comparison of spline- and loess-based approaches for the ... · 2.2 loess-based approach the...

Comparison of spline- and loess-based approaches for the estimation of

child mortality

Richard Silverwood and Simon Cousens

London School of Hygiene and Tropical Medicine

16th April 2008

Contents

1 Introduction 2

2 Overview of methods 2

2.1 Spline-based approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Loess-based approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 General issues with the methods 4



4 Comparison of spline- and loess-based approaches 6

4.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Incorporation of uncertainty 22


5.1.1 Random draw simulation approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.1.2 Analytic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.1.3 Comparison with uncertainty intervals obtained using the loess-based approach . . . . . . . . . 23


5.2.1 Analytic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6 Further extensions and alternative approaches 38

6.1 Incorporation of sampling variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6.2 Multilevel modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7 Summary 41

Appendix: Comparison of the datasets 42

1

1 Introduction

There are currently (at least) two approaches towards the estimation of childhood mortality: a ‘spline-based approach’

favoured by the Inter-agency Coordination Group on Child Mortality Estimation and detailed by Hill et al (the ‘Green

Book’) [1], and a ‘loess-based approach’ described by Murray et al [2].

This report aims to provide a brief overview of these methods (Section 2), discussing any outstanding issues

with their application (Section 3), and to compare the results obtained under each (Section 4). The incorporation of

uncertainty into both estimating procedures is examined (Section 5), and further extensions and alternative approaches

discussed (Section 6). The findings are briefly summarised, and the key issues remaining to be addressed highlighted

(Section 7).

2 Overview of methods

The spline-based approach (Section 2.1) and the loess-based approach (Section 2.2) are briefly summarised.

2.1 Spline-based approach

For each country, the spline-based approach proceeds as follows:

1. Assign weights to each observed value of infant or under-5 mortality:

• Assign weights based on data/study type, length of time before survey to which estimate refers, age group

of mother, etc.;

• If one survey contributes two types of estimates then reduce both sets of estimates to half their standard

weight.

2. Define knots:

• Work backwards in time from the most recent observation;

• Weights summed and a knot defined every time the sum of the weights reaches a multiple of 5;

• For last knot defined (i.e. earliest knot), remaining weights must sum to at least 5.

3. Using weighted least squares regression fit the linear spline model,

log(y) = β0 + β1x +K∑

k=1

bk(x− κk)+ + ε, (2.1)

where y is childhood mortality, x is year, κ1, . . . , κK are the K knot locations, (x− κk)+ is equal to 0 if x < κk

and x− κk if x ≥ κk, and ε is an error term which is assumed to be normally distributed.

4. Critically examine results:

• Identify any datasets that are clearly aberrant;

• Reduce weights for the entire aberrant dataset(s) by a constant factor — generally 0.5, 0.25 or 0.

5. Refit spline model using revised weights.

2

6. Decide whether to select the infant or under-5 mortality sequence of estimates as the more consistent series.

7. Corresponding values of the other indicator (the derived indicator) can be obtained using a model life table.

2.2 Loess-based approach

The general approach is to fit loess regression curves to the data using a variety of smoothing parameters to vary the

sensitivity to recent data trends.

The basic loess function is

log(y) = β0 + β1x + β2z + ε, (2.2)

where y is under-5 mortality, x is calendar year, z is an indicator variable taking value 1 if the observed value comes

from a vital registration system and value 0 otherwise, and ε is an error term which is assumed to be normally

distributed.

The loess function (2.2) is fitted using weighted least squares regression, with the weights corresponding to each

observed under-5 mortality value calculated using a separate weighting function. This weighting function is tuned by

a single parameter, α.

Let x0 be the time point at which a fitted loess curve is required and define ψ = ||x − x0|| to be the separation

between a time point x and x0. For α < 1, the weighting function is calculated using only the 100α% of observations

closest to x0 (i.e. with smallest values of ψ), using

w =

(1−

(ψ

Ψ

)3)3

, (2.3)

where w is the weight corresponding to the time point x, and Ψ is the maximum value of ψ among the 100α% of

observations with the smallest values of ψ. For α ≥ 1, all data are included and the weighting function becomes

w =

(1−

(ψ

Ψα1/2

)3)3

. (2.4)

These weighting functions are illustrated in Fig. 1 for Armenia.

For each country, the loess-based approach proceeds as follows:

1. Decide upon the minimum value of α which will be used (αmin):

• Calculate the minimum value of α which will ensure that at least 3 data points are always included in the

loess regression (as is required for the variance-covariance matrix associated with the regression coefficients

to be estimable);

• Examine the fitted loess curve corresponding to this α value. If this α value does not provide a sufficiently

smooth fit to the data then increase α until a sufficiently smooth fit is achieved.

• The resulting α value should be used as αmin, unless the country under consideration has fewer than 100,000

children younger than 5 years, in which case αmin is the maximum of this α value and 0.4.

2. Decide upon the maximum value of α which will be used (αmax):

3

1980 1985 1990 1995 2000 2005

020

4060

80

Year

Und

er−

5 m

orta

lity

(per

100

0)

1980 1985 1990 1995 2000 2005

0.0

0.2

0.4

0.6

0.8

1.0

Year

Wei

ght

Fig. 1: Data (left-hand plot; blue points represent vital registration data, black points represent non-vital registration data) and weight

functions calculated at 1990 (right-hand plot; red lines represent smaller α values, yellow lines represent larger α values) for Armenia.

• Examine the correlations between the fitted loess curves and the ordinary least squares fits for various

values of α;

• If correlation becomes almost perfect once a given value of α is passed, set αmax to be this value. Otherwise

use αmax = 2.

3. For each value of α in {αmin, αmin + 0.05, αmin + 0.10, . . . , 1.0, 1.1, . . . , αmax}:

• Calculate the weights associated with each observed under-5 mortality value using (2.3) or (2.4) as appro-

priate;

• Fit the loess function (2.2) using weighted least squares regression;

• Simulate 1000 random draws from the multivariate normal distribution defined by the estimated regression

coefficients and their variance-covariance matrix;

• For each of the 1000 random draws, calculate the estimated/predicted under-5 mortality at the required

time point, assuming non-vital registration data.

4. Pool the 1000 estimates/predictions per α value across the set of α values.

5. Calculate the final estimated/predicted under-5 mortality at the required time point as the median of these

pooled estimates/predictions, with an uncertainty interval corresponding to the 2.5th and 97.5th centiles of these

pooled estimates/predictions.

3 General issues with the methods

There remain some outstanding issues with both the spline- and loess-based approaches.

4


In the spline-based approach the fitted models are restricted to being piecewise linear on the log scale, which is

unlikely to be an accurate representation of the actual trend. The down-weighting of aberrant datasets in step 4 of

the procedure described in Section 2.1 is rather ad-hoc and introduces an element of subjectivity into the procedure,

as does the decision over whether the infant or under-5 mortality sequence of estimates is the more consistent series

in step 6. For a truly transparent and reproducible method, these areas of subjectivity would ideally be formalised.

Additionally, the spline-based approach does not currently include any quantification of the uncertainty surrounding

each estimate, for example through the use of uncertainty intervals.


There are several issues regarding the α values which are used. For example, setting αmin as the first acceptable value

above 0.10 and αmax as the last acceptable value below 2.00 is somewhat arbitrary, as is increasing α in increments

of 0.05 when less than 1 and 0.1 when greater than 1. The selection of αmin and αmax, as described in Section 2.2,

is also a little subjective. Additional subjectivity is introduced by the manner in which ‘extreme outliers that clearly

differ from the rest of the datapoints’ [2] are excluded from the analysis.

The effects of varying αmin and αmax are illustrated in Fig. 2–Fig. 5. In each, loess curves are fitted to the same

dataset (Belize) for α = {αmin, αmin+0.05, αmin+0.10, . . . , 1.0, 1.1, . . . , αmax} and under-5 mortality in 2015 predicted.

This dataset shows an overall decreasing trend, but with an apparently more recent increasing trend. In each case, the

left-hand plot shows the fitted loess curves and the right-hand plot shows the distribution of the simulated random

draws with the final predicted under-5 mortality and uncertainty interval in 2015.

In Fig. 2, αmin = 0.30 and αmax = 2.00. The loess curves corresponding to the lower values of α follow the more

recent increasing trend in the data. The loess curves corresponding to the higher α values ignore the more recent

increasing trend and are all very similar to one another. This results in a skewed distribution of the simulated random

draws and a wide, skewed uncertainty interval.

In Fig. 3, αmax is still 2.00, but αmin is increased to 0.60. The loess curves corresponding to the lower values of α

in Fig. 2 which follow the more recent increasing trend in the data are no longer present. This results in a less skewed

distribution of the simulated random draws and a narrower, less skewed uncertainty interval. Although the upper

bound of the uncertainty interval is reduced drastically, relatively little change is seen in the median and no change is

seen in the lower bound.

In Fig. 4, αmin = 0.30, similarly to in Fig. 2, but now αmax is decreased to 1.50. Some of the loess curves

corresponding to the higher α values in Fig. 2 which ignore the more recent increasing trend are no longer present.

This results in a lower density of simulated random draws at the lower end of the distribution, although the distribution

still remains highly skewed. Neither bound of the uncertainty interval changes greatly, but the median value increases

somewhat.

In Fig. 5, αmin = 0.60 and αmax = 1.50. The loess curves corresponding to both the lower α values and the higher

α values in Fig. 2 are no longer present. This again results in a less skewed distribution of the simulated random draws

and a narrower, less skewed uncertainty interval. Although the upper bound of the uncertainty interval is reduced

drastically, relatively little change is seen in either the median or the lower bound.

Thus it can be seen that varying αmin and αmax can in some circumstances make a large difference to the final

5

1970 1980 1990 2000 2010

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Year

Log

unde

r−5

mor

talit

y (p

er 1

000)

2

3

4

5

Density

Pre

dict

ed lo

g un

der−

5 m

orta

lity

(per

100

0) in

201

5

0.0 0.5 1.0 1.5

(12.6)

(56.3)

(8.6)

Fig. 2: Fitted loess curves (left-hand plot; all data are vital registration data) and distribution of simulated random draws with final

predicted under-5 mortality and uncertainty interval in 2015 (right-hand plot; values in brackets are transformed back to the original scale)

using α = {0.30, 0.35, . . . , 1.00, 1.10, . . . , 2.00} in Belize.

predicted under-5 mortality and associated uncertainty interval. Generally, the lower the αmin value is set, the more

likely it is that (potentially spurious) trends in the most recent data which differ from the overall trend will be picked

up. The higher the αmax value is set, the more likely it is that the loess curves fitted for α values close to αmax are

essentially the same. When both low α values and high α values are included then the result may be wide, highly

skewed uncertainty intervals. As the selection of αmin and αmax is somewhat subjective, the potential consequences

should be borne in mind.

4 Comparison of spline- and loess-based approaches

4.1 Method

Estimated/predicted under-5 mortality under the spline- and loess-based approaches in the years 2000, 2005, 2010 and

2015 are compared in the 60 country datasets available in the UNICEF database. The countries included are shown

in Table 1.

The spline-based approach proceeds as described in Section 2.1. The final weights (those used in step 5) are

provided with the datasets and used here. For some countries (Lesotho and Zimbabwe), recently published spline-

based estimates are set to be constant, but this is ignored here.

The loess-based approach ignores the weights used for the spline-based approach. Instead, data which come from

a vital registration system are identified using the documentation provided with the data. The loess-based approach

then proceeds as described in Section 2.2.

For illustration, fitted spline and loess curves are shown in Fig. 6 and Fig. 7 for Armenia and Zimbabwe, respectively.

The fitted curves are seen to be very similar for Armenia, but for Zimbabwe they differ greatly, particularly since the

most recent knot in the spline-based approach.

