Statistical Notes, No 6 Revised (March 1995), Exams of Statistics

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Direct Standardization
(Age-Adjusted Death Rates)
Lester R. Curtin, Ph.D. and Richard J. Klein, M.P.H.
Introduction
Most population-based mortality objectives and
subobjectives in Healthy People 2000 are tracked using
age-adjusted rates from the National Vital Statistics System
(appendix table I). The exceptions are deaths from
alcohol-related motor vehicle crashes, all motor vehicle
crashes, and work-related injuries (objectives 4.1, 9.3, and
10.1), which are monitored with crude death rates from other
data systems. In addition, objectives that refer to specific age
groups are tracked with age-specific rather than age-adjusted
rates.
Although the age-adjusted death rate (ADR) is one of
the most frequently used indexes of mortality, there is often
confusion concerning the basic concepts of its construction,
use, and interpretation. Some of the persistent issues include
the appropriateness of the ADR as a summary measure, the
validity of comparisons between ADRs, the method of
calculation, and the appropriateness of alternate summary
measures.
Why use age-adjusted death rates?
The total number of health events (for example, the
number of deaths) occurring in a population is useful for
determining the magnitude of a public health problem.
However, the absolute number of deaths is seldom useful for
comparisons between population groups (for example,
comparing males and females) or for comparing trends.
Assuming equal risk, a larger population group will tend to
generate more events (deaths) than a smaller group simply
because of its size. Therefore, to compare relative differences
in mortality among population groups, or for a given
population group over time, the number of deaths must be
related to the ‘‘population at risk’ of dying to produce death
rates. The population of interest may be the entire population
of an area or a population subgroup (for example, people in
a certain age group).
The simplest death rate is the crude death rate (CDR),
defined as the total number of deaths divided by the midyear
population. CDRs are usually expressed as a rate per 1,000
or 100,000 population. CDRs for individual age cohorts,
called age-specific death rates (ASDRs), are the ratio of the
number of deaths in a given age group to the population of
that age group, again usually expressed per 1,000 or 100,000
population.
To compare the relative health of population groups or
to assess change in mortality over time, two criteria must be
considered. First, rates should relate the number of events to
the population at risk. Second, because many health
outcomes vary by age, the effect of the population’s age
distribution must be taken into account.
Although it does relate the number of events to the
population, the crude rate does not take into account the age
distribution of the population. As such, it is not an
appropriate measure for comparing differences between
population groups or for assessing change in mortality over
time. Because death rates for most diseases generally
increase with age, a population group with a relatively
Number 6—Revised
March 1995
From the CENTERS FOR DISEASE CONTROL AND PREVENTION/ National Center for Health Statistics
U.S. DEPARTMENT OF HEALTH AND HUMAN SERVICES
Public Health Service
Centers for Disease Control and Prevention
National Center for Health Statistics CENTERS FOR DISEASE CONTROL
AND PREVENTION
Revised to clarify computations in tables III and V
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Download Statistical Notes, No 6 Revised (March 1995) and more Exams Statistics in PDF only on Docsity!

Direct Standardization

(Age-Adjusted Death Rates)

Lester R. Curtin, Ph.D. and Richard J. Klein, M.P.H.

Introduction

Most population-based mortality objectives and

subobjectives in Healthy People 2000 are tracked using

age-adjusted rates from the National Vital Statistics System

(appendix table I). The exceptions are deaths from

alcohol-related motor vehicle crashes, all motor vehicle

crashes, and work-related injuries (objectives 4.1, 9.3, and

10.1), which are monitored with crude death rates from other

data systems. In addition, objectives that refer to specific age

groups are tracked with age-specific rather than age-adjusted

rates.

Although the age-adjusted death rate (ADR) is one of

the most frequently used indexes of mortality, there is often

confusion concerning the basic concepts of its construction,

use, and interpretation. Some of the persistent issues include

the appropriateness of the ADR as a summary measure, the

validity of comparisons between ADRs, the method of

calculation, and the appropriateness of alternate summary

measures.

