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10 Rates vs. risks

A rate is a the number of events occurring per unit of time

The incidence rate of disease occurrence is the

occurrence of new cases per unit of time in a well-

defined population

e.g. cases of a rare disease in Sweden per year

Incidence rate vs. mortality rate

When death is the event whose incidence we are

measuring, we refer to the mortality rate.

2

Reporting of rates

3

It is common to express rates per 100,000 individuals

e.g. If 32 children develop diabetes in a population of

200,000 children in a year. The incidence is

” 32 per 200,000 persons per year

or

32/200,000person-years = 16/100,000 person-years.

Sometimes other denominators used: e.g.

Per 100 or 1000 or 10,000 individuals

Reporting of rates (you saw in Epi 1)

4

http://seer/cancer.gov/statfacts/html/prost.html

http://seer/cancer.gov/statfacts/html/colorect.html

http://seer/cancer.gov/statfacts/html/pancreas.html

What about Swedish rates ( www.sos.se)?

3

Rates (continued)

Commonly used rates are incidence rate (IR) and mortality rate .

Person time is the summed time the people in the sample have

been observed, before an event.

Rates require observations of incidence in time. Thus, they are

estimated from cohort studies.

IR and MR are usually expressed in terms of number of events

per 1000 (or 10 000 or 100 000) person-years.

Other terms for IR: Incidence density, hazard

timeperson

incidencesof#

PT

IIR

Other terms for Incidence Rate

“instantaneous risk” (or probability);

Incidence density

“hazard” (especially for mortality rates);

“person-time incidence rate”;

a“force of morbidity”.

Related Terms that are not rates:

Cumulative incidence: The proportion of people having

an incidence during the follow up period.

Prevalence: The proportion of people having the

disease at a certain time.

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Calculating Incidence Density

(with exact individual follow-up)

A hypothetical cohort of 12 subjects.

Followed for the period of 5.5 years.

7 withdrawals among non-cases

three (7,8,12) lost to follow-up;

two (3,4) due to death;

two (5,10) due to study termination.

PT = 2.5+3.5+…+1.5 = 26.

ID=5/26=0.192 per (person-) year

or 1.92 per 10 (person-)years.

Example *of Incidence Rates (from Epi I)

* Kesavan et al. Prospective Study of Magnesium and Iron Intake and Pancreatic Cancer

in Men. Am J Epidemiol 2010;171:233–241

How are these

calculated?

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Population-Time Without Individual Data

e.g., some population-based registries.

Person-years computed using the mid-year population.

For rare events, periods of several years may be used.

Need to remove those not at risk (e.g., women for

prostate cancer incidence).

Incidence Density: Remarks

Any fluctuations in the instantaneous rate are obscured and can lead to

misleading conclusions. e.g.,

1000 persons followed for 1 year

100 persons followed for 10 years

produce the same number of person-years. If the average time to

disease onset is 5 years, ID in the first cohort will be lower.

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Confidence interval for a rate….

Before we learn about confidence intervals for rates we need to

learn about a probability distribution called Poisson.

Recall the Binomial Distribution

If an experiment consists of n ”trials” each with probability ”p” of

success, we can find the probability of any numner of events

(0,1,2….n)

The formula for the probability involves n and p

The expected number of successes =np

If n is large and p is small, we can actually calculate the probabilities using

only ”np” in an alternative formula, called the Poisson formula

!

)(x

eeventsxP

x

P(0 events) = e-λ

λ = np

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Example:

Comparison of Binomial and Poisson probabilities

http://www.graphpad.com/quickcalcs/probability1.cfm

Example: 100 trials, each with p=.02

No. successes Prob (Binomial) Prob (Poisson)

0 0.133=(.98)100 0.135 =e-2

1 0.271 0.271

2 0.273 0.271

3 or more 0.323 0.323

Check Poisson tables

( in book or page provided)

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Poisson Distribution

Describes the probability of the number of occurences of an event

in a specified time, provided these events occur

At random

Independently

using only the “expected number” of events in the calculation

Example of Poisson in text book

(slightly modified)