6

1970 1980 1990 2000 2010

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Year

Log

unde

r−5

mor

talit

y (p

er 1

000)

2

3

4

5

DensityP

redi

cted

log

unde

r−5

mor

talit

y (p

er 1

000)

in 2

015

0.0 0.5 1.0 1.5 2.0

(11.5)

(22.4)

(8.5)



using α = {0.60, 0.65, . . . , 1.00, 1.10, . . . , 2.00} in Belize.

1970 1980 1990 2000 2010

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Year

Log

unde

r−5

mor

talit

y (p

er 1

000)

2

3

4

5

Density

Pre

dict

ed lo

g un

der−

5 m

orta

lity

(per

100

0) in

201

5

0.0 0.5 1.0

(14.2)

(59.1)

(8.9)



using α = {0.30, 0.35, . . . , 1.00, 1.10, . . . , 1.50} in Belize.

7

1970 1980 1990 2000 2010

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Year

Log

unde

r−5

mor

talit

y (p

er 1

000)

2

3

4

5

DensityP

redi

cted

log

unde

r−5

mor

talit

y (p

er 1

000)

in 2

015

0.0 0.5 1.0 1.5

(12.2)

(23.1)

(8.7)



using α = {0.60, 0.65, . . . , 1.00, 1.10, . . . , 1.50} in Belize.

Armenia Fiji Maldives Somalia

Belarus Gambia Mexico Syria

Belize Georgia Micronesia, Fed. States Tajikistan

Brazil Ghana Moldova Thailand

Burkina Faso Guinea Mongolia Timor Leste

Burundi Guinea Bissau Nepal Togo

Cambodia Haiti Palau Trinidad & Tobabgo

Central African Republic Honduras Papua New Guinea Turkmenistan

Chad India Peru Tuvalu

China Iraq Russian Federation Ukraine

Colombia Jamaica Rwanda United Arab Emirates

Congo Kazakhstan Sao Tome & Principe Uruguay

Cote d’Ivoire Kyrgyzstan Senegal Uzbekistan

Egypt Lesotho Sierra Leone Venezuela

Ethiopia Malawi Solomon Islands Zimbabwe

Table 1: Countries included in the comparison of the spline- and loess-based approaches.

8

1980 1990 2000 2010

020

4060

8010

0

Year

Und

er−

5 m

orta

lity

(per

100

0)

1980 1990 2000 2010

020

4060

8010

0Year

Und

er−

5 m

orta

lity

(per

100

0)

Fig. 6: Fitted spline curve (left-hand plot; dashed vertical line indicates knot location) and loess curve (right-hand plot; blue points

represent vital registration data, black points represent non-vital registration data, dashed lines represent uncertainty intervals) for Armenia.

1960 1970 1980 1990 2000 2010

050

100

150

200

Year

Und

er−

5 m

orta

lity

(per

100

0)

1960 1970 1980 1990 2000 2010

050

100

150

200

Year

Und

er−

5 m

orta

lity

(per

100

0)

Fig. 7: Fitted spline curve (left-hand plot; dashed vertical lines indicate knot locations) and loess curve (right-hand plot; all data are

non-vital registration data, dashed lines represent uncertainty intervals) for Zimbabwe.

9

4.2 Results

Estimated/predicted under-5 mortality in the years 2000, 2005, 2010 and 2015 under the two different methods are

presented in Tables 2–4. Although it is clear that there are often discrepancies between the spline- and loess-based

estimated/predicted values, these differences are more easily interpreted when the results are presented graphically.

An important aspect of the spline-based approach is the down-weighting of aberrant datasets in step 4 of the

procedure described in Section 2.1, referred to henceforth as ‘ad-hoc weight adjustment’. As this ad-hoc weight ad-

justment is conducted on a largely subjective basis, it is perhaps likely that for countries with a great deal of ad-hoc

weight adjustment the final estimates/predictions of under-5 mortality will be less comparable with the loess-based

equivalents, for which no additional weighting has been applied. Thus when analysing the results, countries with no

ad-hoc weight adjustments and countries with more complex ad-hoc weight adjustments are examined separately. In

some countries the ad-hoc weight adjustment involves only a down-weighting of data coming from vital registration

systems. As the loess-based approach also effectively downweights the influence of vital registration data, countries in

which this is the case are included with those where there is no ad-hoc weight adjustment.

Fig. 8–15 show the loess-based estimates/predictions of under-5 mortality plotted against the spline-based esti-

mates/predictions for each year separately. Countries where the estimated/predicted under-5 mortality using one

approach is 20–33.3% greater than the mean estimated/predicted under-5 mortality for the two different approaches

are highlighted with solid markers. This corresponds to the larger estimate/prediction being 1.5–2 times greater than

the smaller. Countries where the estimated/predicted under-5 mortality using one approach is more than 33.3% greater

than the mean estimated/predicted under-5 mortality for the two different approaches are labelled. This corresponds

to the larger estimate/prediction being more than twice the smaller.

It can be seen that in countries with either no ad-hoc weight adjustments or merely a down-weighting of the

vital registration data (Fig. 8, Fig. 10, Fig. 12 and Fig. 14) the estimated/predicted under-5 mortality under the two

different approaches is generally more similar than in countries with more complex ad-hoc weight adjustments (Fig. 9,

Fig. 11, Fig. 13 and Fig. 15). In countries with either no ad-hoc weight adjustments or merely a down-weighting of the

vital registration data there appears to be a reasonably similar proportion of countries where the spline-based method

provides the greater estimate/prediction and where the loess-based method provides the greater estimate/prediction.

This is not so true for countries with more complex ad-hoc weight adjustments, though interpretation of this is more

difficult. Also, within each category of ad-hoc weight adjustment, the estimations/predictions under the two different

approaches tend to get less similar as time progresses.

Table 5 summarises the observations from Fig. 8–15 by tabulating, for each category of ad-hoc weight adjustment

and each year separately, the differences between the spline- and loess-based estimates/predictions.

For countries with no ad-hoc weight adjustments or merely a down-weighting of the vital registration data, in

2000 all estimates/predictions are within 10% of the mean, showing good agreement between the two approaches.

As time progresses, however, predictions become a little less similar — by 2015 only 56% remain within 10% of the

mean, with 8% more than 20% from the mean. In 2000 approximately as many countries have a greater spline-based

estimate/prediction as have a greater loess-based estimate/prediction. The proportion of countries where the loess-

based prediction is greater than the spline-based prediction increases so that by 2015 this is true for 68% of countries.

Thus, although the numbers involved are small, there is some evidence of a tendency for the loess-based approach to

10

Cou

ntry

2000

.520

05.5

2010

.520

15.5

Splin

eLoe

ssSp

line

Loe

ssSp

line

Loe

ssSp

line

Loe

ss

Arm

enia

36.3

35.3

(32.

2,39

.0)

25.5

27.1

(21.

8,30

.8)

17.9

21.3

(14.

3,25

.3)

12.5

16.7

(9.1

,21

.2)

Bel

arus

17.3

13.1

(11.

1,17

.8)

13.9

10.7

(8.9

,13

.0)

11.2

8.8

(6.0

,11

.5)

9.0

7.4

(3.8

,10

.4)

Bel

ize1

19.3

21.2

(19.

8,23

.3)

12.8

15.3

(13.

7,18

.5)

8.5

11.1

(9.5

,14

.8)

5.6

8.0

(6.6

,11

.8)

Bra

zil

29.6

34.2

(30.

6,38

.5)

21.2

26.7

(22.

9,31

.1)

15.3

20.9

(17.

1,25

.1)

11.0

16.3

(12.

6,20

.3)

Bur

kina

Faso

193.

919

6.9

(184

.9,21

2.7)

202.

319

1.1

(173

.6,22

8.0)

211.

118

4.7

(162

.8,24

8.0)

220.

217

8.3

(152

.3,27

1.7)

Bur

undi

180.

818

3.8

(171

.4,19

6.0)

181.

018

1.5

(165

.5,20

4.1)

181.

117

8.9

(158

.8,21

4.2)

181.

217

5.9

(152

.5,22

5.3)

Cam

bodi

a10

4.2

104.

8(9

6.3,

114.

1)85

.491

.0(6

3.3,

107.

3)70

.080

.9(3

8.3,

102.

5)57

.472

.1(2

2.9,

98.0

)

Cen

tral

Afr

ican

Rep

ublic

186.

118

6.3

(165

.2,20

1.7)

176.

618

7.6

(158

.1,21

5.6)

167.

618

8.4

(150

.7,23

2.8)

159.

018

9.3

(143

.9,25

1.9)

Cha

d220

5.3

203.

6(1

93.0

,21

4.5)

208.

720

3.5

(185

.3,22

3.7)

212.

220

3.7

(175

.4,23

6.1)

215.

720

3.9

(162

.6,24

8.0)

Chi

na36

.631

.1(2

6.4,

36.7

)25

.427

.5(2

2.1,

35.0

)17

.624

.3(1

7.2,

35.5

)12

.321

.5(1

2.9,

36.2

)

Col

ombi

a25

.924

.5(2

1.4,

28.8

)21

.419

.9(1

6.4,

25.7

)17

.716

.0(1

2.7,

23.1

)14

.612

.9(9

.8,

20.8

)

Con

go11

7.0

114.

0(1

00.7

,12

8.2)

124.

811

5.8

(95.

3,14

7.2)

133.

211

6.1

(89.

2,17

6.9)

142.

211

6.9

(83.

0,21

3.1)

Cot

ed’

Ivoi

re2

137.

913

8.8

(125

.5,15

4.6)

131.

612

8.8

(102

.9,15

1.2)

125.

512

0.3

(80.

3,15

0.5)

119.

711

2.6

(61.

7,14

9.8)

Egy

pt2

50.5

54.4

(48.

8,63

.2)

37.7

40.3

(32.

7,50

.5)

28.1

30.2

(21.

7,40

.6)

21.0

22.6

(14.

4,32

.7)

Eth

iopi

a15

0.6

159.

5(1

46.1

,17

2.3)

127.

114

6.2

(117

.1,16

5.2)

107.

313

4.9

(91.

5,15

8.4)

90.6

124.

4(7

2.2,

152.

2)

Fiji

113

.115

.0(1

2.9,

20.4

)9.

811

.8(9

.6,

19.5

)7.

49.

3(7

.2,

18.2

)5.

67.

3(5

.3,

17.4

)

Gam

bia

131.

912

9.6

(119

.5,13

8.3)

116.

311

5.6

(103

.5,12

8.9)

102.

510

2.6

(89.

0,12

1.5)

90.3

91.2

(76.

8,11

5.2)

Geo

rgia

36.6

37.6

(30.

0,46

.7)

32.7

36.4

(28.

3,51

.2)

29.3

34.9

(24.

9,68

.5)

26.2

33.3

(21.

6,93

.1)

Gha

na11

2.7

114.

2(1

08.2

,12

0.6)

118.

710

9.2

(101

.0,13

2.4)

124.

910

3.8

(94.

1,14

8.8)

131.

598

.7(8

7.4,

168.

1)

Gui

nea

184.

318

5.5

(178

.9,19

3.1)

164.

816

5.3

(144

.7,17

5.9)

147.

314

8.1

(110

.2,16

2.8)

131.

713

2.6

(83.

2,15

1.2)

Table

2:

Est

imate

d/pre

dic

ted

under

-5m

ort

ality

(per

1000).

Spline-

base

dm

ethod

pro

vid

espoin

tes

tim

ate

sonly

.Loes

s-base

dm

ethod

pro

vid

espoin

tes

tim

ate

sand

95%

unce

rtain

tyin

terv

als

.