Why use age-adjusted death rates?

The total number of health events (for example, the

number of deaths) occurring in a population is useful for

determining the magnitude of a public health problem.

However, the absolute number of deaths is seldom useful for

comparisons between population groups (for example,

comparing males and females) or for comparing trends.

Assuming equal risk, a larger population group will tend to

generate more events (deaths) than a smaller group simply

because of its size. Therefore, to compare relative differences

in mortality among population groups, or for a given

population group over time, the number of deaths must be

related to the ‘‘population at risk’’ of dying to produce death

rates. The population of interest may be the entire population

of an area or a population subgroup (for example, people in

a certain age group).

The simplest death rate is the crude death rate (CDR),

defined as the total number of deaths divided by the midyear

population. CDRs are usually expressed as a rate per 1,

or 100,000 population. CDRs for individual age cohorts,

called age-specific death rates (ASDRs), are the ratio of the

number of deaths in a given age group to the population of

that age group, again usually expressed per 1,000 or 100,

population.

To compare the relative health of population groups or

to assess change in mortality over time, two criteria must be

considered. First, rates should relate the number of events to

the population at risk. Second, because many health

outcomes vary by age, the effect of the population’s age

distribution must be taken into account.

Although it does relate the number of events to the

population, the crude rate does not take into account the age

distribution of the population. As such, it is not an

appropriate measure for comparing differences between

population groups or for assessing change in mortality over

time. Because death rates for most diseases generally

increase with age, a population group with a relatively

Number 6—Revised

March 1995

From the CENTERS FOR DISEASE CONTROL AND PREVENTION/National Center for Health Statistics

U.S. DEPARTMENT OF HEALTH AND HUMAN SERVICES

Public Health Service

Centers for Disease Control and Prevention

National Center for Health Statistics CENTERS FOR DISEASE CONTROL

AND PREVENTION

Revised to clarify computations in tables III and V

younger age distribution will tend to have fewer total deaths

from a given disease than a comparably sized population

group with an older age distribution. Similarly, even if the

age-specific risks of dying for a group remain unchanged

between two time points, the number of deaths will increase

as the population ages.

As an alternative to crude rates, ASDRs can be used.

The most comprehensive and reliable method of comparing

death rates over time or between different population groups

is to compare individual ASDRs for all age groups of

interest. However, this method often requires an extremely

large number of comparisons and tends to overwhelm both

the investigator and the intended audience.

Because the crude death rate is not appropriate and

ASDRs provide too much detailed information, a summary

measure that controls for a population’s age distribution is

needed. A commonly used measure is the ADR (1,2).

Age-adjusted rates were developed in 1841 for the

analysis of mortality data (3). In the 19th century, mortality

data provided the most useful, and often the only, measure

of the health of a population. About that time, it was

observed that a community could have ASDRs that were

lower than the national average at each age interval, but,

because the community’s population was older, the overall

CDR was higher for the community than for the Nation.

Table A presents a hypothetical comparison to illustrate

this situation.

In each of the two comparably-sized communities in

table A, the ASDRs increase with age; at each age the

age-specific rates are higher for community B than for

community A. Yet, the total (or crude) death rate is lower for

community B. This occurs because community A is an older

population; 60 percent of its citizens are 65 years and over.

In contrast, only 10 percent of community B’s population is

in the oldest age group. Because the death rate is highest in

the oldest age group, there are fewer total deaths in

community B.

Direct standardization

There are two basic methods of standardization, or

age-adjustment; both were introduced in the 19th century.

These two methods have become known as the direct and

indirect methods. (Indirect standardization is discussed in a

later section.) When the direct standardization method is

applied to ASDRs, the resultant summary index is called the

ADR. Two assumptions are made when this index is

computed for a population: The population’s observed

age-specific rates are assumed to be valid, and the age

distribution of the population is assumed to be that of a

standard, or reference, or population.

Table B illustrates the calculation of the ADR using the

hypothetical data from table A. Specific computational

formulae for the ADR are given in the technical appendix.