If expected number of admissions at maternity unit per day =6,

we can find the probability of any number

P(X = 3) = 0.0892

P(X < 2) = P(X = 0) + P(X = 1) = .0025 + .0149 = 0.0174

P(X ≤ 14) = 1- (.0009+.0003+.0001) = .9987

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Use the Poisson distribution to assess whether the

following report in Metro (25/10/2011) indicates a

significant rise in infant deaths:

”Plötslig spädsbarnsdöd ökar igen i Sverige. Förra året dog 27 barn,

jmfort med 13 barn 2007”

”Infant mortality may be increasing in Sweden. Last year there were

27 infant deaths compared to 13 in 2007”

The Poisson distribution can also be applied to events that occur

within some area or volume instead of time interval, e.g.

cases of some disease per squared-kilometer,

white cells count in a certain amount of blood

Provided the assumptions of random and independent occurrences

is reasonable

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11 Testing and comparing rates

Mean = λ

Variance = λ

Std. Deviation =

Remark: A peculiar characteristic of the Poisson distribution is that

the mean is equal to the variance. For any given set of observed

count data, the goodness of the Poisson assumption can be

assessed, to some extent, by looking at how close the mean and

the variance are.

Mean and variance of Poisson

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Example: if expected number of patients at an emergency

department is 2 per minute, then the probability distribution

(check that you agree)

x | Probability

-----+------------

0 | .1353

1 | .2706

2 | .2706

3 | .1804

4 | .0902

5 | .0360

6 | .0120

7 | .0034

8 | .0008

9 | .0001

10 | .0000

... | ...

------------------

0.1

.2.3

Pro

ba

bili

ty

0 1 2 3 4 5 6 7 8 9 10

Poisson(lambda = 2)

Quite skewed: but for large expected number , the distribution

becomes approximately normal

Remember: the number of events d has:

Mean = λ

Variance = λ

Std. Deviation =

So we can construct a CI using the mean and SE

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Confidence Interval for a rate

We use the poisson distribution as a model for calculating

confidence intervals for rates. We assume the number of events,

d~Poisson(λ)

T

d

nobservatio oftime-personTotal

eventsofNumberRate

T

Rateevents) of rs.e.(Numberates.e.

T

d

T)(

However, it is better to do the CI on the log-scale

Note this only depends on the number and not on T

6

d

1)( rate logs.e.

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CI for rate example (22.1 from text)

Guatemala study, 57 lower respiratory tractinfections in 873 child-

years i.e., rate of 65.3/1000py.

estimate λ= 65.3, log(λ )=4.179, d=57

95% CI for log rate = 4.179 1.96 (0.132) = 3.919 to 4.438

95% CI for rate = exp( 3.919) to exp(4.438)

= 50.36 to 84.65 per 1000 child years

132.057

11)(log d

SE

Comparing Two Incidence Rates

Assume data from

a cohort study:

Exposed Unexposed Total

Cases I1 I0 I

Person.-

time

PT1 PT0 PT

We get two estimates, one each for exposed and non-exposed

IR0=I0/PT0 and IR1=I1/PT1.

As with risks, we can compare them using the difference or ratio

Incidence rate difference: IRD = IR1 - IR0.

Incidence rate ratio: IRR = IR1 / IR0. (more commonly used)

10

11)SE(ln

IIIRR

Note similarity to

Poisson

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Comparing Two Incidence Densities: Example

Postmenopausal Hormones and Coronary Heart Disease Cohort Study:

Stampfer et al., NEJM (1985).

Involving female nurses:

Hormone use

Yes No Total

CHD 30 60 90

Person-years 54308.7 51477.5 105786.2

IR1=30/54308.7=0.00055; IR0=60/51477.5=0.00116

IRD = IR1 - IR0 = -0.00061

IRR = IR1 / IR0 = 0.474

Comparing two incidence rates (IRR) continued

Hormone use

Yes No Total

CHD 30 60 90

Person-

years

54308.7 51477.5 105786.2

224.060

1

30

1)SE(ln IRR

95% CI for ln IRR: ln(0.474) 1.96 (0.224) = (-1.178, -0.315).

95% CI for IRR: (e-1.178, e-0.315) = (0.308, 0.729).