1Spline-

base

des

tim

ate

sdo

not

corr

espond

topublish

edvalu

esdue

topublish

edvalu

esbei

ng

der

ived

from

the

fitt

edin

fant

mort

ality

spline.

2Spline-

base

des

tim

ate

sdiff

ersl

ightl

yfr

om

publish

ed

valu

es.

3R

ecen

tpublish

edsp

line-

base

des

tim

ate

sare

const

ant

—th

isis

ignore

dher

e.

11

Cou

ntry

2000

.520

05.5

2010

.520

15.5

Splin

eLoe

ssSp

line

Loe

ssSp

line

Loe

ssSp

line

Loe

ss

Gui

nea

Bis

sau

217.

521

6.0

(211

.4,22

0.7)

202.

920

4.2

(195

.9,21

3.6)

189.

219

3.1

(179

.4,20

9.0)

176.

418

2.5

(164

.1,20

4.8)

Hai

ti10

9.2

116.

7(1

08.0

,12

6.2)

84.2

101.

6(8

2.3,

114.

3)64

.988

.9(6

1.8,

103.

4)50

.177

.8(4

6.5,

93.6

)

Hon

dura

s39

.640

.2(3

8.2,

42.5

)28

.732

.3(3

0.1,

35.4

)20

.825

.9(2

3.6,

29.9

)15

.120

.7(1

8.5,

25.2

)

Indi

a188

.294

.8(8

6.9,

114.

3)77

.286

.0(7

6.4,

130.

2)67

.577

.9(6

7.2,

146.

8)59

.170

.4(5

9.1,

162.

6)

Iraq

47.5

46.8

(42.

3,51

.9)

46.6

41.2

(35.

4,47

.9)

45.7

36.5

(27.

7,45

.5)

44.9

32.4

(21.

4,43

.8)

Jam

aica

31.9

21.9

(17.

2,30

.6)

31.3

19.5

(14.

1,35

.4)

30.6

17.2

(11.

3,42

.2)

29.9

15.1

(9.0

,51

.1)

Kaz

akhs

tan

42.9

39.1

(34.

4,45

.2)

31.0

31.0

(22.

7,37

.8)

22.3

25.3

(13.

3,33

.2)

16.1

20.7

(7.8

,29

.0)

Kyr

gyzs

tan

51.4

51.6

(45.

5,57

.7)

42.5

42.5

(35.

5,49

.5)

35.2

35.4

(27.

3,43

.3)

29.1

29.6

(20.

7,37

.9)

Les

otho

310

8.4

93.1

(85.

3,11

1.0)

131.

886

.3(7

6.2,

127.

2)16

0.2

79.8

(68.

1,14

7.5)

194.

773

.9(6

1.0,

171.

1)

Mal

awi

155.

316

8.9

(155

.0,17

9.4)

125.

315

0.0

(121

.8,16

2.5)

101.

113

3.5

(93.

9,14

7.5)

81.6

118.

8(7

3.0,

134.

1)

Mal

dive

s54

.153

.6(4

8.9,

58.8

)33

.335

.6(2

7.0,

42.3

)20

.424

.3(1

4.3,

30.8

)12

.616

.5(7

.6,

22.5

)

Mex

ico1

35.8

35.3

(31.

8,40

.4)

29.7

29.4

(25.

7,35

.9)

24.7

24.4

(20.

8,31

.9)

20.5

20.3

(16.

7,28

.2)

Mic

rone

sia,

Fed.

Stat

es46

.547

.1(4

2.6,

52.3

)41

.842

.9(3

6.5,

50.5

)37

.539

.1(3

1.0,

49.0

)33

.735

.6(2

6.6,

47.5

)

Mol

dova

24.3

27.5

(25.

3,29

.9)

19.8

22.6

(17.

7,25

.4)

16.1

18.9

(11.

8,22

.0)

13.1

15.9

(7.8

,19

.1)

Mon

golia

61.5

57.1

(51.

5,64

.8)

45.2

39.0

(29.

0,50

.2)

33.3

27.1

(15.

0,39

.7)

24.5

18.8

(7.8

,31

.1)

Nep

al86

.196

.5(8

6.6,

110.

8)63

.181

.8(6

5.8,

99.5

)46

.269

.8(4

9.7,

89.4

)33

.959

.5(3

7.5,

80.7

)

Pal

au1

9.9

3.1

(1.4

,6.

6)6.

71.

5(0

.4,

7.0)

4.5

0.7

(0.1

,9.

4)3.

00.

3(0

.0,

14.1

)

Pap

uaN

ewG

uine

a279

.683

.3(7

2.5,

95.5

)72

.977

.5(6

3.5,

95.8

)66

.872

.2(5

5.6,

94.9

)61

.267

.2(4

8.0,

94.7

)

Per

u41

.348

.0(4

2.6,

53.0

)27

.337

.7(2

9.7,

43.3

)18

.029

.6(2

0.5,

35.4

)11

.923

.2(1

4.2,

29.0

)

Rus

sian

Fede

rati

on23

.922

.0(1

7.0,

27.1

)16

.919

.4(1

3.5,

24.3

)11

.517

.3(1

1.5,

22.0

)7.

815

.5(9

.7,

19.8

)

Table

3:

Est

imate

d/pre

dic

ted

under

-5m

ort

ality

(per

1000)

conti

nued

.Spline-

base

dm

ethod

pro

vid

espoin

tes

tim

ate

sonly

.Loes

s-base

dm

ethod

pro

vid

espoin

tes

tim

ate

sand

95%

unce

rtain

ty

inte

rvals

.1

Spline-

base

des

tim

ate

sdo

not

corr

espond

topublish

edvalu

esdue

topublish

edvalu

esbei

ng

der

ived

from

the

fitt

edin

fant

mort

ality

spline.

2Spline-

base

des

tim

ate

sdiff

ersl

ightl

yfr

om

publish

edvalu

es.

3R

ecen

tpublish

edsp

line-

base

des

tim

ate

sare

const

ant

—th

isis

ignore

dher

e.

12

Cou

ntry

2000

.520

05.5

2010

.520

15.5

Splin

eLoe

ssSp

line

Loe

ssSp

line

Loe

ssSp

line

Loe

ss

Rw

anda

183.

418

6.3

(175

.4,19

8.9)

162.

718

2.5

(168

.1,20

3.6)

144.

417

8.9

(158

.0,21

0.2)

128.

117

5.3

(147

.6,21

6.8)

Sao

Tom

e&

Pri

ncip

e97

.271

.1(5

4.1,

88.7

)95

.962

.3(3

2.5,

86.4

)94

.655

.5(1

9.5,

84.7

)93

.349

.5(1

1.5,

83.3

)

Sene

gal

132.

613

2.0

(122

.0,14

3.1)

118.

711

9.5

(106

.9,13

7.6)

106.

310

7.8

(93.

6,13

3.2)

95.2

97.1

(81.

7,12

9.9)

Sier

raLeo

ne27

6.5

273.

9(2

62.6

,29

0.5)

271.

126

4.1

(249

.5,30

4.3)

265.

925

4.1

(236

.7,32

5.0)

260.

724

4.4

(224

.6,34

8.7)

Solo

mon

Isla

nds1

79.2

56.5

(42.

4,75

.2)

68.8

57.8

(38.

8,91

.8)

59.9

58.7

(34.

9,11

7.2)

52.1

59.3

(31.

5,15

1.5)

Som

alia

164.

815

8.6

(143

.6,17

5.2)

148.

513

3.8

(112

.5,16

1.8)

133.

811

4.7

(82.

1,15

4.2)

120.

698

.5(5

9.2,

145.

0)

Syri

a19

.921

.7(1

6.3,

25.4

)14

.516

.1(1

0.4,

19.7

)10

.711

.9(6

.5,

15.1

)7.

88.

8(4

.2,

11.7

)

Taj

ikis

tan

93.3

80.6

(71.

2,90

.9)

71.4

67.4

(54.

0,81

.6)

54.6

57.1

(36.

1,76

.0)

41.8

48.5

(23.

8,69

.8)

Tha

iland

12.9

14.4

(12.

4,16

.4)

8.4

10.3

(7.9

,12

.2)

5.5

7.4

(4.9

,9.

1)3.

65.

3(3

.0,

6.8)

Tim

orLes

te10

6.8

113.

7(9

8.6,

131.

8)61

.392

.5(6

3.1,

122.

9)35

.275

.4(3

7.9,

115.

8)20

.262

.5(2

2.8,

110.

6)

Tog

o12

4.0

129.

0(1

23.5

,13

5.6)

110.

611

9.6

(112

.5,13

1.5)

98.5

110.

5(1

01.8

,12

9.6)

87.8

102.

3(9

1.8,

127.

0)

Tri

nida

d&

Tob

ago

34.2

30.5

(25.

1,38

.4)

37.0

28.5

(22.

1,42

.9)

39.9

26.6

(19.

2,49

.4)

43.1

24.8

(16.

4,55

.8)

Tur

kmen

ista

n70

.664

.2(5

5.8,

74.0

)54

.142

.2(3

3.0,

52.7

)41

.428

.8(1

6.1,

40.2

)31

.719

.8(7

.7,

30.5

)

Tuv

alu1

35.7

34.9

(17.

2,69

.0)

30.3

29.2

(11.

2,73

.7)

25.7

24.5

(7.5

,78

.3)

21.8

20.6

(4.8

,82

.7)

Ukr

aine

22.8

22.3

(20.

4,25

.3)

23.6

20.6

(18.

2,23

.3)

24.4

18.8

(14.

8,21

.7)

25.2

17.2

(11.

7,20

.7)

Uni

ted

Ara

bE

mir

ates

10.3

9.5

(7.3

,11

.8)

8.5

6.9

(4.9

,9.

1)7.

14.

9(3

.2,

7.2)

5.8

3.5

(2.1

,5.

8)

Uru

guay

116

.019

.5(1

7.2,

25.1

)16

.216

.4(1

4.2,

22.4

)16

.513

.8(1

1.6,

20.0

)16

.811

.5(9

.5,

17.7

)

Uzb

ekis

tan

62.3

55.9

(49.

2,66

.0)

46.0

46.1

(40.

9,51

.3)

34.0

36.9

(31.

0,43

.6)

25.1

30.0

(21.

8,38

.0)

Ven

ezue

la24

.525

.6(2

3.6,

29.3

)21

.322

.1(1

9.9,

28.1

)18

.419

.1(1

6.8,

27.2

)16

.016

.5(1

4.1,

26.0

)

Zim

babw

e313

5.3

73.0

(64.

9,85

.0)

185.

568

.5(5

8.6,

88.2

)25

4.2

63.9

(52.

3,92

.6)

348.

559

.5(4

6.8,

96.0

)

Table

4:

Est

imate

d/pre

dic

ted

under

-5m

ort

ality

(per

1000)

conti

nued

.Spline-

base

dm

ethod

pro

vid

espoin

tes

tim

ate

sonly

.Loes

s-base

dm

ethod

pro

vid

espoin

tes

tim

ate

sand

95%

unce

rtain

ty

inte

rvals

.1

Spline-

base

des

tim

ate

sdo

not

corr

espond

topublish

edvalu

esdue

topublish

edvalu

esbei

ng

der

ived

from

the

fitt

edin

fant

mort

ality

spline.

2Spline-

base

des

tim

ate

sdiff

ersl

ightl

yfr

om

publish

edvalu

es.

3R

ecen

tpublish

edsp

line-

base

des

tim

ate

sare

const

ant

—th

isis

ignore

dher

e.