To calculate the ADR, the standard population and the

age-specific death rate for each age interval are multiplied

and these products are summed. In this example, the total

for community A is 420. This sum is divided by the total

standard population (10,000 in this case) to obtain the ADR.

As with crude rates, the ADR is usually expressed in terms

of a rate per 1,000 or per 100,000 population. Thus, the

ADR for community A is 42 deaths per 1,000 population

and the ADR for community B is 52 per 1,000. Note that,

although the crude rate for community A was larger than that

for community B, the ADR for community A is smaller than

the ADR for community B. This is consistent with each of

Table A. Crude death rate comparison

Age

Community A

Rate^1

Community B

Deaths Population Deaths Population Rate^1

0–34 years.................................... 20 1,000 20 180 6,000 30 35–64 years................................... 120 3,000 40 150 3,000 50 65 years and over............................... 360 6,000 60 70 1,000 70

Total........................................ 500 10,000 50 400 10,000 40

(^1) Per 1,000 population.

Table B. Age-adjusted death rate calculation

Age

Community A Community B

Standard population Rate^1

Rate × population Rate

Rate × population

0–34 years.................................... 3,000 20 60 30 90 35–64 years................................... 3,000 40 120 50 150 65 years and over............................... 4,000 60 240 70 280

Total........................................ 10,000 42 420 52 520

(^1) Per 1,000 population.

valid. Kitagawa illustrates this situation with an example of

the mortality of white males living in metropolitan counties

compared with those residing in nonmetropolitan counties in

1960 (13). In this case, ASDRs for white males under age 40

were lower in metropolitan counties than in nonmetropolitan

counties. After age 40, the reverse was true. A summary

index, such as the ADR, does not adequately describe the

mortality differentials in the two groups. In cases such as

these, the ADR is an imprecise indicator of mortality; the

age-specific comparisons would be a better choice.

Indirect standardization

Because of concerns with the use of ADR, some

mortality analysts prefer indirect standardized rates. Indirect

standardization is generally thought of as an approximation

to direct standardization. That is, when data needed to

compute a direct measure (e.g., ASDRs) are not available,

there may still be enough information to compute an

indirectly standardized measure. However, indirect

standardization has intrinsic value and should be considered

on its own merits, not solely as an approximation to direct

standardization (14,15).

For indirect standardization, a standard set of

age-specific death rates are assumed to apply to the observed

population. For example, the age-specific U.S. death rates

could be applied to the age-specific local area population.

This technique yields an ‘‘expected’’ number of deaths in a

population, assuming the standard set of ASDRs was

operating in the population.

An indirect adjusted death rate (IADR) can be computed

from the expected number of deaths, but the index most

often used is the ratio of the expected to the actual observed

number of deaths. This ratio is called the standardized

mortality ratio (SMR). The mathematics of indirect

standardization and an example of the calculation of an

SMR are given in the appendix.

Summary

This paper describes some of the issues related to the

computation and use of age-adjusted rates. The following

points were made:

+ The age-adjusted rate is an index measure, the magnitude

of which has no intrinsic value. It should be used for

comparison purposes only.

+ If it is appropriate to use age-adjustment, then the

comparison should not be affected by the selection of a

standard population. Conversely, if the comparison can be

affected by the choice of a standard population, then it is

not appropriate to age-adjust for that comparison.

+ The standard population should not be ‘‘abnormal’’ or

‘‘unnatural’’ when compared to populations under study.

Considering the amount of published material, there are

advantages to using the U.S. standard population.

+ Standardization is not a substitute for the examination of

age-specific rates.

While standardization is most often applied to a series

of age-specific death rates, direct or indirect standardization

can also be applied to variables other than age. For example,

infant mortality rates can be adjusted for birthweight

distribution (16). Age-adjustment can also be used to

monitor other measures of health at the local level, such as

incidence or prevalence of disease.