CI allows us to reject the null hypothesis of no difference.

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12 Standardisation

Incidence rates: Remarks

If applied to the whole cohort/population, sometimes called crude rate.

However, sex, age, race etc. can have substantial influence on the

incidence of disease.

Comparing crude rates for two populations, which differ for example in

age, can be misleading (confounding!).

Therefore, usually standardized rates are compared.

e.g., for cancer, age- and sex-standardized rates are used.

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Standardized Rates

We will introduce standardization using age.

We will assume that our population is stratified by age (i.e., subdivided

into age-groups).

need to define age-groups (e.g., 0-4, 5-9,…).

compute age-specific incidence rates (IR).

Person-time and no. of cases for each age-group are required.

There are two methods of standardization:

Direct;

Indirect.

Standardization

Direct method

Age-specific rates of study population are applied to the age-

distribution of a “standard population”.

Indirect method

Age-specific rates from a reference population are applied to the

study population to obtain expected numbers of events.

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Direct Standardization

So we can report the rate that would have occurred if our population had the same age (and sex) distribution as the standard population

This provides a measure of disease rates in the different populations to be compared, adjusted for age

The idea is to apply the age-specific rates from our population to a ”standard population”

Standard world population

Standard European population

Standard US population

Standard world population (from Statistical Methods for Registries, Chapter 11 of IARC

publication 128)

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Data files of standard populations (World,

European, US, Canadian) available from SEER:

http://seer.cancer.gov/stdpopulations/

(use 1,000,000 rather than 100,000)

Direct standardisation of stomach cancer 1993-1997 in males

in Denmark (Chapter 8, Cancer Incidence in Five Continents Vol.

VIII,, IARC Publications No. 155

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IARC publications available on-line

Previous table from

http://www.iarc.fr/en/publications/pdfs-online/epi/sp155/ci5v8-

chap8.pdf

Another good Reference for standardisation:

http://www.iarc.fr/en/publications/pdfs-online/epi/sp95/sp95-

chap11.pdf

Comments on direct standardized rate (DSR)

If there is no confounding, crude rate is adequate.

DSR by itself is not meaningful – it makes sense only when comparing two or

more populations.

If possible, compare age-specific rates.

The rates should exhibit more or less similar trends (also in the standard).

DSR depends on the choice of the standard population.

The age-distribution of the latter should not be radically different from the compared

populations.

There are several standard populations (e.g., for the world, European, US etc.).

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New WHO standard world population : Implications?

(www.who.int/healthinfo/paper31.pdf )

Indirect Standardization

Direct standardization requires age-specific rates for both (all)

compared populations.

If these are not available, or they are imprecise, the indirect

method is preferred.

Both should lead to similar conclusions; if they do not, the

reason should be investigated.

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Indirect Standardization:

Compare the observed number of events in our population to what would be expected if rates another “reference population” applied

The ratio of observed to expected is called the

Incidence Rate Ratio IRR or

Standardized Incidence Rate SIR

If the ratio is of observed and expected deaths, it is called

Mortality Rate Ratio MRR or Standardized Mortality Rate SMR

Indirect Standardization: example

MRR=458/220.9= 2.07

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Indirect Standardization: examples from

your own work e.g. with register data

Good examples of Direct and Indirect standardization

25.2, 25.3 (text book)

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Comparison of Directly Standardized Rates:

If we have two standardized rates, we may want to compare them. Assume we have DSR1 and DSR2. It is usual to compute the ratio DSR1/DSR2, And obtain a confidence interval for either of these two measures (using statistical software)

Comparison of Indirectly Standardized Rates

D

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Comments on standardization of rates

Standardization is a simple way to remove effect of confounding.

We used age in all examples, but can be extended to more than one

confounder.

Same idea as for relative risk, where we construct estimates of common

(i.e. adjusted) OR (like ORMH).

Importance of standardized rates

From :

Taylor P. Standardized mortality ratios, Int Jour Epi, 2013; 42: 1882-1890

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(continued) Taylor P. Standardized mortality ratios, Int Jour

Epi, 2013; 42: 1882-1890