent of them had rates of at least 2,700 (see Table 3.17).

Of course, the gross reproduction rate measures only fertility. It makes no allowance for the fact that some women may die during the childbearing years. For a more accurate measure of the replacement of women by their daughters in the hypothetical cohort, we must use the net reproduction rate.

The net reproduction rate (NRR) is a measure of the number of daughters who will be born to a hypothetical cohort of women, taking into account the mortality of the women from the time of their birth. Hence, the net reproduction rate estimates the average number of daughters who will replace a cohort of 1,000 female infants by the time the cohort has been subjected to the risk of mortality from ages 0 to 49 and the risks of live birth from ages 15 to 49. Like the TFR and the GRR, the NRR may be expressed per woman or per 1,000 women. We start with a hypothetical cohort of 1,000 girls just born. Only a certain proportion of these 1,000 girls


wil l live to reach the childbearing period. Within the childbearing period mortality will also take its toll, so that a given woman may bear daughters through age 20 or age 30 say, but not live to age 50. The net reproduction rate provides an estimate of replacement in the hypothetical cohort, given mortality levels taken from an appropriate life table.

We show how to calculate the net reproduction rate in Table 3.18. First we enter the age-specific fertility rates, in-

Table 3.16. Estimated crude birth rates and gross reproduction rates for world regions: 1985-90

Crude Gross birth reproduction

Region rate rate

All regions 27.0 1,671 Less developed regions 30.7 1,900 More developed regions 14.8 936 Africa 44.5 3,045

East Africa 48.1 3,343 Central Africa 46.6 3,182 North Africa 36.9 2,485 Southern Africa 33.7 2,193 West Africa 48.2 3,338

Latin America 27.9 1,657 Caribbean 25.2 1,442 Central America 31.8 1,910 South America 26.8 1,593

North America 15.9 921 Asia (excluding former USSR) 27.5 1,681

East Asia 20.5 1,121 Southeast Asia 29.9 1,818 South Asia 33.9 2,271 West Asia 35.2 2,456

Europe (excluding former USSR) 12.8 833 Eastern Europe 14.9 1,023 Northern Europe 13.6 897 Southern Europe 11.6 750 Western Europe 12.2 770

Former USSR 18.9 1,184 Oceania 19.5 1,228

Source: United Nations Statistical Office (1992, tables A-7, A-13, pp. 150-52, 174—76). Note: Source values for TFR have been multiplied by 0.4873, a rough average proportion of births that are female, calculated from United Nations Statistical Office, Demographic Yearbook 1990(1992, table 10).

Table 3.17. Distribution of countries by world region and level of gross reproduction rate: 1985-90

Asia Europe Less More (excluding (excluding

Level All developed developed North Latin former former Former of GRR regions regions regions Africa America America USSR) USSR) USSR Oceania

Total 131 99 32 43 2 22 34 26 3 1 Under 900 21 4 17 0 1 1 4 15 0 0 900-1,299 21 7 14 1 1 3 3 10 2 1 1,300-1,599 8 7 1 0 0 5 2 1 0 0 1,600-1,999 12 12 0 1 0 6 5 0 0 0 2,000-2,399 11 11 0 3 0 2 6 0 0 0 2,400-2,699 9 9 0 4 0 2 2 0 1 0 2,700-2,999 11 11 0 4 0 3 4 0 0 0 3,000-3,299 17 17 0 13 0 0 4 0 0 0 3,300-3,599 16 16 0 13 0 0 3 0 0 0 3,600+ 5 5 0 4 0 0 1 0 0 0

Source. United Nations Statistical Office (1992, table A-14.) Note: Countries with populations under 1 million are excluded. More developed regions, as defined by United Nations Statistical Office (1990), include North America, Japan, Europe, Australia, New Zealand, and the former USSR. Less developed regions include all regions of Africa, Latin America, and Asia except Japan and all regions of Oceania except Australia and New Zealand. Figures are'from estimates prepared by the United Nations Population Division. Source values for the TFR have been multiplied by 0.4873, a rough average proportion of births that are female, calculated from United Nations Demographic Yearbook 1990 (1992, table 10).


eluding only female live births (column 3). Next, we enter the values for the number of person-years lived in each age interval, using the ^Lx column from an abridged life table for females in the current period (column 4). Then we take the product of the 5 L x values and the age-specific rates (column 3 multiplied by column 4). The 5 L values already refer to a five-year period so that we do not need to multiply by 5 to derive the NRR as we do with the GRR. The NRR is simply the sum of the products of columns 3 and 4, or 1,165.92 per 1,000 in the present example.

Expressed in a formula, the calculation of the net reproduction rate is as follows: NRR = the sum of the multiplications of (a) each five-year

age-specific fertility rate including only female live births and (b) the number of person-years lived by

Table 3.18. Calculation of the gross and net reproduction rates and the length of a generation for the nonwhite population: United States, 1988

Person- Female years births per

Female lived in 1,000 women births age inter for 5-year

Midpoint per 1,000 val (per period of age women per female) - col. (4) Col. (5)

Ages interval year* - SLJ 100,000 x col. (3) x col. (2) (1) (2) (3) (4) (5) (6)

10-14 12.5 2.06 4.90814 10.11 126.375 15-19 17.5 45.76 4.89889 224.17 3,922.975 20-24 22.5 74.23 4.88317 362.48 8,155.800 25-29 27.5 59.73 4.85954 290.26 7,982.150 30-34 32.5 38.06 4.82719 183.72 5,970.900 35-39 37.5 16.34 4.78262 78.15 2,930.625 40-44 42.5 3.41 4.72086 16.10 684.250 45-49 47.5 0.20 4.63498 0.93 44.175

Sum na 239.79 na 1,165.92 29,817.250

Gross reproduction rate = sum of col. (3) x 5 - 1,198.95. Net reproduction rate = sum of col. (5) - 1,165.92. Length of a generation - 29, £ 117.250/1,165.92 = 25.574 years.

Source: United States, National Center for Health Statistics (1990, tables 1 -6, 1 -59, 6-4). na—not applicable. a. Calculated by multiplying the proportion female of births in each five-year age group by the age-specific fertility rate for that age group.


the average female in the stationary population for the age interval corresponding to the fertility rate

= X U ' ) | f L ^

S'-x

where NRR = net reproduction rate, £ means one should sum the products for every age group. X

5Fx

f = the age-specific fertility rates (per 1,000 women5) including only female live births in the numerator, and

( A Y the number of person- years lived per woman in the age interval x referring to the exact age] at the beginning of the age interval

Keyfitz and Flieger (1990, 65) estimate that the average NRR in 1985 was 1,430 for the world's total population, 900 for the more developed regions, and 1,590 for the less developed regions. They also cite rates in 1985 as high as 3,250 (Kenya) and as low as 620 (Federal Republic of Germany). It is difficult to interpret the precise meaning of these net rates unless we compare them with the gross reproduction rates. A country may have a low net reproduction rate because fertility rates are low, because mortality rates are high, or both. To take two examples from a single country, Japan during 1930-34 had a GRR of 2,320 and an NRR of 1,620 per 1,000 women. With these data we can infer that fertility was moderately high and mortality was high. By 1980, however, Japan had a gross reproduction rate of 840 and a net reproduction rate of 830. Both fertility and mortality rates were low. You may find it useful to interpret other values shown in Tables 3.19, 3.20, and 3.21 or to refer to a more complete listing such as that given in Keyfitz and Flieger (1990).

The net reproduction rate is a measure of how many daughters would replace 1,000 women if age-specific fertility and mortality rates remained constant indefinitely. Con-

5. Using fertility rates per 1,000 women is equivalent to multiplying their sum by 1,000, as we did above for the TFR and the GRR.

Table 3.19. Measures of reproduction: selected countries, recent years

Vital rates

Gross Net Mean Life ex Intrinsic ratesb Crude rates

reproduc reproduc age at pectancy Natural Country and date tion rate tion rate birth at birth" increase Births Deaths Births Deaths

Mauritius 1960 2,800 2,370 28.22 57.7 31.40 41.19 9.78 39.31 10.70 1970 1,820 1,650 28.71 63.4 27.06 9.45 17.62 25.97 7.78 1980 1,420 1,340 27.40 66.2 20.79 10.07 10.72 26.40 7.21

Chile 1960 2,300 1,870 29.30 56.7 21.80 33.53 11.73 33.79 12.12 1970 1,650 1,450 28.35 62.9 13.34 24.12 10.78 25.24 8.78 1980 1,210 1,150 27.08 69.7 5.30 16.75 11.45 21.13 6.67

Mexico 1960 3,140 2,540 29.06 57.2 33.10 44.11 11.01 46.14 11.32 1970 2,400 2,210 29.20 66.7 28.04 34.51 6.47 41.67 9.49 1980 2,290 2,140 28.95 68.8 27.12 33.12 5.99 34.85 6.24

Canada 1961 1,860 1,790 27.79 71.4 21.26 27.58 6.32 25.76 7.77 1970 1,150 1,110 27.14 72.8 3.93 15.28 11.35 17.47 7.32 1980 850 830 26.92 75.1 -6.78 9.47 16.25 15.42 7.13 1985 810 880 27.41 76.4 -8.18 8.74 16.92 14.82 7.15

United States 1960 1,780 1,710 26.37 70.2 20.75 27.31 6.56 23.65 9.39 1970 1,200 1,170 26.00 71.0 5.90 16.72 10.82 18.32 9.43 1980 890 870 25.97 73.9 -5.14 10.36 15.50 15.92 8.77 1985 900 880 26.31 74.9 -4.81 10.40 15.21 15.75 8.74

Taiwan 1960 2,810 2,540 29.33 64.2 32.31 38.93 6.72 39.62 6.96 1970 1,940 1,860 27.58 69.1 22.75 29.19 6.44 27.16 4.90 1980 1,220 1,190 26.17 72.1 6.56 17.12 10.56 23.35 4.76 1985 910 890 26.42 73.6 -4.23 11.07 15.30 17.92 4.80

Japan 1960 980 920 27.88 67.9 -2.89 12.71 15.60 17.24 7.48 1970 1,000 980 27.83 72.2 -0.89 12.89 13.78 18.53 6.83 1980 840 830 27.77 76.3 -6.65 9.52 16.17 13.51 6.19

Peninsular Malaysia 1970 2,380 2,170 29.43 65.5 26.97 34.11 7.14 32.49 6.99 1980 1,880 1,790 29.30 69.1 20.11 27.28 7.17 30.29 5.55 1985 1,890 1,820 27.77 70.4 20.48 27.12 6.64 31.34 5.27

Denmark 1960 1,230 1,190 26.95 72.4 6.54 17.22 10.68 16.48 9.40 1970 960 940 26.67 73.5 -2.51 11.88 14.39 14.37 9.79 1980 750 740 26.83 74.3 -11.14 7.96 19.11 11.18 10.92 1985 710 690 27.73 74.6 -13.05 7.23 20.20 10.51 11.42

France 1961 1,370 1,320 28.10 70.9 9.97 19.39 9.42 18.07 11.32 1970 1,210 1,180 27.12 72.4 5.97 16.50 10.53 16.75 10.63 1980 950 930 26.83 74.4 -2.66 11.40 14.07 14.85 10.15 1985 890 870 27.48 75.5 -4.90 10.20 15.09 13.93 10.01