Throughout the history of the ADR, the utility of the

measure has often come into question. Any summary index,

including direct or indirect standardization, will mask

age-specific differences. Therefore, some authors have

stressed the importance of comparing individual age-specific

rates rather than attempting to summarize differences among

the age-specific rates (17,18). A summary index, however, is

more easily compared than an entire table of age-specific

rates. Thus, the age-adjusted rate continues to be an integral

part of the analysis of mortality trends and differentials.

Accepting this, the need for a summary index must be

balanced with recognition of the limitations of summary

measures.

References

1. Shyrock HS, Siegel JS. The methods and materials of demog-

raphy, vol 2. U.S. Bureau of the Census. Washington: U.S.

Government Printing Office. 1971.

2. Spiegelman M. Introduction to demography. Rev. ed. Cam-

bridge, MA. Harvard University Press. 1968.

3. Neison FGP. On a method recently proposed for conducting

inquiries into the comparative sanatory condition of various

districts. Journal of the Royal Statistical Society of London (now

the Royal Statistical Society), vol 7, pp 40–68. 1844.

4. Kalton G. Standardization: A technique to control for extraneous

variables. Applied Statistics, 17:118–36. 1968.

5. National Center for Health Statistics. Vital Statistics of the

United States, 1989, vol II, mortality, part A. Washington: Public

Health Service. 1992.

6. Johnson R. Proposed new standard population. Proceedings of

the social statistics section, American Statistical Association, pp

7. Feinleib MF, Zarate AO, eds. Reconsidering age adjustment

procedures: Workshop proceedings. National Center for Health

Statistics. Vital Health Stat 4(29). 1992.

8. Spiegelman M, Marks HH. Empirical testing of standards for the

age adjustment of death rates by the direct method. Human

Biology, 38:280–92. 1966.

9. Curtin LR, Maurer J, Rosenberg HM. On the selection of

alternative standards for the age-adjusted death rate: Proceed-

ings of the social statistics section, American Statistical Associa-

tion, pp 218–23. 1980.

10. Wolfenden HH. On the theoretical and practical considerations

underlying the direct and indirect standardization of death rates.

Population Studies, 16:188–90. 1962.

11. Kleinman JC. Age-adjusted mortality indices for small areas:

Applications to health planning. American Journal of Public

Health, 67:834–40. 1977.

12. Fleiss JL. Statistical methods for rates and proportions. John

Wiley and Sons, New York. 1973.

13. Kitagawa EM. Theoretical considerations in the selection of a

mortality index and some empirical comparison. Human Biology,

14. Inskip H, Beral V, Fraser P. Methods for age-adjustment of rates.

Statistics in Medicine 2:455–66. 1983.

15. Tukey JW. Statistical mapping: What should and should not be

plotted. Proceedings of the 1976 workshop on automated cartog-

raphy and epidemiology. National Center for Health Statistics.

DHEW (PHS) 79–1254. 1979.

16. Foster JE, Kleinman JC. Adjusting neonatal mortality rates for

birthweight. National Center for Health Statistics. Vital Health

Stat 2(94). 1982.

17. Woosley TD. Adjusted death rates and other indices of mortality.

Chapter 4 in Vital Statistics Rates in the United States,

1900–1940.Washington: U.S. Government Printing Office. 1959.

18. Elveback LR. Discussion of indexes of mortality and tests of

their statistical significance. Human Biology 38:322–24. 1966.

19. Chiang CL. Standard error of the age-adjusted death rate. U.S.

Department of Health, Education, and Welfare: Vital Statistics

Special Reports 47:271–85. 1961.

20. Keyfitz N. Sampling variance of standardized mortality rates.

Human Biology 38:309–17. 1966.

This appendix presents examples of the computation of

the age-adjusted death rate (ADR), indirect adjusted death

rate (IADR), and the standard mortality ratio (SMR). These

examples demonstrate that each standardized index is a

weighted average of the age-specific rates. For the ADR, the

weights are determined by the standard population. A

discussion of the variability of the ADR is also included.

Suppose the data are aggregated into i = 1, 2, ..., I age

groups. Let:

di = the number of deaths in the i-th age interval, and

p i = the population size in the i-th age interval.