13

010

020

030

0Lo

ess−

base

d es

timat

e/pr

edic

tion

0 100 200 300Spline−based estimate/prediction

Fig. 8: Comparison of loess- and spline-based estimates/predictions of under-5 mortality in the year 2000.5 in countries with either

no ad-hoc weight adjustments or merely a down-weighting of the vital registration data. Countries where estimated/predicted under-5

mortality using one approach is 20–33.3% greater than the mean estimated/predicted under-5 mortality for the two different approaches

are highlighted with solid markers. Countries where the estimated/predicted under-5 mortality using one approach is more than 33.3%

greater than the mean estimated/predicted under-5 mortality for the two different approaches are labelled.

Palau010

020

030

0Lo

ess−

base

d es

timat

e/pr

edic

tion

0 100 200 300Spline−based estimate/prediction

Fig. 9: Comparison of loess- and spline-based estimates/predictions of under-5 mortality in the year 2000.5 in countries with more

complex ad-hoc weight adjustments. Countries where estimated/predicted under-5 mortality using one approach is 20–33.3% greater than

the mean estimated/predicted under-5 mortality for the two different approaches are highlighted with solid markers. Countries where

the estimated/predicted under-5 mortality using one approach is more than 33.3% greater than the mean estimated/predicted under-5

mortality for the two different approaches are labelled.

14

010

020

030

0Lo

ess−

base

d ap

proa

ch

0 100 200 300Spline−based approach






Palau

Zimbabwe

010

020

030

0Lo

ess−

base

d ap

proa

ch

0 100 200 300Spline−based approach






15

050

100

150

200

250

Loes

s−ba

sed

appr

oach

0 50 100 150 200 250Spline−based approach






Lesotho

Palau

ZimbabweTimor Leste

050

100

150

200

250

Loes

s−ba

sed

appr

oach







16

010

020

030

040

0Lo

ess−

base

d es

timat

e/pr

edic

tion

0 100 200 300 400Spline−based estimate/prediction






Jamaica

Lesotho

Palau

ZimbabweTimor Leste

010

020

030

040

0Lo

ess−

base

d es

timat

e/pr

edic

tion

0 100 200 300 400Spline−based estimate/prediction






17

produce slightly higher predictions. This is borne out by p-values in the region 0.05 to 0.1 depending on the statistical

test used.

For countries with more complex ad-hoc weight adjustments, in 2000 83% of estimates/predictions are within

10% of the mean, but the 17% not within 10% of the mean all have a lower loess-based estimate/prediction than

spline-based. As time progresses, predictions get further from the mean until by 2015 only 35% are within 10% of the

mean and 34% are more than 20% from the mean. However, by 2015 approximately as many countries have a greater

spline-based estimate/prediction as have a greater loess-based prediction.

Loess-based estimate/prediction as a % of the

mean of the loess- and spline-based estimates/predictions

YearAd-hoc weighting

<80% 80–90% 90–100% 100–110% 110–120% >120%adjustment?

2000.5None/VR only 0 (0%) 0 (0%) 12 (48%) 13 (52%) 0 (0%) 0 (0%)

More complex 2 (6%) 4 (11%) 12 (34%) 17 (49%) 0 (0%) 0 (0%)

2005.5None/VR only 0 (0%) 1 (4%) 7 (28%) 16 (64%) 1 (4%) 0 (0%)

More complex 5 (14%) 3 (9%) 9 (26%) 14 (40%) 3 (9%) 1 (3%)

2010.5None/VR only 0 (0%) 1 (4%) 7 (28%) 12 (48%) 4 (16%) 1 (4%)

More complex 5 (14%) 6 (17%) 6 (17%) 9 (26%) 6 (17%) 3 (9%)

2015.5None/VR only 1 (4%) 1 (4%) 6 (24%) 8 (32%) 8 (32%) 1 (4%)

More complex 7 (20%) 6 (17%) 3 (9%) 9 (26%) 5 (14%) 5 (14%)

Table 5: Loess-based estimate/prediction as a % of the mean of the loess- and spline-based estimates/predictions. ‘None/VR only’ is

either no ad-hoc weight adjustments or merely a down-weighting of the vital registration data. ‘More complex’ is more complex ad-hoc

weight adjustments.

Fig. 16–21 plot the spline-based estimate/prediction for each country and each year alongside the loess-based

estimate/prediction and associated uncertainty interval. Again, countries with no ad-hoc weight adjustments or

merely a down-weighting of the vital registration data (Fig. 16–18) are examined separately to countries with more

complex ad-hoc weight adjustments (Fig. 19–21). In most countries, the spline-based estimates/predictions lie within

the loess-based estimated uncertainty intervals. However, for several countries with more complex ad-hoc weighting

adjustments there are more severe discrepancies.

4.3 Conclusions

When there are complex ad-hoc weight adjustments, it is understandable that discrepancies between the spline- and

loess-based estimates/predictions occur, thus it is perhaps more informative to concentrate on the results obtained in

countries where these is little or no ad-hoc weight adjustments. In these countries the estimated/predicted under-5

mortality is often very similar for both the spline- and loess-based approaches, particularly at time points closer to

the range of years over which data are observed. However, there is some evidence of a tendency for the loess-based

approach to produce slightly higher estimates than the spline-based approach, particularly at time points further from

18

010

2030

4050

Und

er−

5 m

orta

lity

(per

100

0)

Arm

enia

Bel

ize

Col

ombi

a

Fiji

Hon

dura

s

Mex

ico

Mic

rone

sia

Mol

dova

Fig. 16: Comparison of loess- and spline-based estimates/predictions of under-5 mortality in countries with either no ad-hoc weight

adjustments or merely a down-weighting of the vital registration data. Black points are loess-based estimates/predictions with uncertainty

intervals displayed as bars. Red points are spline-based estimates/predictions. For each country the estimates/predictions correspond,

from left to right, to the years 2000.5, 2005.5, 2010.5 and 2015.5.

050

100

150

Und

er−

5 m

orta

lity

(per

100

0)

Cam

bodi

a

Gam

bia

Geo

rgia

Mal

dive

s

Per

u

Tog

o

Tur

kmen

ista

n

Tuv

alu





19

010

020

030

040

0U

nder

−5

mor

talit

y (p

er 1

000)

Bur

undi

C. A

f. R

ep.

Con

go

Gui

nea

Gui

nea

Bis

sau

Rw

anda

Sen

egal

Sie

rra

Leon

e

Som

alia





010

2030

40U

nder

−5

mor

talit

y (p

er 1

000)

Bel

arus

Bra

zil

Chi

na

Pal

au

Rus

sian

Fed

.

Syr

ia

Tha

iland

Ukr

aine

U.A

.E.

Uru

guay

Ven

ezue

la

Fig. 19: Comparison of loess- and spline-based estimates/predictions of under-5 mortality in countries with more complex ad-hoc weight

adjustments. Black points are loess-based estimates/predictions with uncertainty intervals displayed as bars. Red points are spline-based

estimates/predictions. For each country the estimates/predictions correspond, from left to right, to the years 2000.5, 2005.5, 2010.5 and

2015.5.

20

050

100

Und

er−

5 m

orta

lity

(per

100

0)

Egy

pt

Iraq

Jam

aica

Kaz

akhs

tan

Kyr

gyzs

tan

Mon

golia

Nep

al

Pap

ua N

. G.

S. T

. & P

.

Taj

ikis

tan

T. &

T.

Uzb

ekis

tan




2015.5.

010

020

030

040

0U

nder

−5

mor

talit

y (p

er 1

000)

Bur

kina

F.

Cha

d

Cot

e d’

Iv.

Eth

iopi

a

Gha

na

Hai

ti

Indi

a

Leso

tho

Mal

awi

Sol

omon

Is.

Tim

or L

este

Zim

babw

e




2015.5.

21

the range of years over which data are observed.

One potential explanation for this observation could be the apparent shift of births and deaths in DHS birth history

data from the most recent 5-year period to the previous period in some countries acknowledged elsewhere. This can

lead to an underestimate of mortality for the most recent 5-year period and an overestimate for the previous period,

resulting in an overestimate of the trend between these two time points. The spline-based approach, using a fixed set

of weights, is likely to be more sensitive to this feature of the data than the loess-based approach, using a variety of

smoothing parameters.

The loess- and spline-based approaches are likely to provide similar estimates/predictions if the data points lie on

a very obvious trajectory, if there are no vital registration data, and if, in the spline-based approach, the data points

are (finally, if not initially) weighted similarly to the average weighting in the loess-based approach. The loess-based

approach places more emphasis on the long-term trend than the spline-based approach. In the instances where there

is a large difference between the estimated/predicted under-5 mortality under the two approaches it is often because

a more recent deviation from the long-term trend is having a large effect on the fitted spline, but the deviation is only

acknowledged in smaller α values in the loess-based approach so the overall effect is diluted.

5 Incorporation of uncertainty


One disadvantage of the spline-based method as it currently stands is the lack of any indication of the level of

uncertainty associated with any estimate/predication of child mortality. One possible way to incorporate uncertainty

into the spline-based approach is through a similar random draw simulation method to that used in the loess-based

approach. This is detailed in Section 5.1.1. An alternative approach is to calculate ‘analytic’ uncertainty intervals

directly, as described in Section 5.1.2. In Section 5.1.3 uncertainty intervals are calculated using the random draw

simulation method and compared to those found under the loess-based approach.

5.1.1 Random draw simulation approach

For each country, a possible method for the incorporation of uncertainty into the spline-based approach proceeds as

follows:

1. Fit the infant or under-5 mortality spline as per steps 1–5 in Section 2.1.

2. Simulate 10,000 random draws from the multivariate normal distribution defined by the estimated coefficients

and their variance-covariance matrix.

3. For each required time point:

• For each of the 10,000 random draws, calculate the estimated/predicted mortality at the required time

point;

• Pool the 10,000 estimates/predictions;

• The uncertainty interval corresponds to the 2.5th and 97.5th centiles of these pooled estimates/predictions.

22

5.1.2 Analytic approach

As an alternative to the random draw simulation approach detailed in Section 5.1.1, ‘analytic’ uncertainty intervals

can be calculated directly.

Consider the linear spline model (2.1),

log(y) = β0 + β1x +K∑

k=1

bk(x− κk)+ + ε.

From this expression it is possible to calculate for any fitted point, log(y), the corresponding variance, var(log(y)). A

95% confidence interval for the fitted point can then be constructed as

log(y)± 1.96√

var(log(y)), (5.1)

or using the t-distribution analogously if the sample size is small.

For example, if there are K = 2 knots then (2.1) becomes

log(y) = β0 + β1x + b1(x− κ1)+ + b2(x− κ2)+ + ε.

Fitted values are then

log(y) = β0 + β1x + b1(x− κ1)+ + b2(x− κ2)+ (5.2)

with variance

var(log(y)) = var(β0) + x2var(β1) + (x− κ1)2var(b1) + (x− κ2)2var(b2) + 2xcov(β0, β1)

+ 2(x− κ1)cov(β0, b1) + 2x(x− κ1)cov(β1, b1) + 2(x− κ2)cov(β0, b2)

+ 2x(x− κ2)cov(β1, b2) + 2(x− κ1)(x− κ2)cov(b1, b2),

(5.3)

and a 95% confidence interval can be constructed by substituting (5.2) and (5.3) into (5.1).

However, as the number of knots K increases, the length of the expression for var(log(y)) increases rapidly.

As the approach detailed in Section 5.1.1 is simulating random draws from the multivariate normal distribution

defined by the same estimated coefficients, variances and covariances used in (5.2) and (5.3), with sufficient random

draws the estimated uncertainty intervals under the two approaches should be identical.

5.1.3 Comparison with uncertainty intervals obtained using the loess-based approach

Method

Estimated/predicted under-5 mortality uncertainty intervals under the spline- and loess-based approaches in the years

2000, 2005, 2010 and 2015 are compared in the same 60 datasets detailed in Section 4.1.