Hungary 1960 970 910 25.78 68.1 -3.47 12.42 15.89 14.62 10.08 1970 950 910 25.47 69.3 ^.57 12.04 15.60 14.69 11.63 1980 940 910 24.63 69.2 ^.74 11.85 15.59 13.88 13.57 1985 890 870 25.04 69.2 -5.65 10.89 16.54 12.23 13.86

Netherlands 1960 1,530 1,490 29.75 73.4 13.54 21.53 7.99 21.07 7.64 1970 1,260 1,230 28.16 73.7 7.44 17.30 9.86 18.32 8.41 1980 780 770 27.73 76.0 -9.53 8.28 17.81 12.81 8.08 1985 740 730 28.42 76.5 -11.21 7.59 18.80 12.29 8.47

Romania 1960 1,120 1,010 26.43 65.2 0.38 15.12 14.75 18.93 9.23 1970 1,400 1,310 26.63 68.1 10.29 20.50 10.20 21.09 9.54 1980 1,190 1,140 25.27 69.3 5.20 16.86 11.67 17.97 10.44

Former USSR 1959 1,370 1,270 28.43 68.1 8.41 19.05 10.65 25.21 7.39 1970 1,180 1,120 27.46 67.7 4.10 16.27 12.17 17.48 8.74 1979 1,110 1,050 26.49 67.5 1.95 14.83 12.87 18.19 10.05 1987 1,230 1,180 26.42 69.5 6.28 17.13 10.86 19.92 9.85

Australia 1960 1,700 1,640 27.49 70.8 18.27 25.28 7.01 22.62 8.65 1971 1,400 1,360 26.92 71.5 11.54 20.30 8.76 21.15 8.47 1980 920 900 27.14 74.7 -3.76 10.92 14.68 15.35 7.40 1985 940 920 27.69 75.5 -2.97 11.22 14.19 15.67 7.53

Source: Keyfitz and Flieger (1990, 64—101). a. Simple average of male and female values. Intrinsic vital rates, much like the GRR and NRR, refer to what would happen if the ASFRs and ASDRs were to continue indefinitely into the future. b. Female population.

Table 3.20. Gross and net reproduction rates: Europe, Great Depression years, post World-War II, and recent past

Region and E a r l y 1 9 3 0 s

country GRR NRR

Northwestern and Western Europe Austria 890 740 Denmark 1,040 920 England and Wales 930 810 France 1,100 920 Germany' 800 720 Netherlands 1,310 1,190 Norway 1,040 960 Sweden 820 730

Southern and Eastern Europe Greece 1,870 1,250 Hungary 1,390 1,040 Italy 1,580 1,220 Poland 1,710 1,240 Portugal 1,870 1,290 Former Yugoslavia 2,200 1,390

Middle 1960s Middle 1980s

GRR NRR GRR NRR Date

1,300 1,240 714 702 1985 1,270 1,240 708 697 1985 1,330 1,290 869 856 1985 1,350 1,320 888 873 1985 1,220 1,170 625 604 1985 1,480 1,430 739 729 1985 1,410 1,370 814 801 1985 1,150 1,130 840 828 1985

1,090 1,000 806 784 1985 910 860 892 867 1985

1,300 1,220 777 765 1980-85b

1,220 1,150 1,132 1,100 1985 1,520 1,350 830 810 1985 1,280 1,150 1,000 959 1980-85b

Sources: Office of Population Research (1950, 172-78, 1968,249-54; United Nations Statistical Office, Demographic Yearbook 1986, (1988, table 22, pp. 548-72). a. Federal Republic of Germany after World War II. b. Estimate prepared by the UN Population Division.


Table 3.21. Gross and net reproduction rates, by color: United States, 1905-10 to 1988

GRR NRR

Non- Non-Year Total White white Total White white

1905-10 1,790 1,740 2,240 1,340 1,340 1,330 1930-35 1,110 1,080 1,340 980 970 1,070 1935-40 1,100 1,060 1,410 980 960 1,140 1946-49 1,510 1,480 1,780 1,420 1,400 1,540 1950-54 1,630 1,560 2,070 1,550 1,500 1,840 1955-59 1,800 1,730 2,330 1,730 1,670 2,110 1960-64 1,690 1,620 2,160 1,620 1,570 1,980 1965-69 1,280 1,220 1,700 1,240 1,190 1,570 1970-74 1,030 980 1,330 1,000 960 1,250 1975 864 819 1,120 841 800 1,076 1980 896 850 1,143 876 833 1,106 1985 898 853 1,112 881 838 1,082 1988 943 883 1,208 924 868 1,175

Sources: 1905-40: Office of Population Research (1950, 172); 1946-74: Office of Population Research [1979, 353); 1975-88: United, States, National Center for Health Statistics (1990, table 1-4, p. 5|. Note: Source data for 1905-10 through 1970-74 are multiplied by 1,000.

sequently, rates above 1,000 mean that eventually the population would increase and rates below 1,000 mean that eventually the population would decrease, provided that the age-specific rates remained the same and no migration occurred. Rates such as 3,127 imply an eventual speedy rate of natural increase if the age-specific fertility rates do not decline.

T H E M E A N L E N G T H Another measure of replacement that follows easily from the O F A G E N E R A T I O N calculations performed for the net reproduction rate is the

mean length of a generation. This measure answers the question, On the average, how many years after birth does a woman replace herself with female children? The mean length of a generation indicates the speed with which each woman replaces herself with potential new mothers.

The length of a generation is a weighted sum of the female births per 1,000 women for each five-year period, all divided by the net reproduction rate—that is, it is the average age of women at the birth of their children. (More pre-


cisely, it is the average age at which the 1,000 women in our hypothetical cohort give birth. The average age at birth for women in the actual population is influenced by the age structure of the population.) The weights used are the ages of the women. We illustrate the computation of this measure in Table 3.18 using data for the U.S. nonwhite population in 1988. To obtain the mean length of a generation, we multiply the midpoints of each age interval (column 2) by the number of female births per 1,000 women for the five-year period (column 5) and then divide the result by the net reproduction rate (the sum of column 5).

From the calculation procedure, you can see that the length of a generation is affected by two components: (1) the overall fertility and mortality levels and (2) the proportion of fertility that occurs in each age group. This is true because every age-specific female birth rate is affected by the overall fertility and mortality level (e.g., columns 3 and 4 would have lower entries in general if the overall level of fertility were lower and mortality were higher) and also because higher age-specific fertility rates at younger ages lead to a lower value for the mean length of the generation than do higher ASFRs at older ages.6

For the countries and years listed in Table 3.19, values of the mean length of a generation have ranged between 24.63 (Hungary, 1980) and 29.75 (Netherlands, 1960). This means that, in the absence of migration and of changes in age-specific fertility and mortality rates, the average woman will replace herself with daughters in no fewer than 24 and no more than 30 years.

6. Technically there is a slight difference between the length of a generation (T) and the mean age at childbirth (called the mean age of the net maternity function by Keyfitz and Flieger 1990]. The former is the average time between the births of members of one cohort and the births of their daughters, whereas the latter is "the average age |u) at which women bear their girl children. Generally, Tis slightly smaller than \i in increasing populations, while in decreasing populations . . . the opposite holds true" (Keyfitz and Flieger 1990, 30—31). The values we cite are actually p., not T.


The length of a generation is important because it affects the growth rate of a population independently of the number of children born as measured by the net reproduction rate. The net reproduction rate tells us how much a population is growing per generation. It does not tell us how long the generation is. The more rapidly a generation replaces itself, the more rapidly it will add new members to the population (at whatever rate per generation prevails).

An example should help you to understand how to use the net reproduction rate in combination with the length of generation. The United States has a shorter generation length (as shown by the mean age at childbirth) than Western Europe because the average age at marriage is earlier and childbearing occurs at younger ages in the United States (Table 3.19). Therefore, even if the NRRs were the same in the United States and Western Europe, the U.S. population growth rate would be higher because the cycle of reproduction is repeated more rapidly.

For these reasons the age pattern of fertility decline in countries with high fertility is important. If the net reproduction rate falls by 10 percent as a result of fertility declines among older women, it will have less effect than an equal decline among younger women. To illustrate, women in India have their children at early ages (as compared, for example, with Chinese women in Singapore or Malaysia or Taiwan). This means that the population growth rate for India is likely to be higher even if the total number of children born per woman is no greater there than in other populations where childbearing takes place at older ages.

Changing the age at which women bear children can, in itself, have an effect on the growth rate. Information on the length of generation should therefore be important to policymakers and family planning program administrators. They should know that, in the long run, preventing births among young women is more important for reducing population growth than preventing births among older women.


E X E R C I S E i Alter the fertility rates in column 3 of Table 3.18 in such a way as to retain the same sum (i.e., keep the same gross reproduction rate). Do this by increasing the rates for younger women and decreasing the rates for older women. What effect does this have on the net reproduction rate? What effect does this have on the mean length of a generation?

E X E R C I S E 2 How would you interpret the following information for various countries? Assume that there is no net migration affecting the age structure in any of these countries.

Gross Net Crude Crude reproduction reproduction birth death

Country rate rate rate rate

A 1,000 985 14 14 B 1,000 985 17 6 C 3,000 2,950 Not available Not available D 3,000 1,000 45 45 E 1,500 1,485 Not available Not available F 3,000 1,500 45 22

Determine whether each of the following statements is true or false.

1. The net reproduction rate can never be higher than the gross reproduction rate.

2. If the gross reproduction rate declines in any given year, it inevitably means that at least a minority of the women in the childbearing years will end up with fewer children than they would have had prior to the decline.

3. Regardless of which fertility measure we use, we will find that fertility is higher in most of the less developed areas of the world than in the more developed areas.

4. For all practical purposes, the gross reproduction rate is equal to the product of the total fertility rate times the proportion of live births that are female.

5. A gross reproduction rate of 1,500 is very high.

T H I R D S E T O F

T R U E / F A L S E

Q U E S T I O N S


6. A total fertility rate of 2,350 is very high. 7. For most of human history, it is likely that net repro

duction rates close to 1,000 were common.

T H I R D S E T O F

M U L T I P L E - C H O I C E 1. The net reproduction rate in the United States is now ap-Q U E S T I O N S proximately:

(a) 500-1,000 per 1,000 women. (b) 1,000-1,500 per 1,000 women. (c) 2,500-3,500 per 1,000 women. (d) 4,000-5,000 per 1,000 women. (e) none of the above. What is the net reproduction rate in your own country?

2. A net reproduction rate of more than 1,000 means that: (a) a population will certainly increase in the future. (b) a population will certainly decrease in the future. (c) a population will eventually increase if age-specific fer

tility and mortality rates remain fixed and there is no migration.

(d) a population will eventually decrease if age-specific fertility and mortality rates remain fixed and there is no migration.

(e) a population will remain at about the same size if age-specific fertility and mortality rates remain fixed and there is no migration.

3. The American Hutterites had a gross reproduction rate of 4,000 and a net reproduction rate of 3,660 during one period. This indicates that: (a) both mortality and fertility were very high. (b) both mortality and fertility were very low. (c) fertility was very high and mortality was moderately

low. (d) mortality was moderately high and fertility was very

low. (e) mortality was very low and fertility was only moder

ately high.


4. The net reproduction rate is a measure of the: (a) annual excess of births over deaths. (b) annual rate at which women are replacing themselves

on the basis of prevailing fertility and mortality, assuming no migration.

(c) decennial growth rate of the population. (d) per generation growth rate assuming current age-spe

cific fertility and mortality rates and no net migration. (e) none of the above.

5. In two countries, A and B, the age-specific fertility rates per 1,000 women are as follows:

Age group Country A Country B

10-19 25 25 20-29 100 100 30-39 50 50 40-49 25 25

In country A, 60 percent of the population is female whereas only 50 percent of the population is female in country B. In country A, 35 percent of the females are between the ages of 10 and 49, whereas 40 percent of the females are between the ages of 10 and 49 in country B. (a) Country A has a higher gross reproduction rate than

country B. (b) Country B has a higher gross reproduction rate than

country A. (c) Country A has the same gross reproduction rate as coun

try B. (d) Country A has the same net reproduction rate as coun

try B. (e) The crude birth rates are the same in both countries. (f) The general fertility rates are the same in both coun

tries. (g) Two of the above are correct. (h) Three of the above are correct.

6. Populations with net reproduction rates of 1,000 per 1,000 women: (a) invariably have low age-specific fertility rates.


(b) have low crude birth rates but may have high age-specific fertility rates.

(c) have declining age-specific fertility rates. (d) may have either high or low age-specific fertility rates. (e) invariably have low crude birth rates.

C E N S U S M E A S U R E S Thus far we have discussed an interrelated system of mea-O F F E R T I L I T Y sures that usually require census data for the denominators

and vital statistics data for the numerators.7 In many countries, vital registration systems either do not exist or are inaccurate: they underregister the number of vital events and often misclassify the characteristics (e.g., age, place of residence) of the persons who gave birth or died. In countries where this is true, other measures of fertility based on census information have been used as substitutes for the measures we have already discussed. Such census measures have an advantage over vital-statistics measures because censuses usually collect much more information than do birth certificates on many characteristics of individuals—such as income, education, rural-urban residence—which are important because of their effects on fertility. Therefore, census measures allow a much more thorough study of differential fertility.

Most nations have had at least one census in the last 10 years, and the data collected can be used to calculate various indirect measures of fertility. The most common indirect measures are the following:

1. the ratio of children 0-4 years old to women of ages 15-49 or 15-44 years

2. the ratio of children 5-9 years old to women of ages 15-49 or 15-44 years

3. the percentage of the total population 0-4 years old 4. the percentage of the total population 5-9 years old 5. the percentage of the total population 0-14 years old

7. Sample surveys are sometimes used to collect data for both the numerators and the denominators, and a census can collect the information for both numerators and denominators. It is difficult, however, to obtain accurate reporting on births in a census, given the levels of training and supervision normally employed.


6. the number of children ever born to women, by five-year age groups of the women

7. the number of women's own (as opposed to adopted) children under age 5, by five-year age groups of women

We discuss only the first and sixth measures here because the problems of interpretation and use are similar for the other measures.

The ratio of children 0-4 years old to women of ages 15-44 or 15-49 is often called the child-woman ratio (CWR). It can be expressed algebraically as follows:

CWR=*-^V or k-^y pf pf 30M5 35M5

where 5PQ = population 0 - 4 years old

,P/5 (or J5P/5) = number of women 15 - 44 (15 - 49) years old, and

it = 1,000.

The child-woman ratio is based not on births, but on the survivors of births occurring in the last five years. One drawback of this measure is that the deaths of children in those five years are not accounted for; and although the deaths of women in the childbearing years compensate partly for the deaths to children, the ratio understates fertility. Moreover, because the ratio deals with survivors instead of actual births, two populations may have the same fertilty rates but the child-woman ratios will not reflect this fact if one population has higher child mortality rates than the other. The population with the higher death rates will have a lower child-woman ratio.

Using the child-woman ratio poses several other problems. One is that, if the ratio has surviving children of ages 0-4 in the numerator, it measures past fertility—fertility 2.5 years, on the average, before the census date. Another is that young children are more likely than older people to be underenumerated in a census. For that reason demographers sometimes select children 5-9 years old for the numerator, but this aggravates the problem of measuring current fertil-


ity because the ratio then refers, on the average, to fertility 7.5 years before the census date.

Even with all these problems, any reasonable measure is better than none, and the child-woman ratio has often been used when vital registration data are lacking for a country or for subdivisions of a country. Palmore (1978), for example, summarized the child-woman ratios for most nations and territories, using data from the 1970 round of censuses. He found that the the child-woman ratio ranged from 313 (in Sweden) to 928 (in Western Samoa) per 1,000 women and that the ratio was well correlated with more direct measures of fertility in countries with reliable data. For 56 nations with reliable data the child-woman ratio had the following correlations with direct measures of fertility: .961 with the crude birth rate, .975 with the general fertility rate, and .970 with the total fertility rate. On the basis of this information, Palmore developed a series of equations for determining the level of direct measures of fertility using such census mea-

Table 3.22. Percentage ever married and number of children ever born for women of ages 40-44 and 30-34: United States, selected years 1940-1990

Children ever born

Percentage Per 1,000 Age of women Percentage childless among Per 1,000 ever-married in given year ever married those ever married women women

40-44 1990 92 11.3 2,045 2,167 1980 95 6.6 2,988 3,105 1970 95 8.6 2,952 3,096 1960 94 14.1 2,409 2,564 1950 92 20.0 2,170 2,364 1940 90 17.4 2,490 2,754

30-34 1990 82 16.8 1,589 1,788 1980 90 13.7 1,826 1,970 1970 93 8.3 2,640 2,804 1960 93 10.4 2,445 2,627 1950 91 17.3 1,871 2,059 1940 85 23.3 1,678 1,964

Sources: United States, Bureau of the Census (1966, table 1, pp. 11-12; 1979, tables 6, 7, pp. 32-34; 1982:, table 12, pp. 41-43|; Bachu (1991, table 1, p. 17|.


sures as the child-woman ratio and other selected facts about each population. This material is beyond the scope of the present Guide, but you may want to refer to those techniques.

Data on children ever born (CEB) are collected in fewer censuses than the data required for calculating the child-woman ratio. For measures of this type, the census must contain a question for each woman asking her how many live births she has ever had. This information can then be tabulated by the woman's age, yielding measures of the cumulative fertility of women up to specified points in their childbearing years. Like the child-woman ratio, statistics on children ever born measure past fertility and are subject to the additional problem that children who die young may not be remembered. Nevertheless, this type of data has been used widely. Tables 3.22 and 3.23 provide examples from the United States and Table 3.24 provides an example from Indo-

Table 3.23. Number of children ever born per 1,000 women and per 1,000 ever-married women, by age: United States, selected years, 1940-90

Ages 1940 1950 1960 1970 1980 1990

All women 15-44 1,238 1,395 1,746 1,918" 1,506' 1,248 15-19 68 105 127 206b 179b 101 20-24 522 738 1,032 736 554 574 25-29 1,132 1,436 2,006 1,790 1,177 1,089 30-34 1,678 1,871 2,445 2,640 1,826 1,589 35-39 2,145 2,061 2,523 3,015 2,457 1,909 40-44 2,490 2,170 2,409 2,952 2,988 2,045 45-49 2,740 2,292 2,245 2,707 3,091 u

ver-married women 15-44 1,904 1,859 2,314 2,357 1,965 1,757 15-19 572 604 792 633 628 718 20-24 987 1,082 1,441 1,064 930 993 25-29 1,463 1,654 2,241 1,978 1,397 1,329 30-34 1,964 2,059 2,627 2,804 1,970 1,788 35-39 2,414 2,247 2,686 3,167 2,572 2,048 40-44 2,754 2,364 2,564 3,096 3,105 2,167 45-49 2,998 2,492 2,402 2,840 3,185 u

Sources: United States, Bureau of the Census 11966, table 1, p. 12; 1979, table 6, pp. 32-33; 1982, table 12, pp. 41-44, Bachu (1991, table 1, p. 17| u—data unavailable. a. Numbers for ages 18-44. b. Numbers for ages 18 and 19.


Table 3.24. Average number of children ever born per woman among ever-married women of ages 45-49, by province and urban/rural residence: Indonesia, 1980 and 1990

Urban Rural Urban and Rural

Province 1980 1990 1980 1990 1980 1990

Aceh 5.66 5.32 5.21 5.17 5.24 5.19 North Sumatra 6.61 5.86 6.92 6.30 6.84 6.15 West Sumatra 6.91 5.62 6.38 5.86 6.43 5.82 Riau 6.72 5.35 6.48 5.55 6.53 5.49 Jambi 6.40 5.73 5.93 5.31 5.99 5.39 South Sumatra 6.71 5.85 6.34 5.77 6.43 5.79 Bengkulu 6.97 6.21 6.58 5.91 6.61 5.96 Lampung 6.47 5.73 6.09 5.77 6.14 5.77 Jakarta 5.46 4.78 6.29 u 5.50 4.78 West Java 6.20 5.48 5.89 5.41 5.95 5.43 Central Java 5.08 4.79 5.25 4.83 5.22 4.82 Yogyakarta 5.04 4.28 4.99 4.30 5.00 4.29 East Java 4.58 4.17 4.56 4.05 4.57 4.08 Bali 5.51 4.50 5.06 4.36 5.12 4.40 West Nusa Tenggara 6.19 4.94 6.76 6.40 6.68 6.32 East Nusa Tenggara 6.62 5.77 5.63 5.48 5.67 5.50 East Timor u 4.63 u 4.18 u 4.20 West Kalimantan 6.22 5.34 6.16 5.52 6.15 5.49 Central Kalimantan 6.02 5.35 5.94 5.34 5.95 5.34 South Kalimantan 6.19 5.27 5.27 5.00 5.44 5.07 East Kalimantan 5.87 5.24 5.32 4.98 5.54 5.09 North Sulawesi 6.08 4.36 6.87 5.26 6.74 5.06 Central Sulawesi 6.31 5.58 6.32 5.91 6.32 5.86 South Sulawesi 5.99 5.39 5.51 5.16 5.58 5.21 Sulawesi Tenggara 6.69 5.59 5.92 5.63 5.97 5.62 Maluku 6.58 5.24 6.38 5.56 6.40 5.50 Irian Jaya 5.80 5.51 4.09 4.44 4.33 4.65 Indonesia 5.55 5.00 5.46 5.03 5.48 5.02

Source: Indonesia, Central Bureau of Statistics (1993, table 8, p. 24). u—data unavailable.

nesia. One reason for the wide use of this measure is related to the notion of cohort fertility, a concept we discuss next.

When we discussed the life table in the mortality chapter, we pointed out that there were two types of life table, the period or cross-sectional life table and the generation or longitudinal life table. A similar distinction can be made among fertility measures. Thus far we have discussed mostly.what are known as period or calendar-year fertility rates. When

C O H O R T

F E R T I L I T Y

M E A S U R E S


we discussed the total fertility rate, however, we introduced the idea of a cohort, albeit a hypothetical or synthetic cohort. We can also construct fertility rates for real cohorts, and measures so constructed are called cohort fertility measures.