The spline-based approach proceeds as described in Section 2.1, with uncertainty intervals created using the random

draw simulation approach detailed in Section 5.1.1. The final weights (those used in step 5 of Section 2.1) are provided

23

1980 1990 2000 2010

020

4060

8010

0

Year

Und

er−

5 m

orta

lity

(per

100

0)

1980 1990 2000 2010

020

4060

8010

0

Year

Und

er−

5 m

orta

lity

(per

100

0)Fig. 22: Fitted spline curve (left-hand plot; dashed vertical line indicates knot location, dashed curves represent uncertainty intervals)

and loess curve (right-hand plot; blue points represent vital registration data, black points represent non-vital registration data, dashed

curves represent uncertainty intervals) for Armenia.

with the dataset and used here. For some countries (Lesotho and Zimbabwe), recently published spline-based estimates

are set to be constant, but this is ignored here.

The loess-based approach ignores the weights used for the spline-based approach. Instead, data which come from

a vital registration system are identified using the documentation provided with the data. The loess-based approach

then proceeds as described in Section 2.2.

For illustration, fitted spline and loess curves with uncertainty intervals are shown in Fig. 22 and Fig. 23 for

Armenia and Papua New Guinea, respectively. The estimated uncertainty intervals under the two approaches are seen

to be reasonably similar for Armenia, but for Papua New Guinea they differ greatly, particularly since the most recent

knot in the spline-based approach.

Results

Estimated under-5 mortality uncertainty intervals, as well as the corresponding estimates/predictions, in the years

2000, 2005, 2010 and 2015 under the two different methods are presented in Tables 6–8. Although it is clear that there

are often discrepancies between the spline- and loess-based estimated uncertainty intervals, these differences are more

easily interpreted when the results are presented graphically.

As in Section 4.2, countries with no ad-hoc weight adjustments or merely a down-weighting of data coming from

vital registration systems are examined separately to countries with more complex ad-hoc weight adjustments.

Fig. 24–29 plot the spline-based estimate/prediction and associated uncertainty interval for each country and each

year alongside the loess-based estimate/prediction and associated uncertainty interval. Countries with no ad-hoc

weight adjustments or merely a down-weighting of the vital registration data are shown in Fig. 24–26 and countries

with more complex ad-hoc weight adjustments in Fig. 27–29. There is much variability in the relative sizes of the

estimated uncertainty intervals, with the spline-based uncertainty interval being much wider in some cases and much

24

Countr

y2000.5

2005.5

2010.5

2015.5

Spline

Loes

sSpline

Loes

sSpline

Loes

sSpline

Loes

s

Arm

enia

36.3

(33.1

,39.9

)35.3

(32.2

,39.0

)25.5

(20.2

,32.3

)27.1

(21.8

,30.8

)17.9

(11.9

,27.0

)21.3

(14.3

,25.3

)12.5

(7.0

,22.7

)16.7

(9.1

,21.2

)

Bel

aru

s17.3

(15.8

,18.8

)13.1

(11.1

,17.8

)13.9

(11.7

,16.5

)10.7

(8.9

,13.0

)11.2

(8.5

,14.7

)8.8

(6.0

,11.5

)9.0

(6.1

,13.1

)7.4

(3.8

,10.4

)

Bel

ize1

19.3

(15.4

,24.0

)21.2

(19.8

,23.3

)12.8

(8.6

,18.9

)15.3

(13.7

,18.5

)8.5

(4.8

,14.9

)11.1

(9.5

,14.8

)5.6

(2.7

,11.8

)8.0

(6.6

,11.8

)

Bra

zil

29.6

(24.5

,35.7

)34.2

(30.6

,38.5

)21.2

(16.0

,28.3

)26.7

(22.9

,31.1

)15.3

(10.5

,22.3

)20.9

(17.1

,25.1

)11.0

(6.8

,17.7

)16.3

(12.6

,20.3

)

Burk

ina

F.

193.9

(185.1

,202.9

)196.9

(184.9

,212.7

)202.3

(177.3

,230.7

)191.1

(173.6

,228.0

)211.1

(167.7

,266.0

)184.7

(162.8

,248.0

)220.2

(158.3

,307.4

)178.3

(152.3

,271.7

)

Buru

ndi

180.8

(167.9

,194.2

)183.8

(171.4

,196.0

)181.0

(151.9

,215.9

)181.5

(165.5

,204.1

)181.1

(135.2

,244.0

)178.9

(158.8

,214.2

)181.2

(119.9

,275.7

)175.9

(152.5

,225.3

)

Cam

bodia

104.2

(93.5

,115.4

)104.8

(96.3

,114.1

)85.4

(67.2

,108.2

)91.0

(63.3

,107.3

)70.0

(47.5

,102.6

)80.9

(38.3

,102.5

)57.4

(33.4

,97.8

)72.1

(22.9

,98.0

)

C.A

f.R

ep.

186.1

(172.5

,201.1

)186.3

(165.2

,201.7

)176.6

(146.7

,212.4

)187.6

(158.1

,215.6

)167.6

(123.7

,226.4

)188.4

(150.7

,232.8

)159.0

(104.3

,242.7

)189.3

(143.9

,251.9

)

Chad2

205.3

(194.3

,216.8

)203.6

(193.0

,214.5

)208.7

(185.3

,235.0

)203.5

(185.3

,223.7

)212.2

(176.1

,255.6

)203.7

(175.4

,236.1

)215.7

(167.2

,278.5

)203.9

(162.6

,248.0

)

Chin

a36.6

(34.0

,39.4

)31.1

(26.4

,36.7

)25.4

(21.9

,29.5

)27.5

(22.1

,35.0

)17.6

(13.1

,23.7

)24.3

(17.2

,35.5

)12.3

(7.7

,19.1

)21.5

(12.9

,36.2

)

Colo

mbia

25.9

(22.4

,30.0

)24.5

(21.4

,28.8

)21.4

(16.3

,28.1

)19.9

(16.4

,25.7

)17.7

(11.7

,26.7

)16.0

(12.7

,23.1

)14.6

(8.3

,25.4

)12.9

(9.8

,20.8

)

Congo

117.0

(108.1

,126.7

)114.0

(100.7

,128.2

)124.8

(110.8

,140.8

)115.8

(95.3

,147.2

)133.2

(112.8

,157.5

)116.1

(89.2

,176.9

)142.2

(114.7

,176.9

)116.9

(83.0

,213.1

)

Cote

d’Iv.2

137.9

(126.1

,151.0

)138.8

(125.5

,154.6

)131.6

(114.8

,150.9

)128.8

(102.9

,151.2

)125.5

(104.4

,151.1

)120.3

(80.3

,150.5

)119.7

(94.8

,151.2

)112.6

(61.7

,149.8

)

Egypt2

50.5

(46.1

,55.4

)54.4

(48.8

,63.2

)37.7

(28.9

,49.1

)40.3

(32.7

,50.5

)28.1

(17.4

,45.2

)30.2

(21.7

,40.6

)21.0

(10.4

,41.9

)22.6

(14.4

,32.7

)

Eth

iopia

150.6

(140.9

,161.1

)159.5

(146.1

,172.3

)127.1

(113.3

,142.6

)146.2

(117.1

,165.2

)107.3

(90.8

,126.9

)134.9

(91.5

,158.4

)90.6

(72.7

,112.9

)124.4

(72.2

,152.2

)

Fiji1

13.1

(11.4

,15.1

)15.0

(12.9

,20.4

)9.8

(8.1

,11.9

)11.8

(9.6

,19.5

)7.4

(5.8

,9.4

)9.3

(7.2

,18.2

)5.6

(4.2

,7.4

)7.3

(5.3

,17.4

)

Gam

bia

131.9

(122.9

,141.6

)129.6

(119.5

,138.3

)116.3

(98.6

,137.1

)115.6

(103.5

,128.9

)102.5

(78.4

,134.2

)102.6

(89.0

,121.5

)90.3

(62.4

,131.8

)91.2

(76.8

,115.2

)

Geo

rgia

36.6

(33.2

,40.2

)37.6

(30.0

,46.7

)32.7

(27.7

,38.7

)36.4

(28.3

,51.2

)29.3

(22.8

,37.4

)34.9

(24.9

,68.5

)26.2

(18.8

,36.4

)33.3

(21.6

,93.1

)

Ghana

112.7

(108.2

,117.5

)114.2

(108.2

,120.6

)118.7

(105.2

,134.0

)109.2

(101.0

,132.4

)124.9

(99.7

,156.7

)103.8

(94.1

,148.8

)131.5

(94.1

,183.5

)98.7

(87.4

,168.1

)

Guin

ea184.3

(175.6

,193.7

)185.5

(178.9

,193.1

)164.8

(145.4

,186.1

)165.3

(144.7

,175.9

)147.3

(119.5

,180.2

)148.1

(110.2

,162.8

)131.7

(98.0

,174.7

)132.6

(83.2

,151.2

)

Table

6:

Est

imate

d/pre

dic

ted

under

-5m

ort

ality

(per

1000).

Spline-

base

dm

ethod

pro

vid

espoin

tes

tim

ate

sonly

.Loes

s-base

dm

ethod

pro

vid

espoin

tes

tim

ate

sand

95%

unce

rtain

tyin

terv

als

.

1Spline-

base

des

tim

ate

sdo

not

corr

espond

topublish

edvalu

esdue

topublish

edvalu

esbei

ng

der

ived

from

the

fitt

edin

fant

mort

ality

spline.

2Spline-

base

des

tim

ate

sdiff

ersl

ightl

yfr

om

publish

ed

valu

es.

3R

ecen

tpublish

edsp

line-

base

des

tim

ate

sare

const

ant

—th

isis

ignore

dher

e.

25

Countr

y2000.5

2005.5

2010.5

2015.5

Spline

Loes

sSpline

Loes

sSpline

Loes

sSpline

Loes

s

Guin

eaB

issa

u217.5

(211.1

,224.1

)216.0

(211.4

,220.7

)202.9

(189.4

,217.5

)204.2

(195.9

,213.6

)189.2

(169.2

,211.9

)193.1

(179.4

,209.0

)176.4

(151.0

,206.7

)182.5

(164.1

,204.8

)

Hait

i109.2

(101.6

,117.8

)116.7

(108.0

,126.2

)84.2

(72.8

,98.0

)101.6

(82.3

,114.3

)64.9

(51.5

,82.4

)88.9

(61.8

,103.4

)50.1

(36.4

,69.4

)77.8

(46.5

,93.6

)

Hondura

s39.6

(36.6

,42.8

)40.2

(38.2

,42.5

)28.7

(22.4

,36.7

)32.3

(30.1

,35.4

)20.8

(13.5

,32.3

)25.9

(23.6

,29.9

)15.1

(8.0

,28.3

)20.7

(18.5

,25.2

)

India

188.2

(77.2

,100.8

)94.8

(86.9

,114.3

)77.2

(62.9

,94.6

)86.0

(76.4

,130.2

)67.5

(51.4

,88.9

)77.9

(67.2

,146.8

)59.1

(41.8

,83.4

)70.4

(59.1

,162.6

)

Iraq

47.5

(43.0

,52.5

)46.8

(42.3

,51.9

)46.6

(37.1

,58.7

)41.2

(35.4

,47.9

)45.7

(27.7

,74.8

)36.5

(27.7

,45.5

)44.9

(20.7

,96.1

)32.4

(21.4

,43.8

)

Jam

aic

a31.9

(27.8

,36.8

)21.9

(17.2

,30.6

)31.3

(24.1

,40.7

)19.5

(14.1

,35.4

)30.6

(20.5

,45.6

)17.2

(11.3

,42.2

)29.9

(17.5

,51.1

)15.1

(9.0

,51.1

)