Two types of cohorts commonly used in fertility measurement are marriage cohorts and birth cohorts. If we discuss data for a birth cohort, we refer to the fertility rates for a group of women all born in the same year or group of years. For example, we might talk about the 1935-39 birth cohort of women. If discussing marriage cohorts, we refer to the fertility rates for a group of women all married in the same year or group of years (e.g., women of the 1940-44 marriage cohort). Usually, use of the word "cohort" by itself refers to a birth cohort, and we devote our discussion here to data for birth cohorts of women.

One rationale for using birth cohorts to measure fertility hinges on the fact that childbearing in a particular year is determined in part by how many children women have had in preceding years, and that number, in turn, is determined in part by their age. Another rationale for using birth-cohort measures is based on the argument that cultural ideas about family size may change over time. Furthermore, other changes in a society may occur that lead to new patterns of childbearing in successive generations. Examples of such changes are a war that disrupts family formation during one generation (e.g., World War II) and the development of new methods for controling fertility that were not previously available (e.g., the birth control pill, which was introduced into many societies in the 1960s).

The fertility of a population may be influenced by both cohort effects and period effects. For example, an economic depression may affect many cohorts simultaneously (although at different stages of their reproductive histories), causing a low level of period fertility during the depression. Once the depression has passed, period fertility may rise, and women who postponed having children during the depression may


make up for the postponement. Some cohorts, however, will have reached the end of their reproductive years by the time the depression has ended and will no longer be able to bear children. Such a depression-induced fall in fertility is an example of a period effect.

A cohort effect may also be the product of a depression. Children born during a depression may tend to be conservative about their own fertility, preferring the certainty of being able to provide for a few children to the risks of having many children when the economic weather might worsen once more. In spite of economic good times, they may still have small completed families. Another cohort living through the same economic good times may take advantage of the prosperous conditions to have larger families. The resulting overall period fertility could be high, low, or average, but it would be composed of cohorts having various patterns and levels of fertility.

There is no guarantee that measures of period fertility and measures of cohort fertility referring to the same time span will show the same trends. An interesting example was presented by Barclay, using data for Taiwan in the 1933-52 period (Barclay 1958, 184-88). During that period, the total fertility rate changed as follows:

Period Total fertility rate

1933-42 7,400 1938-47 6,850 1948-52 6,250

These period rates clearly show declining fertility. Cohort measures for the same time period, however, show a different pattern:

Birth cohort Average number of women of children ever born

1888-92 6.90 1893-97 6.90 1898-1902 7.25 1903-07 7.35

The cohort measures clearly indicate rising fertility! How is this possible? The answer lies in recognizing that


both sets of measures are correct but refer to different groups of women.

How discrepancies like the one for Taiwan occur is clarified by a simple artificial example. Suppose we have the following age-specific rates:

Age-specific fertility rates at ages:

Birth cohort 15-24 25-34 35-44

1891-1900 50 300 200 1901-10 70 300 180 1911-20 110 300 180 1921-30 90 300 220

If we assume that all the births occurred to women 15-44 years old and that there was no mortality, we can make the following statements:

1. In 1935 the women born in 1911-20were 15-24 years of age; the women of the 1901-10 cohort were 25-34; and the women of 1891-1900 cohort were 35-44. Hence, the three figures on the left-to-right upward diagonal (110, 300, 200) represent the period fertility for the year 1935. The total fertility rate for that year was 6,100.8 Similarly, the total fertility rate for 1945 was 5,700. These two rates indicate a decrease in fertility.

2. Although the period total fertility rates declined between 1935 and 1945, the cohort rates were successively higher:

1891-1900 5,500 1901-10 5,500 1911-20 5,900 1921-30 6,100

This example demonstrates that it is possible to have period rates that change in one direction and, at the same time, cohort cumulative rates that change in the opposite direction. Such a paradox results from differences in the timing of births for the separate cohorts, which can produce unusually low or high points while the basic cohort trend is in a direction not indicated by the period rates.

8. Since we are using 10-year age groups, the total fertility rate is the sum of the age-specific rates multiplied by 10.


In the example just given, the age pattern of childbearing changed during successive birth cohorts, and the change produced a discrepancy between cohort rates and period rates. Such discrepancies are quite possible if there have been shifts in the ages at which women marry, or if women plan and control their fertility and there have been outside causes (the economic depression mentioned earlier, for example) that lead to either a postponment or an acceleration of childbearing. That is, women born between 1891 and 1920 had an increasing proportion of their children in the earlier childbearing years. Women born after 1920 began having more children in their later childbearing years.

1. Cohort fertility analyses: (a) have essentially the same use as the net reproduction

rate. (b) have essentially the same use as the gross reproduction

rate. (c) have the advantage of linking current and future fertil

ity rates to past fertility histories of each cohort. (d) are useful only for populations in which contraception

is not widely used. (e) refer to the experience of Roman military cohorts.

2. The chief difficulty with the net reproduction rate as a predictive device for population growth is that it: (a) excludes the influences of fertility. (b) makes inadequate allowance for mortality. (c) is based on the rates of a single year. (d) overlooks the type of culture possessed by the popula

tion. (e) includes only survivors of births in some past period.

3. Period birth rates and cohort birth rates may exhibit large differences under which of the following conditions? (a) When most couples plan their fertility. (b) When the mean age at marriage is increasing. (c) When the mean age at marriage is decreasing.

F O U R T H S E T O F

M U L T I P L E - C H O I C E

Q U E S T I O N S


(d) Two of the above. (e) Includes (a), (b), and (c) above.

4. Assuming there were no deaths to children or women in the past five years, one-fifth of the child-woman ratio should be approximately equal to: (a) the average general fertility rate for the past five years. (b) the total fertility rate for 7.5 years ago. (c) the crude birth rate for some indeterminate past pe

riod. (d) the average gross reproduction rate for the past five

years. (e) the average net reproduction rate for the past five years.

A N A L Y S I S O F A final type of fertility analysis that is now prominent in the B I R T H I N T E R V A L S demographic literature is measuring the length of time be

tween each birth and the next (a birth interval). As stated by Rindfuss, Palmore, and Bumpass (1982, 5), these measures are important because

the fertility process is itself a sequential and time-dependent process. Birth interval analysis allows more precision in investigating many fundamental questions; it allows the assessment of the effects of intermediate variables, like contraceptive use or lactation, and the explication of the effects of various socioeconomic variables in terms of intermediate variables.

Methods for properly analyzing birth intervals are still under development because three complex methodological problems are associated with birth interval analysis.

The first problem has to do with the quality of the data available for studying birth intervals. Misdating of births or failure to remember their occurrence can reduce the data quality.

The second problem has to do with what has become known as censoring. Censoring occurs when birth intervals are incomplete, or "open." When a sample survey is conducted, for example, many respondents have not yet completed all of their birth intervals. Some of the open intervals


will eventually be closed by a subsequent birth, but when that will happen is unknown. If you analyze only the open intervals or only the closed intervals, an analytic bias results because open intervals tend to be longer than closed intervals. The open intervals tend to be longer partly because some of the open intervals will never be closed by another birth. The usual solution to this problem has been to use life table techniques, with the next birth treated as a "death" and the initial cohort consisting of women who have had the immediately preceding birth. That is, women of parity n who have not yet given birth to a child of order n + 1 are like persons of age x who have not yet died.

We provide an illustration of this life-table approach in Table 3.25, which is adapted from an article by Bumpass et al. (1982) that analyzes data from the 1973 National Demographic Survey of the Philippines and the 1974 Korean World Fertility Survey. In the top panel of the table the proportion of women who gave birth is tabulated by whether or not contraception had been used during the interval and by the duration of the interval. These figures are the equivalent of qx

values. For example, the figure. 14 in the first cell of the table is the probability that a Korean woman who used contraception would have a second birth within the first 20 months after having her first birth.

Comparing women who used contraception with those who did not (i.e., by looking at the proportionate reduction in birth probabilities in the second panel of the table), we can make several observations:

1. In the second interval for Koreans and in all intervals shown for Filipinas, the effect of having used contraception is usually distinctly lower in the first duration segment (0-20 months) than in the second duration segment. Proximate determinants other than contraceptive use were probably the dominant influences in this duration segment. Lactation (and its effect on fecundability) and lower coital frequency immediately after the birth of a child are likely candidates.

2. After the first duration segment, the proportionate

Table 3.25. Birth probabilities within successive birth intervals 2, 3, and 4-8, by duration of interval and contraceptive use status: Philippines and Republic of Korea, 1973-74

Interval 2 Interval 3 Intervals 4-8 Duration of interval and whether — contraception used or not Korea Philippines Korea Philippines Korea Philippines

Proportion giving birth during interval segment <21 months

Yes .14 .28 .03 .23 .02 .19 No .23 .43 .13 .32 .09 .26

21-26 months Yes .21 .20 .08 .14 .04 .11 No .41 .40 .31 .36 .19 .29

27^2 months Yes .28 .21 .15 .17 .07 .11 No .42 .35 .44 .35 .31 .28

33-44 months Yes .45 .48 .36 .36 .12 .20 No .59 .50 .67 .48 .46 .42

Proportionate reduction in birth probability due to contraceptive use" ( |P n o -P y J/PJ

<21 months .39 .35 .74 .29 .84 .29 21-26 months .48 .50 . 73 .60 . 79 .62 27-32 months .34 .38 .66 .53 .77 .61 33-44 months .22 .03 .47 .26 .75 .52

Number of cases < 21 months

Yes 348 141 577 216 1,928 740 No 1,724 1,779 1,308 1,608 2,917 5,491

21-26 months Yes 288 96 526 156 1,862 569 No 1,304 1,005 1,114 1,081 2,622 4,001

27-32 months Yes 202 62 439 118 1,685 441 No 743 570 751 677 2,073 2,716

33-44 months Yes 132 40 312 77 1,395 325 No 404 351 399 411 1,361 1,840

Source: Bumpass et al. (1982, 248]. Note: The first interval is between marriage and the first birth, the second interval is between the first and second births, etc. a. Proportions in the second panel calculated from unrounded figures.


reduction in birth probabilities due to contraceptive use declines after the second duration segment (21-26 months). Contraceptive use for spacing births is probably the explanation for this effect.

3. Among Korean women at higher parities, there is an impressive reduction in fertility with contraceptive use.

The third problem in analyzing birth intervals has to do with the selectivity of the birth intervals available for analysis. Selectivity is particularly evident with data from sample surveys because such surveys typically restrict the respondents to specific age ranges and restrict them by marital status or other criteria.

A hypothetical survey conducted in 1990 illustrates the problem. Table 3.26 shows the birth intervals that would be available for analysis in a survey of all ever-married women between ages 15 and 49. The horizontal dimension of the table indicates the ages of the women at the time of the interview; the vertical dimension represents the ages of the women at the start of a birth interval. Each cell of the table represents the year in which a birth interval began. (The years are shown with the leading 19s omitted; for example, "46" is 1946.} Diagonals from the top left to the bottom right of the table represent the birth intervals begun in a given year.

The solid triangle encloses the intervals actually available for analysis. Notice, first, that intervals begun at age 15 could have been initiated in any year between 1956 and 1990, whereas intervals begun at age 49 could start only in 1990. The time periods represented are different for various ages at initiation. Second, by comparing birth cohorts (each column in the table) you will notice that the cohorts vary considerably in the possible ages at the beginning of the interval. At the extremes, women of age 49 could have initiated an interval at any age from 15 to 49. Women of age 15 could have initiated an interval only at 15. Third, look at the time periods during which the birth intervals were begun. Intervals begun before 1966, for example, had to be initiated at age 25 or younger. Intervals begun before 1959 had to be initiated at age 18 or younger.