Kaza

khst

an

42.9

(37.2

,49.4

)39.1

(34.4

,45.2

)31.0

(22.7

,42.6

)31.0

(22.7

,37.8

)22.3

(13.4

,37.8

)25.3

(13.3

,33.2

)16.1

(7.9

,33.8

)20.7

(7.8

,29.0

)

Kyrg

yzs

tan

51.4

(45.9

,57.4

)51.6

(45.5

,57.7

)42.5

(36.5

,49.4

)42.5

(35.5

,49.5

)35.2

(29.1

,42.6

)35.4

(27.3

,43.3

)29.1

(23.0

,36.8

)29.6

(20.7

,37.9

)

Les

oth

o3

108.4

(99.6

,117.6

)93.1

(85.3

,111.0

)131.8

(108.6

,158.8

)86.3

(76.2

,127.2

)160.2

(117.5

,216.1

)79.8

(68.1

,147.5

)194.7

(127.2

,294.2

)73.9

(61.0

,171.1

)

Mala

wi

155.3

(146.3

,164.7

)168.9

(155.0

,179.4

)125.3

(109.2

,143.6

)150.0

(121.8

,162.5

)101.1

(81.1

,126.0

)133.5

(93.9

,147.5

)81.6

(60.1

,110.7

)118.8

(73.0

,134.1

)

Mald

ives

54.1

(49.6

,58.9

)53.6

(48.9

,58.8

)33.3

(26.9

,40.9

)35.6

(27.0

,42.3

)20.4

(14.1

,29.2

)24.3

(14.3

,30.8

)12.6

(7.4

,21.0

)16.5

(7.6

,22.5

)

Mex

ico1

35.8

(28.9

,44.2

)35.3

(31.8

,40.4

)29.7

(21.5

,41.1

)29.4

(25.7

,35.9

)24.7

(15.9

,38.1

)24.4

(20.8

,31.9

)20.5

(11.8

,35.4

)20.3

(16.7

,28.2

)

Mic

rones

ia46.5

(37.9

,57.4

)47.1

(42.6

,52.3

)41.8

(30.0

,58.7

)42.9

(36.5

,50.5

)37.5

(23.8

,60.0

)39.1

(31.0

,49.0

)33.7

(18.8

,61.5

)35.6

(26.6

,47.5

)

Mold

ova

24.3

(20.7

,28.5

)27.5

(25.3

,29.9

)19.8

(15.8

,24.8

)22.6

(17.7

,25.4

)16.1

(11.9

,21.7

)18.9

(11.8

,22.0

)13.1

(9.0

,19.0

)15.9

(7.8

,19.1

)

Mongolia

61.5

(53.6

,70.5

)57.1

(51.5

,64.8

)45.2

(32.5

,62.4

)39.0

(29.0

,50.2

)33.3

(19.5

,56.5

)27.1

(15.0

,39.7

)24.5

(11.6

,51.0

)18.8

(7.8

,31.1

)

Nep

al

86.1

(81.2

,91.4

)96.5

(86.6

,110.8

)63.1

(56.1

,71.0

)81.8

(65.8

,99.5

)46.2

(38.4

,55.6

)69.8

(49.7

,89.4

)33.9

(26.2

,43.7

)59.5

(37.5

,80.7

)

Pala

u1

9.9

(4.2

,23.2

)3.1

(1.4

,6.6

)6.7

(1.7

,26.2

)1.5

(0.4

,7.0

)4.5

(0.7

,30.0

)0.7

(0.1

,9.4

)3.0

(0.3

,33.9

)0.3

(0.0

,14.1

)

Papua

N.G

.279.6

(58.6

,107.6

)83.3

(72.5

,95.5

)72.9

(43.4

,121.4

)77.5

(63.5

,95.8

)66.8

(32.2

,137.5

)72.2

(55.6

,94.9

)61.2

(23.9

,155.6

)67.2

(48.0

,94.7

)

Per

u41.3

(37.9

,44.8

)48.0

(42.6

,53.0

)27.3

(22.9

,32.4

)37.7

(29.7

,43.3

)18.0

(13.7

,23.5

)29.6

(20.5

,35.4

)11.9

(8.2

,17.1

)23.2

(14.2

,29.0

)

Russ

ian

Fed

.23.9

(23.1

,24.7

)22.0

(17.0

,27.1

)16.9

(16.0

,17.9

)19.4

(13.5

,24.3

)11.5

(9.9

,13.3

)17.3

(11.5

,22.0

)7.8

(6.1

,10.0

)15.5

(9.7

,19.8

)

Table

7:

Est

imate

d/pre

dic

ted

under

-5m

ort

ality

(per

1000)

conti

nued

.Spline-

base

dm

ethod

pro

vid

espoin

tes

tim

ate

sonly

.Loes

s-base

dm

ethod

pro

vid

espoin

tes

tim

ate

sand

95%

unce

rtain

ty

inte

rvals

.1

Spline-

base

des

tim

ate

sdo

not

corr

espond

topublish

edvalu

esdue

topublish

edvalu

esbei

ng

der

ived

from

the

fitt

edin

fant

mort

ality

spline.

2Spline-

base

des

tim

ate

sdiff

ersl

ightl

yfr

om

publish

edvalu

es.

3R

ecen

tpublish

edsp

line-

base

des

tim

ate

sare

const

ant

—th

isis

ignore

dher

e.

26

Countr

y2000.5

2005.5

2010.5

2015.5

Spline

Loes

sSpline

Loes

sSpline

Loes

sSpline

Loes

s

Rw

anda

183.4

(171.1

,196.6

)186.3

(175.4

,198.9

)162.7

(139.9

,188.8

)182.5

(168.1

,203.6

)144.4

(113.3

,184.1

)178.9

(158.0

,210.2

)128.1

(91.4

,179.4

)175.3

(147.6

,216.8

)

S.T

.&

P.

97.2

(81.9

,115.2

)71.1

(54.1

,88.7

)95.9

(77.3

,118.7

)62.3

(32.5

,86.4

)94.6

(72.8

,122.7

)55.5

(19.5

,84.7

)93.3

(68.8

,126.5

)49.5

(11.5

,83.3

)

Sen

egal

132.6

(125.8

,139.5

)132.0

(122.0

,143.1

)118.7

(106.5

,131.9

)119.5

(106.9

,137.6

)106.3

(89.5

,125.7

)107.8

(93.6

,133.2

)95.2

(75.2

,119.8

)97.1

(81.7

,129.9

)

Sie

rra

Leo

ne

276.5

(262.1

,291.1

)273.9

(262.6

,290.5

)271.1

(241.7

,304.0

)264.1

(249.5

,304.3

)265.9

(221.5

,318.7

)254.1

(236.7

,325.0

)260.7

(202.5

,334.7

)244.4

(224.6

,348.7

)

Solo

mon

Is.1

,479.2

56.5

(42.4

,75.2

)68.8

57.8

(38.8

,91.8

)59.9

58.7

(34.9

,117.2

)52.1

59.3

(31.5

,151.5

)

Som

alia

164.8

(146.3

,184.9

)158.6

(143.6

,175.2

)148.5

(123.3

,177.1

)133.8

(112.5

,161.8

)133.8

(103.4

,171.1

)114.7

(82.1

,154.2

)120.6

(86.2

,165.9

)98.5

(59.2

,145.0

)

Syri

a19.9

(15.5

,25.4

)21.7

(16.3

,25.4

)14.5

(8.9

,23.9

)16.1

(10.4

,19.7

)10.7

(5.0

,22.6

)11.9

(6.5

,15.1

)7.8

(2.9

,21.5

)8.8

(4.2

,11.7

)

Tajikis

tan

93.3

(88.0

,98.9

)80.6

(71.2

,90.9

)71.4

(61.4

,83.3

)67.4

(54.0

,81.6

)54.6

(42.1

,71.4

)57.1

(36.1

,76.0

)41.8

(28.8

,61.2

)48.5

(23.8

,69.8

)

Thailand

12.9

(11.0

,15.0

)14.4

(12.4

,16.4

)8.4

(6.0

,11.9

)10.3

(7.9

,12.2

)5.5

(3.2

,9.6

)7.4

(4.9

,9.1

)3.6

(1.7

,7.8

)5.3

(3.0

,6.8

)

Tim

or

Les

te106.8

(93.8

,121.8

)113.7

(98.6

,131.8

)61.3

(36.2

,105.2

)92.5

(63.1

,122.9

)35.2

(13.3

,94.8

)75.4

(37.9

,115.8

)20.2

(4.8

,86.9

)62.5

(22.8

,110.6

)

Togo

124.0

(118.5

,129.7

)129.0

(123.5

,135.6

)110.6

(100.6

,121.3

)119.6

(112.5

,131.5

)98.5

(84.9

,113.9

)110.5

(101.8

,129.6

)87.8

(71.7

,107.1

)102.3

(91.8

,127.0

)

T.&

T.

34.2

(28.8

,40.6

)30.5

(25.1

,38.4

)37.0

(24.2

,56.5

)28.5

(22.1

,42.9

)39.9

(19.8

,80.7

)26.6

(19.2

,49.4

)43.1

(16.1

,115.8

)24.8

(16.4

,55.8

)

Turk

men

ista

n70.6

(66.1

,75.4

)64.2

(55.8

,74.0

)54.1

(45.1

,64.4

)42.2

(33.0

,52.7

)41.4

(30.3

,55.7

)28.8

(16.1

,40.2

)31.7

(20.4

,48.5

)19.8

(7.7

,30.5

)

Tuvalu

135.7

(17.7

,70.6

)34.9

(17.2

,69.0

)30.3

(12.2

,72.5

)29.2

(11.2

,73.7

)25.7

(8.4

,75.0

)24.5

(7.5

,78.3

)21.8

(5.8

,77.9

)20.6

(4.8

,82.7

)

Ukra

ine

22.8

(20.4

,25.3

)22.3

(20.4

,25.3

)23.6

(17.3

,31.7

)20.6

(18.2

,23.3

)24.4

(14.0

,41.8

)18.8

(14.8

,21.7

)25.2

(11.2

,55.4

)17.2

(11.7

,20.7

)

U.A

.E

.10.3

(9.0

,11.7

)9.5

(7.3

,11.8

)8.5

(7.2

,10.1

)6.9

(4.9

,9.1

)7.1

(5.6

,8.9

)4.9

(3.2

,7.2

)5.8

(4.3

,7.9

)3.5

(2.1

,5.8

)

Uru

guay1

16.0

(14.6

,17.5

)19.5

(17.2

,25.1

)16.2

(14.0

,18.9

)16.4

(14.2

,22.4

)16.5

(11.6

,23.4

)13.8

(11.6

,20.0

)16.8

(9.6

,29.3

)11.5

(9.5

,17.7

)

Uzb

ekis

tan

62.3

(57.7

,67.3

)55.9

(49.2

,66.0

)46.0

(36.7

,58.2

)46.1

(40.9

,51.3

)34.0

(22.4

,52.5

)36.9

(31.0

,43.6

)25.1

(13.7

,47.5

)30.0

(21.8

,38.0

)

Ven

ezuel

a24.5

(22.3

,26.8

)25.6

(23.6

,29.3

)21.3

(18.5

,24.2

)22.1

(19.9

,28.1

)18.4

(15.3

,22.0

)19.1

(16.8

,27.2

)16.0

(12.7

,19.9

)16.5

(14.1

,26.0

)

Zim

babw

e3135.3

(105.4

,173.4

)73.0

(64.9

,85.0

)185.5

(123.2

,279.4

)68.5

(58.6

,88.2

)254.2

(143.6

,448.8

)63.9

(52.3

,92.6

)348.5

(167.0

,721.6

)59.5

(46.8

,96.0

)

Table

8:

Est

imate

d/pre

dic

ted

under

-5m

ort

ality

(per

1000)

conti

nued

.Spline-

base

dm

ethod

pro

vid

espoin

tes

tim

ate

sonly

.Loes

s-base

dm

ethod

pro

vid

espoin

tes

tim

ate

sand

95%

unce

rtain

ty

inte

rvals

.1

Spline-

base

des

tim

ate

sdo

not

corr

espond

topublish

edvalu

esdue

topublish

edvalu

esbei

ng

der

ived

from

the

fitt

edin

fant

mort

ality

spline.