The three points made above clearly illustrate biases introduced by selectivity. If a woman's age at the beginning of an interval, her birth cohort, and the time period were all unrelated to her fertility, these biases could be ignored. Unfortunately, all three of these variables are known to be highly related to fertility.

The principal question introduced by selectivity biases is, Of those birth intervals available for analysis, which should be analyzed? There is no single solution, and a full treatment of the selectivity issue is beyond the scope of this Guide. If you are interested in pursuing the matter further, you should read Rodriguez and Hobcraft (1980) and Rindfuss, Palmore, and Bumpass (1982).

F I F T H S E T O F 1. Censoring refers to the fact that: M U L T I P L E - C H O I C E (a) women often forget the exact dates of birth of their chilQ U E S T I O N S dren.

(b) at the time of data collection, some women have not completed childbearing.

(c) not all women in the population are interviewed. (d) data for some countries are suppressed by the govern

ment. (e) Both (b) and (c) above are correct.

2. Selectivity biases in survey data: (a) are important only in the analysis of birth intervals. (b) may affect studies of the intervals between marriages,

geographic movements, or job changes. (c) arise because women often forget the exact ages of birth

of their children. (d) arise because the survey is. cross-sectional rather than

a complete longitudinal study of each birth cohort of women.

(e) Both (a) and (c) above are correct. (f) Both (b) and (d) above are correct.

3. Open birth intervals are likely: (a) to be longer than closed birth intervals.

Table 3.26. Year in which any birth interval had to begin, given a woman's age at the place in 1990: all intervals

Age at begin-n j n g Qf Age at time of interview interval 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

15 \ 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 16 \ 9 0 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 17 \ 9 0 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 18 \ 9 0 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 19 \90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 20 \ 9 0 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 21 \90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 22 \90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 23 \90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 24 \S0 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 25 \50 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 26 \ 9 0 89 88 87 86 85 84 83 82 81 80 79 78 77 76 27 \90 89 88 87 86 85 84 83 82 81 80 79 78 77 28 \ 90 89 88 87 86 85 84 83 82 81 80 79 78 29 \ 9 0 89 88 87 86 85 84 83 82 81 80 79 30 \ 9 0 89 88 87 86 85 84 83 82 81 80 31 \ 9 0 89 88 87 86 85 84 83 82 81 32 \ 9 0 89 88 87 86 85 84 83 82 33 \ 9 0 89 88 87 86 85 84 83 32 \ 9 0 89 88 87 86 85 84 34 \ 9 0 89 88 87 86 85 36 \ 90 89 88 87 86 37 \ 90 89 88 87 38 N. 90 89 88 39 \ 90 89 40 \ 9 0 41 \ ^ 42 43 44 45 46 47 48 49

beginning of the interval and her age at the time of being interviewed, for a survey taking

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69

64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 76 75 74 73 72 71 70 69 68 67 66 65 64 62 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62

90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64

90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66

90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68

89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70


(b) to be shorter than closed birth intervals. (c) to be about the same length as closed birth intervals.

4. Selectivity biases refer to biases introduced by selectivity on: (a) age at the initiation of a birth interval. (b) time period. (c) birth cohort. (d) all of the above. (e) none of the above.

A D D I T I O N A L The literature on fertility analysis is growing rapidly. Conse-R E A D I N G quently, the works mentioned here are necessarily selective

and do not adequately reflect the diversity of the literature. Further discussion of the methods described in this chapter can be found in several of the sources listed in Chapter 2. In particular, you may want to consult Bogue et al. (1993).

Most books of readings on population have one or more chapters on fertility, as do most demography textbooks. New methods for constructing fertility rates from deficient data are being developed regularly. Bogue and Palmore (1964) and Palmore (1978), mentioned earlier in this chapter, are among the early illustrations. Prominent methods include the following: the "own-children method" developed principally by Lee-Jay Cho and colleagues (see Cho, Retherford, and Choe 1986 for a thorough description of the method and Rao et al. 1993 for one example of the method's use); the "Brass methods" developed by William Brass and others (Natarajan and Singh 1988 discuss the methods and use them to estimate district-level fertility for India in 1980); and the regression techniques developed by James Palmore (1978; for an illustration of which, see Swamy et al. 1993), and by J. R. Rele (1967,1987), Subbiah Gunasekaran and James Palmore (1984), and others (see the Bogue et al. compendium mentioned above for many of the others).

Additional materials on the use of both fertility and mortality measures can be found through judicious sampling


of appropriate journals, some of which are listed below. The list is far from exhaustive. In particular, we include only a small, illustrative sample of the journals published outside of the United States and Europe.

Asian and Pacific Population Forum, published from 1974 through 1993 in Honolulu by the East-West Center under three titles, the other two being the Asian and Pacific Census Newsletter and the Asian and Pacific Census Forum; in English.

Majalah Demografi Indonesia, published in Jakarta by Universitas Indonesia, Lembaga Demografi; most articles in Indonesian, but some in English with Indonesian abstracts.

Demography, a journal of the Population Association of America, published in Providence, Rhode Island, by the Department of Sociology, Brown University; in English.

Family Planning Perspectives, published in New York by The Alan Guttmacher Institute; in English.

Genus, published in Rome by Comitato italiano per lo studio dei problemi della popolazione; in Italian.

International Family Planning Perspectives, published in New York by The Alan Guttmacher Institute; in English.

Journal of BioSocial Science, published in Oxford, England, by Blackwell Scientific Publications for the Galton Foundation; in English.

Philippine Population Journal, published in Manila by the Commission on Population and the University of the Philippines Demographic Research and Development Foundation, Inc.; in English.

Pogon sahoe nonjip (Journal of population, health and social welfare), published in Seoul by Hanguk Ingu Pogon Yonguwon,- in Korean with English abstracts.

Population, published in Paris by the Institut National d'Etudes Demographiques; in French.

Population and Development Review, published in New York by The Population Council; in English.

Population and Environment, published in New York by Human Sciences Press; in English.


Population Index, published by the Population Association of America in cooperation with the Office of Population Research, Princeton University in Princeton, New Jersey; in English. Although mainly an index, this publication also contains one or more articles in each issue.

Population Studies, the journal of the International Union for the Scientific Study of Population, published in London by London University Press; in English.

Social Biology, published in Madison, Wisconsin, by the Society for the Study of Social Biology; in English.

Studies in Family Planning, published in New York by The Population Council; in English.

Theoretical Population Biology, published in New York by Academic Press; in English.

Many journals that do not focus specifically on population studies also carry articles of interest.

Notations and Formulas

Many notation systems are found in the demographic literature. Although the meaning of the symbols used is normally clear from the context, variability in notation can result in confusion. In this Guide we have endeavored to use a consistent system of notation based upon that of the life table. You should be aware that in some cases our system is different from that found elsewhere in the literature. We believe that the consistency we have introduced makes up for this difference.

In demography, we use letters to stand for a number of events or persons. Thus, the capital letter B is used to represent the number of births, D to represent the number of deaths, and P to represent the number of people in the population. (The letter often, but not always, represents the first letter of the word for the concept we are symbolizing.) Lowercase letters are also used; for example, d stands for the number of deaths in a life-table population.

Subscripts and superscripts are common in demographic notation. Perhaps the most common is the subscript x, which usually follows the letter it modifies and stands for exact age at the beginning of an age interval. Dx refers to all deaths to persons x to x + 1 years of age, whereas D 1 8 refers to deaths to all persons who became 18 on their last birthday (which is the same as saying all persons of exact age 18 to exact age 19). Another common subscript is n, which often precedes the letter it modifies and refers to the size of an age interval: „Pf refers to all females of ages x to x + n. Thus 5P£ refers to all females of ages 20 through 24—that is, exact age 20 to exact age 25, an age interval of five years. If n = 1, the n is often not written: Dn = ,D 1 8 .

132 Notations and Formulas

In this volume we use superscripts mainly to designate gender PF refers to the female population, PM to the male population. Superscripts (or subscripts) may also be used to refer to a date or period of time: P 1 9 7 5 would be the 1975 population count or estimate.

Two other symbols need to be mentioned here. One is k, which refers to a constant by which many demographic measures are multiplied to make them easier to understand. For example, the crude birth rate in a country may be 0.012 per year. Although this is an accurate demographic description, it is harder for many people to understand than the same rate multiplied by a constant, k = 1,000. The crude birth rate is then 12, or 12 per 1,000, and that is how it is usually expressed. Similarly, the population growth rate is often expressed as a percentage; in this case, k = 100. The other symbol is the Greek capital letter sigma, S, which is a summation sign. It is used in demography to indicate that the expression following it is to be summed. For example, the notation 1-49

means: Take the sum of the population at each age from 0 through 49—that is, P0 + P, + P2 + . . . + P48 + PA9.

Bearing these conventions and rules in mind, we present a list of concepts defined in this volume and the formulas used to describe them algebraically.

In

Concept Formula

MORTALITY

Crude death rate (CDR) M

Age-specific death rate (ASDR) (for exact ages x to x + n)

Age-standardized death rate (for population B with A as standard) y PA

k

Notations and Formulas 133

Concept Formula

Infant mortality rate (IMR) IMR = —k

B

LIFE TABLE

Probability of dying between exact ages x and x + n nqx

Number of deaths between exact ages x and x + n ndx

Survivors to exact age x ix

Number of years lived between exact ages x and x + n nLx

Total years lived after exact age x Tx

Expectation of life after exact age x ex

RELATIONSHIPS AMONG COLUMNS OF THE LIFE TABLE

d Probability of dying between exact ages x and x + n „qx = M-J-

X

Number of deaths between exact ages x and x+n ndx = lx - £xtn

Number of years lived between n , . L.=—{t )

exact ages x and x + n 2 ' (except at youngest and oldest ages)

x=i

where °° refers to the last (open) age interval

Total number of years lived after exact age x

Expectation of life, or average number of years j lived after exact age x e '

Life-table birth rate (b) = life-table death rate (d) = S-

134 Notations and Formulas

Concept Formula

FERTILITY, N A T U R A L INCREASE, A N D REPRODUCTION RATES

Crude birth rate (CBR)

Crude rate of natural increase (CRNI)

General fertility rate (GFR)

CBR = —k P

GFR = - ^ - r * or -^k pf pf

Age-specific fertility rate (ASFR) (for exact ages xtox + n)

Birth-order-specific fertility rate

Total fertility rate (TFR)

D

F = " ' k

pi F1 = " ' k "r* p f K

TFR = « ^ „ F I

Gross reproduction rate (GRR)

Net reproduction rate (NRR)

GRR = nX„F/ X

NRR = X („F/) |

Child-woman ratio (CWR)

where: B = births D = deaths P = population P1 = female population x = exact age n = size of age interval i = order of birth, and k = a constant

CWR = - ^ V * or pf pf

30* 15 3 5 ' 1 5

APPENDIX 2

Relationship between q and Mx Values

T H E G E N E R A L C A S E Constructing a life table for a real population requires determining values of the qx function from observed values of age-specific death rates, which are symbolized by Mx in our notation. The qx values differ from the age-specific death rates (MJ that we have discussed earlier in the following ways:

1. In the qx values the denominator includes members of only one (hypothetical! birth cohort, whereas in the age-specific death rate (MJ the denominator includes members of more than one (real) birth cohort. For example, the persons in the real population age group of exact age 4 to exact age 5 in midyear 1989 would include some persons born in 1984 and some born in 1985. Hence, the denominator for the age-specific death rate includes parts of both the 1984 and 1985 birth cohorts.