2Spline-

base

des

tim

ate

sdiff

ersl

ightl

yfr

om

publish

edvalu

es.

3R

ecen

tpublish

edsp

line-

base

des

tim

ate

sare

const

ant

—th

isis

ignore

dher

e.4

Insu

ffici

ent

data

poin

tsw

ith

non-z

ero

wei

ghts

for

calc

ula

tion

ofsp

line-

base

dunce

rtain

tyin

terv

al.

27

1960 1970 1980 1990 2000 2010

050

100

150

200

250

Year

Und

er−

5 m

orta

lity

(per

100

0)

1960 1970 1980 1990 2000 2010

050

100

150

200

250

Year

Und

er−

5 m

orta

lity

(per

100

0)Fig. 23: Fitted spline curve (left-hand plot; dashed vertical lines indicate knot locations, dashed curves represent uncertainty intervals)

and loess curve (right-hand plot; all data are non-vital registration data, dashed curves represent uncertainty intervals) for Papua New

Guinea.

narrower in others. However, for every year and every country (except Zimbabwe) the spline- and loess-based uncer-

tainty intervals overlap to some extent.

Fig. 30–37 show the range of the loess-based estimated under-5 mortality uncertainty interval plotted against the

range of the spline-based estimated uncertainty interval for each year separately. Countries where the range of the

estimated uncertainty interval using one approach is 20–33.3% wider than the mean of the ranges of the estimated

uncertainty intervals for the two different approaches are highlighted with solid markers. This corresponds to the

wider estimated uncertainty interval being 1.5–2 times wider than the narrower. Countries where the range of the

estimated uncertainty interval using one approach is more than 33.3% wider than the mean of the ranges of the

estimated uncertainty intervals for the two different approaches are labelled. This corresponds to the wider estimated

uncertainty interval being more than twice the width of the narrower.

It can be seen that in countries with either no ad-hoc weight adjustments or merely a down-weighting of the vital

registration data (Fig. 30, Fig. 32, Fig. 34 and Fig. 36) the ranges of the estimated under-5 mortality uncertainty

intervals under the two different approaches are perhaps generally more similar than in countries with more complex

ad-hoc weight adjustments (Fig. 31, Fig. 33, Fig. 35 and Fig. 37), though this is not so marked as when examining the

estimates/predictions in Section 4. In both countries with no ad-hoc weight adjustments or merely a down-weighting

of the vital registration data and countries with more complex ad-hoc weight adjustments there appears to be a

reasonably similar proportion of countries where the spline-based method provides the wider range of the estimated

uncertainty interval and where the loess-based method provides the wider range. Also, within each category of ad-hoc

weight adjustment, the ranges of the estimated uncertainty intervals under the two different approaches tend to get

somewhat less similar as time progresses.

Table 9 summarises the observations from Fig. 30–37 by tabulating, for each category of ad-hoc weight adjustment

28

010

2030

4050

Und

er−

5 m

orta

lity

(per

100

0)

Arm

enia

Bel

ize

Col

ombi

a

Fiji

Hon

dura

s

Mex

ico

Mol

dova

Per

u



intervals displayed as bars. Red points are spline-based estimates/predictions with uncertainty intervals displayed as bars. For each country

the estimates/predictions correspond, from left to right, to the years 2000.5, 2005.5, 2010.5 and 2015.5.

050

100

150

Und

er−

5 m

orta

lity

(per

100

0)

Cam

bodi

a

Gam

bia

Geo

rgia

Mal

dive

s

Mic

rone

sia

Tim

or L

este

Tur

kmen

ista

n

Tuv

alu





29

010

020

030

040

0U

nder

−5

mor

talit

y (p

er 1

000)

Bur

undi

C. A

f. R

ep.

Con

go

Gui

nea

Gui

nea

Bis

sau

Rw

anda

Sen

egal

Sie

rra

Leon

e

Som

alia





010

2030

4050

Und

er−

5 m

orta

lity

(per

100

0)

Bel

arus

Bra

zil

Chi

na

Kaz

akhs

tan

Pal

au

Rus

sian

Fed

.

Syr

ia

Tha

iland

U. A

. E.

Uru

guay

Ven

ezue

la



estimates/predictions with uncertainty intervals displayed as bars. For each country the estimates/predictions correspond, from left to

right, to the years 2000.5, 2005.5, 2010.5 and 2015.5.

30

050

100

150

Und

er−

5 m

orta

lity

(per

100

0)

Egy

pt

Hai

ti

Iraq

Jam

aica

Kyr

gyzs

tan

Mon

golia

Nep

al

S. T

. & P

.

T. &

T.

Taj

ikis

tan

Ukr

aine

Uzb

ekis

tan





010

020

030

040

0U

nder

−5

mor

talit

y (p

er 1

000)

Bur

kina

F.

Cha

d

Cot

e d’

Iv.

Eth

iopi

a

Gha

na

Indi

a

Leso

tho

Mal

awi

Pap

ua N

. G.

Sol

omon

Is.

Tog

o

Zim

babw

e





31

Belize

MicronesiaFiji

Georgia

010

2030

4050

Loes

s−ba

sed

appr

oach


Fig. 30: Comparison of loess- and spline-based uncertainty interval ranges for estimates of under-5 mortality in the year 2000.5 in countries

with either no ad-hoc weight adjustments or merely a down-weighting of the vital registration data. Countries where the range of the

estimated under-5 mortality uncertainty interval using one approach is 20–33.3% wider than the mean of the ranges of the estimated

under-5 mortality uncertainty intervals for the two different approaches are highlighted with solid markers. Countries where the range of

the estimated under-5 mortality uncertainty interval using one approach is more than 33.3% wider than the mean of the ranges of the

estimated under-5 mortality uncertainty intervals for the two different approaches are labelled.

Palau

Papua N. G.

Belarus

Nepal

Russian Fed.Uruguay

010

2030

4050

Loes

s−ba

sed

appr

oach



with more complex ad-hoc weight adjustments. Countries where the range of the estimated under-5 mortality uncertainty interval using

one approach is 20–33.3% wider than the mean of the ranges of the estimated under-5 mortality uncertainty intervals for the two different

approaches are highlighted with solid markers. Countries where the range of the estimated under-5 mortality uncertainty interval using

one approach is more than 33.3% wider than the mean of the ranges of the estimated under-5 mortality uncertainty intervals for the two

different approaches are labelled. Zimbabwe is excluded to aid clarity.

32

BelizeHonduras

Micronesia

Fiji

Georgia

020

4060

80Lo

ess−

base

d ap

proa

ch

0 20 40 60 80Spline−based approach







Palau

Papua N. G.

Ukraine

Uzbekistan

Nepal

Russian Fed.

020

4060

80Lo

ess−

base

d ap

proa

ch








33

Burundi

HondurasMexico

Micronesia

Fiji

Georgia

020

4060

8010

0Lo

ess−

base

d ap

proa

ch








Iraq

Palau

Papua N. G.

Syria

T. & T.

Ukraine

Uzbekistan

India

Nepal

Russian Fed.

020

4060

8010

0Lo

ess−

base

d ap

proa

ch








34

Burundi

HondurasMexico

Micronesia

Congo

Fiji

Georgia

050

100

150

200

Loes

s−ba

sed

appr

oach








IraqPalau

Papua N. G.

Syria

T. & T.

UkraineUruguayUzbekistan

China

India

Nepal

Russian Fed.

050

100

150

200

Loes

s−ba

sed

appr

oach



with more complex ad-hoc weight adjustments. Countries where the uncertainty interval range estimated under one approach is more than

50% wider than that estimated under the other approach are highlighted with solid markers. Countries where the uncertainty interval

range estimated under one approach is more than 100% wider than that estimated under the other approach are labelled. Zimbabwe is

excluded to aid clarity.

35

and each year separately, the differences between the ranges of the spline- and loess-based estimated under-5 mortality

uncertainty intervals.

For countries with no ad-hoc weight adjustments or merely a down-weighting of the vital registration data, in 2000

40% of estimated uncertainty intervals have ranges within 10% of the mean and 64% are within 20%. Approximately

as many countries have a wider spline-based estimated uncertainty interval as have a wider loess-based estimated

uncertainty interval. As time progresses, estimated uncertainty interval ranges get a little further from the mean. By

2015 only 36% remain within 10% of the mean, with 44% more than 20% from the mean. The proportion of countries

where the loess-based estimated uncertainty interval is narrower than the spline-based estimated uncertainty interval

also increases so that by 2015 almost half the countries have a loess-based estimated uncertainty interval less than

90% the width of the mean. Thus, whilst it may be expected that the loess-based approach would provide wider

uncertainty intervals due it allowing for some ‘model’ uncertainty by considering different α values, this is not seen

here.

For countries with more complex ad-hoc weight adjustments, in 2000 almost 40% of estimated uncertainty intervals

have ranges within 10% of the mean and over 60% are within 20%. Over three quarters of countries have a loess-based

estimated uncertainty interval that is wider than the spline-based estimated uncertainty interval. As time progresses,

estimated uncertainty interval ranges get further from the mean until by 2015 less than 20% are within 10% of the

mean and nearly 60% are more than 20% from the mean. However, by 2015 almost as many countries have a wider

spline-based estimated uncertainty interval as have a wider loess-based estimated uncertainty interval.

Loess-based uncertainty interval range as a % of the

mean of the loess- and spline-based uncertainty interval ranges

YearAd-hoc weighting

<80% 80–90% 90–100% 100–110% 110–120% >120%adjustment?

2000.5None/VR only 4 (16%) 4 (16%) 6 (24%) 4 (16%) 2 (8%) 5 (20%)

More complex 3 (9%) 2 (6%) 3 (9%) 10 (29%) 6 (18%) 10 (29%)

2005.5None/VR only 7 (28%) 4 (16%) 5 (20%) 5 (20%) 1 (4%) 3 (12%)

More complex 8 (24%) 5 (15%) 3 (9%) 5 (15%) 7 (21%) 6 (18%)

2010.5None/VR only 6 (24%) 5 (20%) 4 (16%) 6 (24%) 0 (0%) 4 (16%)

More complex 9 (26%) 7 (21%) 4 (12%) 3 (9%) 4 (12%) 7 (21%)

2015.5None/VR only 7 (28%) 5 (20%) 2 (8%) 7 (28%) 0 (0%) 4 (16%)

More complex 13 (38%) 3 (9%) 4 (12%) 2 (6%) 5 (15%) 7 (21%)

Table 9: Loess-based uncertainty interval range as a % of the mean of the loess- and spline-based uncertainty interval ranges. ‘None/VR

only’ is either no ad-hoc weight adjustments or merely a down-weighting of the vital registration data.‘More complex’ is more complex

ad-hoc weight adjustments.

Conclusions

Although it is again perhaps more informative to concentrate on the results obtained in countries where these is little

or no ad-hoc weight adjustment, there is much variability in the spline- and loess-based uncertainty intervals for both

36

categories of ad-hoc weight adjustment. Whilst some spline-based uncertainty intervals correspond almost exactly to

the loess-based equivalent, for others there is only a small overlap. The relative widths of the uncertainty intervals

gives no real suggestion that one approach may generally provide either wider or narrower intervals, although the

differences between the two approaches generally increase a little as time progresses.

This method for incorporating uncertainty into the spline-based approach only deals with uncertainty about the

fitted spline at the time point of interest, whereas in the loess-based approach both the uncertainty about each fitted

loess curve and the uncertainty surrounding the appropriate level of smoothing to use are handled through the pooling

of the simulated random draws across the set of α values. Although this may lead to the expectation of the loess-based

approach providing wider uncertainty intervals, this is not consistently seen here.


The loess-based approach detailed by Murray et al [2] includes a random draw simulation approach to the estimation

of uncertainty intervals for each estimate of childhood mortality. However, an analytic approach analogous to that

detailed for the spline-based approach in Section 5.1.2 could be utilised instead. This is described in Section 5.2.1.

5.2.1 Analytic approach

For a given value of α, consider the basic loess function (2.2),

log(y) = β0 + β1x + β2z + ε.

From this expression, fitted values can be seen to be

log(y) = β0 + β1x + β2z (5.4)

with variance

var(log(y)) =var(β0) + x2var(β1) + z2var(β2) + 2xcov(β0, β1) + 2zcov(β0, β2) + 2xzcov(β1, β2). (5.5)

However, as non-vital registration data are assumed (i.e. z is set to 0 for prediction purposes), (5.4) and (5.5)

reduce to

log(y) = β0 + β1x (5.6)

and

var(log(y)) = var(β0) + x2var(β1) + 2xcov(β0, β1). (5.7)

Thus log(y) can be thought of as following the normal distribution

log(y) ∼ N(β0 + β1x, var(β0) + x2var(β1) + 2xcov(β0, β1)),

or again using the t-distribution analogously if the sample size is small.

If this distribution is combined across the set of α values then an analytic uncertainty interval can be created. The

probability density function (PDF) of log(y) across the set of α values can be found as the mean of the PDFs for each

37

3.4 3.5 3.6 3.7 3.8

02

46

810

Log under−5 mortality (per 1000)

Pro

babi

lity

dens

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nctio

n

3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75

0.0

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Log under−5 mortality (per 1000)

Cum

ulat

ive

dist

ribut

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func

tion

0.025

0.5

0.975

3.47 (32.2) 3.56 (35.3) 3.67 (39.0)

Fig. 38: Probability density functions (left-hand plot; black curves represent probability density functions for individual α values, red

curve is their mean) and cumulative distribution functions (right-hand plot; black curves represent cumulative distribution functions for

individual α values, red curve is their mean, dashed blue line corresponds to a cumulative distribution of 0.5, dashed red lines correspond

to a cumulative distribution of 0.025 and 0.975, values in brackets are transformed back to the original scale) for Armenia in 2000.

α value. Similarly, the cumulative distribution function (CDF) of log(y) across the set of α values can be found as the

mean of the CDFs for each α value. From this overall CDF, the estimate of the mortality can be simply defined as

the value of log(y) at which the CDF is equal to 0.5, and the limits of the uncertainty interval as the values of log(y)

at which the CDF is equal to 0.025 and 0.975.

This approach is illustrated for Armenia in 2000 in Fig. 38. It can be seen that the estimated under-5 mortality

per 1000 is 35.3, with an uncertainty interval from 32.2 to 39.0. These figures are identical to those calculated using

the random draw simulation approach given in Table 6.

More generally, as the random draw simulation approach is simulating random draws from the multivariate normal

distributions defined by the same estimated coefficients, variances and covariances used in (5.6) and (5.7), with sufficient

random draws the estimated uncertainty intervals under the two approaches should be identical.

6 Further extensions and alternative approaches

6.1 Incorporation of sampling variability

Sampling variability affects any data collection where the sample of subjects is not the entirety of the population.

From the Eritrea 2002 Demographic and Health Survey (DHS) Final Report [3]:

The sample of respondents selected in the [DHS] is only one of many samples that could have been selected

from the same population, using the same design and expected size. Each of these samples would yield

results that differ somewhat from the results of the actual sample selected. Sampling errors are a measure

of the variability between all possible samples. Although the degree of variability is not known exactly, it

can be estimated from the survey results.

38

Currently, neither the spline- nor loess-based approach make any explicit attempt to take sampling variability into

account. One way to incorporate sampling variability into the modelling could be to use inverse variance weighting.

Estimated sampling error can be obtained for each survey, then an appropriate scaled sampling error calculated for

each derived value of child mortality from the survey. Vital registration and census data can be assumed to have

minimal random sampling error. Both the spline- and loess-based approaches need little modification to incorporate

sampling variability in this way. The weights in the weighted least squares regressions would simply become the

product of the existing weights and the inverse of the sampling variance for each data point. Incorporating sampling

variability should, on average, reduce uncertainty.

As an example, predicted under-5 mortality in 2015 using the loess-based approach is recalculated, using inverse

variance weighting to incorporate sampling variability, for three countries (Congo, Eritrea and Turkmenistan). The

results are presented in Table 10. It can be seen that the width of the uncertainty interval is reduced in each instance.

The predicted under-5 mortality is also reduced, though this would not be expected to be generally the case.

Observations Predicted 2015 under-5 mortality (per 1000)

Country VR DHSIgnoring Incorporating Percentage change

sampling variability sampling variability Estimate UI width

Congo 0 12 138 (103, 185) 133 (102, 173) −4% −13%

Eritrea 0 10 43 (26, 62) 41 (28, 55) −5% −25%

Turkmenistan 16 11 49 (33, 73) 38 (23, 58) −22% −13%

Table 10: A comparison of predicted 2015 under-5 mortality using the loess-based approach when sampling variability is ignored and

when sampling variability is incorporated. VR is vital registration data, DHS is Demographic and Health Survey, UI is uncertainty interval.

6.2 Multilevel modelling

Within each country, observed values of child mortality coming from the same surveys are likely to be more highly

correlated than values coming from different studies. However, both the spline- and loess-based approaches to esti-

mating childhood mortality assume independence of the observations. One approach for handling the dependencies

between observations would be to use a multilevel model within each country with individual observations as the level

1 variable and survey as the level 2 variable.

The simplest multilevel approach is a random intercepts model. The general random intercepts spline model is

y = (β0 + ui) + β1x +K∑

k=1

bk(x− κk)+ + ε,

where ui ∼ N(0, σu) is the random intercept for survey i.

As an example, the random intercepts splines model is fitted for three countries (Micronesia, Papua New Guinea and

Armenia) and compared to the fitted conventional spline curve in Fig. 39–41. For Micronesia and Papua New Guinea,

where the within-survey trends in under-5 mortality often differ from the overall trend assuming independence of the

data, the fitted conventional spline and random intercepts spline curves differ greatly. For Armenia, where the within-

survey trends in under-5 mortality (in the surveys with non-zero weighting) are similar the overall trend assuming

independence of the data, the fitted curves differ little.

39

3040

5060

70U

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mor

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000)

1980 1985 1990 1995 2000 2005 2010 2015Year

Conventional spline Random intercepts spline

Fig. 39: Comparison of conventional spline and random intercepts spline approaches for Fed. States of Micronesia.

5010

015

020

0U

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mor

talit

y (p

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000)

1960 1970 1980 1990 2000 2010Year


Fig. 40: Comparison of conventional spline and random intercepts spline approaches for Papua New Guinea.

40

1020

3040

5060

7080

90U

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1975 1980 1985 1990 1995 2000 2005 2010 2015Year


Fig. 41: Comparison of conventional spline and random intercepts spline approaches for Armenia.

7 Summary

The spline- and loess-based approaches as currently implemented both have limitations. Both include some level

of subjectivity, meaning that any estimates are less reproducible and transparent. In the loess-based approach, the

selection of the minimum and maximum values of α to use has an element of subjectivity. Whilst this will not usually

have a major effect on the central point estimate, it may have an important impact on the uncertainty range obtained.

Additional subjectivity is introduced by the manner in which ‘extreme outliers’ are excluded from the analysis. In the

spline-based approach subjectivity arises in the down-weighting of datasets which are deemed to be ‘clearly aberrant’

and in the choosing the ‘more consistent’ of the fitted infant and under-5 mortality curves.

However, the loess-based approach does offer several advantages over the spline-based approach. For example,

under the loess-based approach the estimates lie on a smooth rather than piecewise linear (on the log scale) curve.

Model uncertainty, or at least one aspect of it, is incorporated through the use of different smoothing parameters in

the loess-based approach, whereas in the spline-based approach the weights are fixed, aside from the ad-hoc weight

adjustments. The loess-based approach handles vital registration (VR) data in a more appropriate way, through an

additive parameter which allows VR data to influence the shape of the fitted curve but not the position, if it is believed

that VR coverage is less than 100%. Also, the loess-based approach includes the estimation of an uncertainty interval

whereas the spline-based approach does not currently, although the inclusion of this is a simple extension.

In practice, estimated/predicted child mortality using the spline- and loess-based approaches has been seen to often

be very similar, especially in countries where these is little or no ad-hoc weight adjustment in the spline-based ap-

proach, although discrepancies between the approaches generally increase as time since the last observation increases.

In 2000 there appears to be no systematic difference between the approaches, but by 2015 there is some evidence of

the loess-based approach being more likely to provide slightly greater predicted values. Predicted childhood mortality

is most likely to differ between the two approaches when there is a more recent deviation from a long-term trend in

the data. Incorporating uncertainty into the spline-based approach provides uncertainty intervals which are often rela-

41

tively similar to those obtained under the loess-based approach, though in some countries there are sizeable differences.

Thus there remain several general issues:

• How should sampling variability be incorporated into the estimation process? Through inverse variance weight-

ing?

• How should correlations between data points be incorporated into the estimation process? Through multilevel

modelling?

• How should uncertainty be quantified in the estimation process? Through a random draw simulation approach

or analytic limits?

Appendix: Comparison of the datasets

A comparison of the datasets used with the spline-based approach by UNICEF and with the loess-based approach by

Murray et al [2] (both accessed November 2007) may highlight differences which would cause discrepancies between

estimates of childhood mortality even if the same modelling approach was used.

Comparing the 60 available country datasets used for the spline-based approach (detailed in Section 4.1) with the

equivalent datasets used for the loess-based approach (downloaded from www.healthmetricsandevaluation.org/

mortality.xls) shows that whilst in some cases the datasets appear very similar (for example Brazil, see Fig. 42),

more often there are obvious differences. Sometimes there are data points in the dataset used for the spline-based

approach which are not present in the dataset used for the loess-based approach (for example Ukraine, see Fig. 43)

and sometimes the opposite is the case (for example Mexico, see Fig. 44). For a minority of countries there appears

to be little overlap at all between the two datasets (for example Somalia, see Fig. 45).

Given these differences, it is clear that discrepancies in published childhood mortality estimates under the two

approaches are not entirely due to the modelling approaches utilised.

References

[1] K. Hill, R. Pande, M. Mahy, and G. Jones. Trends in Child Mortality in the Developing World: 1960 to 1996.

UNICEF, New York, 1999.

[2] C. J. Murray, T. Laakso, K. Shibuya, K. Hill, and A. D. Lopez. Can we achieve Millennium Development Goal 4?

New analysis of country trends and forecasts of under-5 mortality to 2015. Lancet, 370(9592):1040–54, 2007.

[3] National Statistics and Evaluation Office (NSEO) [Eritrea] and ORC Macro. Eritrea Demographic and Health

Survey 2002. NSEO and ORC Macro, Calverton, Maryland, USA, 2003.

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Fig. 42: Comparison of the datasets provided for the spline- and loess-based approaches for Brazil.

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Fig. 43: Comparison of the datasets provided for the spline- and loess-based approaches for Ukraine.

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Fig. 44: Comparison of the datasets provided for the spline- and loess-based approaches for Mexico.

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Fig. 45: Comparison of the datasets provided for the spline- and loess-based approaches for Tuvalu.

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