2. Similarly, in the qx values the numerator includes members of only one birth cohort, whereas in the age-specific death rate (MJ the numerator includes members of more than one birth cohort. For example, the persons who died at age 4 in 1989 would include some persons born in 1984 and some persons born in 1985.

3. For the denominator of the age^specific death rate (MJ we use the midyear population as an estimate of the number of person-years lived. The midyear population is a biased estimate of the number of persons exposed to the risk of dying at the beginning of the year (as opposed to exposure, which, as we have seen, the midyear population does estimate satisfactorily in the absence of unusual conditions) to the extent that it excludes persons who died during the first half of the year. Further, for life table purposes it is biased because it includes persons who migrated into the popula-

136 Relationship between qx and M x Values

tion during the year—and migration is expressly omitted in the life table calculations.1

Age-specific death rates [MJ usually overestimate the probabilities of dying during a given exact age interval (<jj because they exclude persons dying in the first half of the year from the denominator and because they refer on the average to persons x + Vi years old instead of x years old.

Rather complex methods have been developed for calculating qx values from Mx values, most of which are well beyond the scope of this Guide. An approximate value for qx

can be found for ages over 4, however, on the assumption that:

Mx

This is the most common formula for calculating values for qx (except for the first few years of life), although various methods of adjusting the values of Mx are often used before the basic formula is applied.2

I N F A N T For the younger ages, particularly infancy (exact age 0 to ex-M O R T A L I T Y act_ age 1), the determination of qx is especially problematic.

Often, the infant mortality rate (IMR) is used directly as the value of q0 in a life table. The defect in the infant mortality rate is the same in principle as that for death rates at other ages—that is, more than one cohort is involved in the numerator—but it is more serious because births (the denominator of the IMR) may fluctuate rather dramatically from year to year. It is sometimes possible to obtain a satisfactory degree of precision by averaging over several years. For example, we might calculate the infant mortality rate for 1985-87 by dividing the number of deaths to infants in years 1985, 1986,

1. Of course, deaths of persons who migrate into the population during the year are also included in the numerator of death rates. If we assume that migration is smoothly distributed over the year and that migrants experience the same ASDRs as others, then including migrants in n M does not pose a problem. 2. Forage groups larger than one year the corresponding formula isnqz - \2n(nMJ]/ \2 + n(nMJ\.

Relationship between qx and Mx Values 137

and 1987 by the number of births occurring in those same years: This provides more accuracy than the usual method of calculating the infant mortality rate because the deaths in the numerator are better matched with the proper set of births in the denominator.

Another method of calculating qQ involves either classifying infant deaths by year of birth or determining the number of deaths of children by age (in months) at death. With such data it is possible to construct a rate so that the numerator and denominator both refer to the same cohort: infant deaths in year z to infants born in year z are divided by the number of births in year z ; infant deaths in year z to infants born in year z - 1 are divided by the number of births in year z - 1; the two are then added together. If the requisite data are not available, there are also methods for estimating this type of rate, but we do not discuss them here. If you are interested in them, you can refer to the more advanced sources cited in the concluding section of Chapter 2. The more precise techniques for estimating infant mortality rates discussed in those sources are similar to those used in the life table, and consequently no further adjustment is required; that is, the adjusted IMR simulates a cohort approach and can be used directly as qQ. For ages 1 to 4, similar techniques are often employed, although the use of the simple formula given above for ages over 4 is often used to convert values of M, through M 4 into the appropriate qx values.

APPENDIX 3

Answers to Selected Exercises

C H A P T E R 2 E X E R C I S E i (page 10)

1. Where did the deaths occur? How many people were exposed to the risk of dying?

2. The population size is given only for the end of the year. This is usually an inadequate measure of the number of persons exposed to the risk of dying during the year.

3. When did they die?

F I R S T S E T O F M U L T I P L E - C H O I C E Q U E S T I O N S (page 18)

1. (d). Cannot tell without knowledge of the age structure.

2. (c). Cannot tell for certain without knowledge of the age structure.

S E C O N D S E T O F M U L T I P L E - C H O I C E Q U E S T I O N S (page 28)

1. (a). The age-specific rates for country A are higher in every age group.

2. (b). See Table 2.1. 3. (a).

4. (a). See Table 1.1.

E X E R C I S E 2 (page 29)

Case 1: The differentials in the crude death rates are not a result of differences in age distributions. City B had about the same average mortality levels as the United States, but

140 Answers to Selected Exercises

City A had substantially higher mortality levels than either

City B or the United States.

Case 2: City A and City B both had substantially higher mortality levels than the United States when age differentials are taken into account, and the two cities were very similar in their mortality levels. City B probably had a younger age distribution than either the United States or City A, and this accounts for its lower crude death rate.

Case 3: The mortality rates of the two cities were on the average closely similar to that of the United States. Since City B apparently had an age structure similar to that of the United States, this mortality similarity to the United States is reflected in either the crude or the standardized rate comparisons. City A must have had an older population than either City B or the United States, however, because its higher mortality, as reflected in comparisons of the crude death rate, disappears in the age-standardized comparisons.

Case 4: When age is taken into account and standardization is used, it appears that City A had mortality levels like those of the United States but City B had substantially higher mortality rates than either City A or the United States. This reverses the comparative mortality levels as measured by the crude death rates. Therefore, it is likely that City A had an old population, which gave it a high crude death rate despite low age-specific mortality rates. By the same logic, City B must have had a very young population, which gave it a relatively low crude death rate even though on the average its age-specific death rates were high.

Case 5: Once age is controlled by standardization, it appears that City B had slightly higher mortality levels and City A even higher mortality levels as compared with the United States. City B must have had a somewhat younger population than the United States, or at least one that had somewhat less concentration in higher mortality age groups, be-

Answers to Selected Exercises 141

cause initially the crude death rate was equal to that of the United States, and standardization makes it a little higher. City A must have had a significantly older population than either City B or the United States because the overall mortality differential as compared with the United States or City B is reduced (but not eliminated) when an age adjustment is made.

Case 6: Age differentials obscure the probable mortality differentials among the three populations. The fact that City A had lower mortality levels than either City B or the United States (in age-standardized comparisons) must be obscured in the crude rate comparisons by the fact that City A's age structure must have been very different from that of the other populations. Presumably it has an old age structure because standardization reduces its rate by more than 50 percent, whereas it only slightly increases the rate for City B.



Crude death rate 38.25 32.25 Death rate standardized on distribution for

Country A 38.25 43.25 Country B 27.25 32.25

Country B's lower crude death rate results from the fact that a large part of its population lives in the metropolitan areas, where death rates are relatively low. Country A initially has a high crude death rate, despite its low mortality within each type of region, because its population is concentrated in the high-mortality rural areas.

F I R S T S E T O F T R U E / F A L S E Q U E S T I O N S (page 33)

1. True. See Table 2.8. 2. False, because of the younger age structure in the de

veloping countries. 3. False.


4. False. It is not a good estimate if events have occurred unevenly throughout the year.

5. False. Death rates are highest at the extreme ages— that is, among both the very young and the old.

T H I R D S E T O F M U L T I P L E - C H O I C E Q U E S T I O N S (page 47)

1. (c). 2. (b). 3. (d). The reason is that a probability has all persons at

the start of a period in the denominator, whereas a death rate has the total number of person-years lived—which, in the absence of migration, must be less than those at the start unless there is no mortality at all.

4. (b).

S E C O N D S E T O F T R U E / F A L S E Q U E S T I O N S (page 49)

1. False. Death rates at these ages are relatively high, and therefore survival ratios are low.

2. False. 3. True. 4. True.

F O U R T H S E T O F M U L T I P L E - C H O I C E Q U E S T I O N S (page 58)

1. (f). 2. (e). The reason is that the life-table death rate = tjT0

and e0 = TjtQ = 1 (life-table death rate). 3. (d). 4. (e), not (d), because (c) is true for the reason that qx de

termines L . x

T H I R D S E T O F T R U E / F A L S E Q U E S T I O N S (page 59)

1. True. See Table 2.13. 2. False. They are equal. 3. True.


4. False. Life tables refer to groups, not individuals, and they refer to real groups (not hypothetical groups) only if mortality rates are not changing or the life table in question is a generational life table.

5. Arguable. Crude death rates do measure the actual rate of mortality of the population as it is at a particular time. They do not measure mortality independently of the effect of the age distribution. Therefore, the standardized rates are better for comparing the underlying population trends. The crude rates are better for measuring the rate at which the population is dying without reference to whether the age distribution has affected it.

6. True.

C H A P T E R 3 F I R S T S E T O F T R U E / F A L S E Q U E S T I O N S (page 78)

1. False. The division is closer to half-and-half. 2. True. At least for the countries included in Table 3.2. 3. True. 4. False. This would result in a crude rate of natural in

crease of 25 per 1,000, and numerous countries have rates that high or higher. The continent of Africa as a whole was recently estimated to have a CBR of 43 and a CDR of 13, giving it a CRNI of 30 (Population Reference Bureau, 1992).

5. True.

F I R S T SET O F M U L T I P L E - C H O I C E Q U E S T I O N S (page 78)

1. (e). 2. (b). 3. (b). 4. (d). 5. (b). 6. (b).


S E C O N D S E T O F M U L T I P L E - C H O I C E Q U E S T I O N S (page 90)

1. (d). The two countries have almost identical percentages of childbearing-age women in the population and identical ASFRs. We need to know more about the age distribution of the women within the age group 15-44.

2. (d). We need to know more about the age distribution of the women within the age group 15-44.

3. (c). 4. (b).

S E C O N D S E T O F T R U E - F A L S E Q U E S T I O N S (page 91)

1. True. 2. True. 3. False. 4. True.

E X E R C I S E i (page 108)

The exact results will depend on the particular changes made. However, any shift that raises birth rates at younger ages and also makes an equal reduction in birth rates at older ages should have the effect of both (a) decreasing the length of a generation because the average age of mothers at the birth of their children will be less and (b) increasing the net reproduction rate because mortality will be less at younger ages. In a population such as that of the United States, this shift will not be of great importance since mortality is low at all ages within the reproductive span and the net and gross reproduction rates are nearly the same.


Country A: Both fertility and mortality are low because both the net and gross reproduction rates are low, and the difference between them is small. If the current age-specific birth


and death rates continue indefinitely, the population size will decline slowly, eventually stabilizing at the rate of 15 per thousand per generation. Since the birth and death rates are equal and the birth rate is low, it appears likely (but not certain) that the age structure is not far from that required for this permanent condition.

Country B: The statements made about country A with respect to the net and gross reproduction rate apply here too. However, the fact that the crude birth rate is much higher than the crude death rate, with a substantial rate of present natural increase, makes it probable that the age structure is young (probably as a result of higher past fertility). Therefore, the attainment of the slow growth decline will take a long time, even if the age-specific vital rates continue at their present level.

Country C: The high gross reproduction rate indicates very high fertility rates. That there is little difference between the net and gross rates means that mortality is very low. This is a population that will grow rapidly (195 percent per generation) if its current vital rates continue indefinitely. Unless it has an unusually old age structure, the crude birth rate is likely to be very high and the crude death rate very low at present. This would be characteristic of a population like the American Hutterites.

Country D: This is a country in which fertility and mortality are both very high. The large gross reproduction rate indicates that fertility is high. The fact that the net reproduction rate is so much lower means that mortality must be very high. The net reproduction rate of 1,000 means that for the long run, mortality is sufficiently high to offset completely the high fertility. Over the long run these rates imply a stationary population. That birth and death rates are currently equal at a high level suggests that a stationary condition is already closely approximated.


Country E: Fertility rates are moderately high and mortality rates low. This inference follows from the fact that the gross reproduction rate is substantially above 1,000 (although there are many higher rates) and the net reproduction rate differs from it only a little. In the long run if the age-specific vital rates remain at their current levels, this population will grow at the rate of about 48 percent per generation. This is a situation rather similar to that in the United States during the period following World War II.

Country F: This country has high fertility and moderate mortality because its gross reproduction rate is very high and the net reproduction rate differs from the gross rate moderately. (If mortality were very low, the net reproduction rate would differ very little from the gross rate. If mortality were very high, the net reproduction rate would be 1,000 or less. Should these vital rates continue indefinitely, the growth rate per generation would be about 50 percent. The actual crude birth rate and death rates given are consistent with a situation in which fertility remains high but mortality has fallen from previously higher levels. This situation is probably representative of a country like Pakistan.

T H I R D S E T O F T R U E / F A L S E Q U E S T I O N S (page 108)

1. True, although the two rates can be equal if there is no mortality before the end of the childbearing ages.

2. False. The GRR refers to a hypothetical cohort, not a real one.

3. True. 4. True. 5. False. 6. False. 7. True.


T H I R D S E T O F M U L T I P L E - C H O I C E Q U E S T I O N S (poge 109)

1. (a). See Table 3.21. 2. (c). 3. (c). 4. (d). 5. (c)., assuming that the sex ratios at birth by age are equal.

The TFRs are certainly equal. 6. (d).

F O U R T H S E T O F M U L T I P L E - C H O I C E Q U E S T I O N S (page 119)

1. (c). 2. (c). 3. (e). 4. (a).

F I F T H S E T O F M U L T I P L E - C H O I C E Q U E S T I O N S (page 125)

1. ( H 2. (f). 3. (a). 4. (d).

Countries with Populations of Fewer than 1 Million: 1990 Estimates or Latest Census

Most of the countries listed below are not included in the

tables in the Guide because demographic measures for small

populations are subject to rather great variability. Figures with

an asterisk (*) are U N estimates.

Country Population

Africa British Indian Ocean Territory Cape Verde Comoros Djibouti Equatorial Guinea Gambia Guinea-Bissau Reunion Sao Tome and Principe Seychelles St. Helena Swaziland Western Sahara

3,000' 370,000 • 551,000' 409,000 * 348,999 .861,000' 965,000 * 599,000' 121,000' 67,000

7,000' 768,000 179,000*

North America Anguilla Antigua and Barbuda Aruba Bahamas Barbados Belize Bermuda British Virgin Islands Cayman Islands Dominica Greenland Grenada Guadeloupe

8,000' 77,000' 60,000 *

253,000 255,000' 188,000 61,000 13,000 27,000 83,000 * 57,000' 85,000'

344,000'

150 Countries with Populations of Fewer than 1 Million

Country Population

Martinique 341,000 * Montserrat 13,000* Netherlands Antilles 189,000* St. Kitts and Nevis 44,000 * Saint Lucia 151,000* St. Perrre and Miquelon 7,000 * St. Vincent and the Grenadines 116,000 * Turks and Caicos Islands 10,000 * U.S. Virgin Islands 117,000 '

South America Falkland Islands 2,000 * French Guiana 99,000' Guyana 796,000 * Suriname 422,000'

Asia Bahrain 503,000 * Brunei Darussalam 266,000 * Cyprus 702,000 East Timor 737,000 * Macau 479,999 * Maldives 215,000* Qatar 368,000 *

Europe Andorra 52,000 Channel Islands 300,000' Faeroe Islands 48,000 * Gibraltar 31,000' Holy See 1,000 * Iceland 255,000 Isle of Man 64,000 * Liechtenstein 29,999 * Luxembourg 373,000 * Malta 354,000 Monaco 29,000 * San Marino 24,000"

Oceania American Samoa 39,000 Cocos Islands 555 Cook Islands 18,000' Fiji 765,000 * French Polynesia 206,000 * Guam 119,000' Johnston Island 1,007 Kiribati 66,000 * Marshall Islands 40,000 * Federated States of Micronesia 99,000 *

Countries with Populations of Fewer than 1 Million 151

Country Population

Nauru 10,000* New Caledonia 168,000* Niue 3,000 * Northern Mariana Islands 26,000' Palau 18,000* Pitcairn 52 Western Samoa 164,000 Solomon Islands 321,000* Tokelau 1,552 Tonga 95,000 * Tuvalu 10,000* Vanuatu 147,000 Wake Island 1,647 Wallis and Futuna Islands 18,000'

Source: United Nations Statistical Office, 1990 Demographic Yearbook (1992, table 3).

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INDEX

[Note: page numbers in boldface type indicate the location of figures and tables.)

age at birth [see mean age at childbirth) exact, 37, 40-41 at last birthday, 41

age at marriage effect on fertility measures, 119

age composition effect of fertility and mortality, 5 In, 51-52 effect on crude birth rate, 81-82, 84-89 (see also standardization of

birth rates) effect on crude death rate, 15-18, 16, 17, 52-53 (see also

standardization of death rates) effect on crude rates, exercises on, 91 effect on fertility measures, exercises on, 110 effect on general fertility rate, 73-74, 89 in life table (see life table, functions, L ) rectangular, 92 of stationary population, 51-52, 52 and voting, 20-21

age distribution (see age composition) age-specific birth rates [see age-specific fertility rates) age-specific death rates (ASDRs, Mr), 10-15, 12

as component of crude death rate, 15-18 correlation among, 22 exercises on, 18 formula for, 12, 132 patterns of, 12 and probability of death, 135-36 for selected countries, 12, 13-14

age-specific fertility rates (ASFRs, Ft), 74-77 correlations among, 94 correlation with total fertility rate, 94 exercises on, 78-79, 90 formula for, 74, .134 of Hutterites, 75-76, 77 pattern of, 74-75, 75

and cohort-period comparisons, 119 and growth rate, 107

for selected countries, 76 world distribution by level, 76-77, 77

ASDR [see age-specific death rates) ASFR (see age-specific fertility rates) average age at childbirth (see mean age at childbirth)

160 Index

birth age at (see mean age at childbirth) live, defined, 64

birth cohort defined, 35 hypothetical (see hypothetical cohort) synthetic (see hypothetical cohort)

birth intervals, analysis of, 120-25 life-table approach, 121-24, 122-23 open intervals, 120-21

exercise on, 125-28 problems with

censoring, 120-24 exercises on, 125

data quality, 120 selectivity, 124-25, 126-27

exercises on, 125, 128 birth-order-specific fertility rates, 82-84, 83

formula for, 82, 134 and general fertility rate, 82-84, 83

birth probabilities

and contraceptive use, 121-24, 122-23 birth rates (see also fertility rates)

age-specific (see age-specific fertility rates) birth-order-specific (see birth-order-specific fertility rates) crude (see crude birth rate) intrinsic, for selected countries, 102-3 in life table, 50

formula for, 133 marital-status specific, 80-82 specific, 79-84 standardized (see standardization of birth rates)

births smooth distribution assumption, 5 timing of, and fertility measures, 118

causes of death in infancy, 31-32

CBR (see crude birth rate) CDR (see crude death rate) censoring

of birth intervals, 120-24 defined, 120 exercise on, 125

census measures of fertility, 111-15 central population, 5 childbearing

average age at (see mean age at childbirth) risk of, 65

childlessness in United States, 113 children ever born (CEB), 65, 114-15

in Indonesia, 115

Index 161

in United States, 113, 114 child-woman ratio (CWR), 112-14

correlations with crude birth rate and total fertility rate, 113 exercises on, 120 formula for, 2, 112, 134 problems with, 112-13 range of, 113

closed population (see under migration assumptions) cohort

birth (see birth cohort) fertility of, 115-19 (see also children ever born)

exercises on, 119-20 and period fertility, 117-19

hypothetical (see hypothetical cohort) life tables for, 35-36 marriage, 116 synthetic (see hypothetical cohort)

cohort effects on fertility, 116-17 composition

age (see age composition) controlling for (see standardization) and crude death rate, 19

confounding factors (see standardization) constants, use of

in demographic measures, 9n, 132 in gross reproduction rate, 95n in net reproduction rate, 97, lOln in percentages, 3 in total fertility rate, 92n

contraceptive use and birth probabilities, 121-24, 122-23 crude birth rate (CBR), 50n, 66

correlations with child-woman ratio, 113 with general fertility rate, 93-94 with total fertility rate, 94

exercises on, 78-79, 90 formula for, 66, 134 for selected countries, 66, 67, 68, 86, 102-3 standardized (see standardization of birth rates) for world and regions, 98

crude death rate (CDR), 9-10 components of, 15-18, 16, 17 (see also standardization of death rates) decomposition of, 15-18 effects of age composition, 15-18, 16, 17, 52-53 (see aiso standardiza

tion of death rates) effects of compositional factors, 19 exercises on, 18-19 formula for, 9, 15, 132 and life-table death rate, 52-53, 53 for selected countries, 10, 11, 102-3 standardized (see standardization of death rates)

162 Index

crude rate of natural increase (CRNI), 66, 69-72 (see also natural increase)

exercises on, 78-79 formula for, 66, 134 historical levels, 70 for selected regions, 66, 69, 69 time needed for population to double, triple, and quadruple, 69-70, 70

cumulative fertility (see children ever born)

dx (deaths in life table) (see under life table, functions) data quality, 15n, 32-33, 64, 120 death

causes of, in infancy, 31-32 risk of, 3, 10, 31, 65, 135

death rates age-specific (see age-specific death rates) crude [see crude death rate) for infants [see infant mortality rate) intrinsic, for selected countries, 102-3 in life table (see life table, death rate) standardized (see standardization of death rates)

deaths in life table, 43

formula for, 43, 133 smooth distribution assumption, 5, 37, 43, 44-45

decomposition, 85n of crude death rate, 15-18

denominator (see also midyear population; person-years lived; see also specific measures)

of age-specific death rates, 135-36 from census data, 111 of child-woman ratio, 112 of general fertility rate, 73 ideal, 3, 7, 66, 135 of infant mortality rate, 31, 136-37 of probabilities, 7 of proportions, 2 of qx (probability of dying), 135-36 of rates, 3-6

density, formula for, 2 dependency ratio, formula for, 2 doubling time, 69-70, 70, 72

exercise on, 79

ex (see life expectancy) events

basic elements of, 1 exercise on, 10

rate of occurrence (see rates) smooth distribution assumption, 5

expectation of life (see life expectancy) exposure to risk, 3-6, 7 (see also denominator; person-years lived)

Index 163

of childbearing, 65 of dying, 3, 10, 31, 65, 135

Ft (see age-specific fertility rates) factors, confounding (see standardization) fecundity

contrasted with fertility, 63-64 fertility, 63-95, 111-25

age range of, 73 of cohort (see cohort, fertility of) cohort effects on, 116-19 cumulative (see children ever born) effect on age composition, 51n, 51-52 factors affecting, 73 and fecundity contrasted, 63-64 measurement problems, 64-65 measures of

based on census, 111-15 (see also children ever born; child-woman ratio)

correlations among, 93-94 data sources for, 111, 11 In estimation techniques, 113-14 indirect (see child-woman ratio) period, 115 (see also age-specific fertility rates; crude birth rate;

general fertility rate) and timing of births, 118-19

period effects on, 116-19 and population growth, 63

fertility decline age pattern of, and growth, 107 components of, 81-82

fertility rates (see also birth rates; crude birth rate; general fertility rate; total fertility rate)

age-specific (see age-specific fertility rates) birth-order-specific, 82-84, 83

formula for, 82, 134 marital-status-specific, 80-82 specific, 79-84

formulas for demographic measures, 132-34 (see also specific measures)

general fertility rate (GFR), 73-74 and birth-order-specific fertility rates, 82-84, 83 correlation with crude birth rate, 93-94 effects of age composition, 73-74, 89 exercises on, 78, 90 formula for, 73, 134 range of, 74 standardized, 87-89, 88, 89

generation, length of [see mean length of generation) GFR (see general fertility rate) gross reproduction rate (GRR), 95-97

calculation of, 96, 100

164 Index

exercises on, 108-10 formula for, 96, 134 interpretation of, 96-97 for selected countries, 102-4, 104 for United States, 105 for world and regions, 97, 98 world distribution by level, 99

growth (see population growth)

Hutterites fertility rates of, 75-76, 77

hypothetical cohort defined, 36 in fertility analysis (see gross reproduction rate; net reproduction rate,-

total fertility rate) in life table, 36-37 (see also life table)

hypothetical population, 15-16, 49-51 (see also stationary population)

IMR (see infant mortality rate| incidence rates (see rates) income composition

and crude death rate, 19 and voting, 20-21

infant, defined, 30 infant death rate, 30n infant mortality rate (IMR), 30-34, 136-37

by age, 3 In, 31-32 for selected countries, 32

exercises on, 33 formula for, 31, 133 levels of, 33, 34 measurement problems, 32-33, 136-37 for selected countries, 33, 34

intermediate variables, 120 intrinsic rates

for selected countries, 102

tt (survivors in life table) (see under life table, functions) Lt (person-years lived in life table) (see under life table, functions) length of generation (see mean length of generation) life expectancy (ej, 46

change over time, 57, 58 comparing, 56-57

problems in, 56 differentials in, 56, 57, 58 formula for, 46, 133 interpretation, 56 range of, 56, 57 for selected countries, 102-3

life table, 35-57 abridged, 46-47

definition of, 46

Index 165

definition of, 46 example of, 48

applications of (see life expectancy, comparing; stationary population; survival ratios)

and birth interval analysis, 121-24 birth rate, 50

formula for, 133 cohort, 36, 115 complete, 37-46

definition of, 46 example of, 38-40

cross-sectional (see life table, period) current (see life table, period) death rate, 50

compared with crude death rate, 52^53, 53 formula for, 133

deaths [dx), 43 formula for, 43, 133

exercises on, 47-49, 58-59 functions

dx (deaths), 43 formula for, 43, 133

ex, 46 (see also life expectancy) formula for, 46, 133

Lx (person-years lived), 43-45 formulas for, 43, 44, 133

at youngest ages, 44 lx (survivors), 41-42

formula for, 42 qx (probability of dying), 41 (see also probability of dying)

formula for, 133 Tx (total person-years lived), 45

formula for, 45, 133 generational, 36, 115 interpretations of, 37, 51 life expectancy (see life expectancy) longitudinal, 36, 115 migration assumption, 37, 49, 56 period, 36-37, 115

basic assumption, 36 defined, 36

person-years lived in [Lx Tx), 43-45 formulas for, 43-45, 133

probability of dying (see probability of dying) radix, 37, 37n, 41

defined, 37 survivors [(g), 41-42 time-specific (see life table, period)

Mx [see age-specific death rates) marital status composition, 79-82

effect on fertility, 81

166 Index

percent ever married in United States, 113 marital-status-specific fertility rates, 80-82 marriage

norms regarding, 80 variations in, 79-80

marriage cohort, 116 marriage squeeze, 80 mean age at childbirth (u), 102-3, 106n (see also mean length of

generation) mean length of generation (7), 102-3, 105-7

calculation of, 100 effect on growth, 106-7 factors affecting, 106 and mean age at childbirth, 106n and migration, 106 and net reproduction rate, 107 for selected countries, 102-3, 106

midyear population (see also denominator; see also specific measures) in age-specific death rates, 135-36 as approximation for person-years lived, 5-6, 7, 30, 66, 135 as estimate of exposure to risk, 135 in general fertility rate, 73 and migration, 135-36, 136n in stationary population, 54

migration and estimating person-years lived, 5-6

migration assumptions closed population

and generation length, 106 and life table, 37, 49, 56 and midyear population, 135-36, 136n and population growth, 69, 105 and probability of dying, 7 and projections, 54 and stationary population, 51 and survival ratios, 55

smooth distribution, 5 models, 37 (see also gross reproduction rate; life table; net reproduction

rate; total fertility rate) mortality, 9-61 (see also age-specific death rates; crude death rate; death

rates; life table; standardization of death rates) differentials, 56, 57, 58 effect on age composition, 51, 51n exercises on, 10 and population growth, 9

u (mu) (see mean age at childbirth)

natural increase (NI), 66, 69, 69-72 defined, 63 intrinsic rate of, for selected countries, 102-3 for world and regions, 66, 69, 69

net reproduction rate (NRR), 97-98, 100-105

Index 167

calculation of, 98, 100, 100 effects of fertility and mortality, 101 exercises on, 108-11, 119 formula for, 100-101, 134 implications, 101, 105 interpretation of, 97-98 range of, 101 for selected countries, 102-3, 104 for United States, 105

notation, 131-32, 132-34 in life table, 41, 133

occupational composition and crude death rate, 19

order-specific fertility rates (see birth-order-specific fertility rates)

percentages, 3 called rates, 6-7

period effect on fertility, 116-19 period fertility

and cohort fertility, 117-18 person-years lived, 3-6, 4

for ages under one, 30, 44 approximations for, 4-6 calculation of, 4 exercises on, 6 ideal denominator for rates, 3, 7, 66, 135 in life table (L j7 TJ, 43-45

formulas for, 43-45, 133 population

central, 5 closed (see under migration assumptions) hypothetical, 15-16, 49-51 (see also hypothetical cohort; stationary

population) midyear (see midyear population) rectangular, 92 standard (see standard population) stationary (see stationary population)

population density formula for, 2

population explosion, 72 population growth (see also crude rate of natural increase; natural

increase) exponential, 72 and fertility, 63, 107 and generation length, 106-7 geometric, 70 and migration, 69, 105 and mortality, 9 rates, for world, 70, 72

population projections, 53-54, 54n, 72

168 Index

population size of countries with fewer than one million people, 149-51 of selected regions, 1650-1990, 71

probabilities, 7 probability of dying [qx], 41

derived from age-specific death rates, 135-36, 136n formula for, 133 and IMR, 31, 31n, 136-37 and migration, 7

projections of population, 53-54, 54n, 72 proportions, 2-3

qx [see probability of dying) quadrupling time, 69-70, 70

radix (of life table), 37, 37n, 41 defined, 37

rates, 3-7 age-specific [see age-specific death rates; age-specific fertility rates) birth [see birth rates,- fertility rates) central, 5 death [see death rates) denominators for, 3-7 [see also denominator) fertility [see fertility rates) infant mortality rate (IMR) [see infant mortality rate) literacy, 6 percentages called rates, 6-7 reproduction [see gross reproduction rate; net reproduction rate) specific, 10, 12, 73 [see also age-specific death rates; age-specific

fertility rates,- birth-order-specific fertility rates; marital-status-specific fertility rates)

standardized [see standardization) survival, 7n

ratios, 1-3 examples and formulas, 2 survival [see survival ratios)

rectangular population, 92 registration system and coverage of births, 64, 111, 113 reproduction rates [see gross reproduction rate; net reproduction rate) residence composition

and crude death rate, 19 risk [see exposure to risk)

SDR (standardized death rate) [see standardization of death rates) selectivity

and analysis of birth intervals, 124-25, 126-27 exercises on, 125, 128

sex composition and crude death rate, 19 and fertility measures, exercises on, 110

sex ratio, formula for, 2

Index 169

standardization standardization of birth rates, 84-89 (see also gross reproduction rate;

net reproduction rate; total fertility rate) for age and sex, 84-87 effects on comparisons, 85

exercises on, 91 exercises on, 90-91 for selected countries, 85, 86 for United States, 85, 87, 87

standardization of death rates, 19-30 for age, example of, 21-22, 23 common factors controlled for, 19 exercises on, 28-30 formula for, 25, 132 for selected countries, 24-28, 26-27

standardization of general fertility rate, 87-89, 88, 89 standard population, 20

choice and effects of, 21, 22-24, 28, 85, 89 stationary population, 49-53 (see also life table)

age structure of, 51-52, 52 defined, 51 exercises on, 58-59 and migration, 51

surveys and censoring, 120-24 completeness of, 33, 64 and selectivity, 124

survival rates, 7n survival ratios, 7n, 53-55

change in, over time, 55, 55 exercises on, 59 formula for, 54 reverse, 54

survivors in life table {lx), 41-42 synthetic cohort (see hypothetical cohort)

T (length of generation) (see mean length of generation) Tx (total person-years lived in life table) (see under life table, functions) total fertility rate (TFR), 91-95

calculation of, 93 constant used in calculation of, 92n correlations

with ASFRs, 94 with child-woman ratio, 113 with crude birth rate, 94

defined, 91 exercises on, 108 formula for, 92, 118n, 134 interpretation of, 92, 95 for periods and cohorts contrasted, 117-18 range of, 93

tripling time, 69-70, 70


FIFTH EDITION

James A. Palmore and Robert W. Gardner

This newly revised edition of Palmore and Gardner's popular introductory textbook presents elementary measures used in demographic analysis, beginning with rates, ratios, percentages, and probabilities and proceeding to the crude death rate and age-specific death rates, standardized rates, the infant mortality rate, the life table, the crude birth rate and age-specific fertility rates, the general fertility rate, total fertility rate, gross and net reproduction rates, period and cohort fertility measures, and the analysis of birth intervals. Written in a direct, conversational style, it includes numerous examples and illustrations that have been updated with data from the 1990 round of censuses. At the end of each section are exercises and quizzes designed to test students' understanding of the material presented. Four appendixes and recommendations for further reading provide readers with additional useful information. Includes an index.

E A S T - W E S T C E N T E R ISBN 0-86638-165-1 $15.00

measuring mortality, fertility, and natural increase

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