The total number of deaths is

d = (^) ∑ i di

the total population is

p = (^) ∑ i pi

Age specific death rates (ASDRs) are defined as

ASDR =

number of deaths for age interval i

midyear population for age interval i

thus,

ASDR = m i = the death rate in the i-th age interval.

The age-specific death rate is given by

m i = d i / p i

In this form, the death rate ( m ) on a unit basis (i.e., per

person) will be between 0 and 1. ASDRs are usually

expressed as a rate per 1,000 or per 100,000 population. For

example, if there are 10 deaths in an age group that has a

total population of 1,000 persons, the ASDR on a unit basis

is 0.01; per 1,000 it is 10, per 100,000 it is 1,000.

The annual crude death rate is defined as the total

number of deaths over all ages divided by the midyear

population. The crude death rate is then

m = total deaths / total population

Again, it is usually expressed per 1,000 or per 100,

population.

Algebraically, the direct standardized (or age-adjusted)

rate is a weighted average of the age-specific death rates. To

compute the ADR, the standard population is used to deter-

mine a set of weights. For convenience, let

psi = population in age group i in the standard population

and let the standard weights be given by

wsi =

psi

i psi

[NOTE: That in this form 0< wsi <1 and the wsi sum to 1. The

weights are often expressed as a standard million so that w si

sum to 1,000,000.]

Then the ADR is given by

ADR = (^) ∑

i wsi * m^ i

The ASDRs used by NCHS to compute the ADR are

rounded to one decimal place. The weights used by NCHS

are based on the 1940 U.S. population and are called the

‘‘standard million.’’ As the name implies, the standard

million weights sum to one million. The standard million is

shown in table II.

The age-adjusted rates shown in most NCHS

publications and those used to track the Healthy People 2000

objectives are computed using the standard million and

ASDRs in 10-year age groups. A specific calculation for

stroke mortality ( Healthy People 2000 objective 15.2) for

males and females is shown in table III. For illustrative

purposes, the deaths and populations are those of a

hypothetical medium-sized State.

In this example, di = deaths in 10-year age-groups,

m i = 10-year ASDR per 100,000, and wsi = psi /Σ i psi

(weights on a unit basis).

Indirect standardization

For direct standardization, the observed ASDRs and a

standard population are used. For indirect standardization,

the observed population and a standard set of ASDRs are

used. Indirect standardized rates are sometimes calculated

and presented, but more often an SMR is presented. The

indirect standardized rate and SMR are defined as follows:

SMR =

number of observed deaths

number of expected deaths

or

SMR =

i di

i m^ si * pi

where m si are the standard ASDRs on a unit basis. The indirect

adjusted death rate is then

IADR = SMR * (crude rate for the standard population)

or

IADR =

M (^) s * (^) ∑ i di

i m^ si * pi

For the data in table A, the age-specific rates for community A

can be used as the standard rates. Then the crude rate and the

indirect standardized rates are the same for community A (

per 1,000); the SMR for community A is 1. The calculation for

the IADR and SMR for community B is shown in table IV.

Then

SMR =

= 1.33 IADR = 1.33 * 50 = 67

Each index has advantages and disadvantages. Indirect rates

can be used when age-specific numbers of deaths are not

available or when the number of deaths is small. Also, the

indirect standardized rates have smaller variability. However,

indirect rates may not be comparable across areas; they can be

used for comparisons of areas only if age and area effects are

independent.

Variability

The numbers of deaths reported for a community

represent complete counts. As such, numbers of deaths and

death rates are not subject to sampling error, although they

are subject to errors in the registration process. However,

when used for analytic purposes, such as comparison of rates

over time or for different areas, the number of events that

actually occurred may be considered as one of a large series

of possible results that could have arisen under the same

circumstances. The probable range of values may be

estimated from the actual figures according to certain

statistical assumptions. From these assumptions the standard

errors of ASDRs and ADRs can be calculated (19,20).

The variance of an ASDR is assumed to be determined

by a binomial distribution. This assumes that the chance of

dying in an age interval is constant within the age interval

and that everyone has the same chance of dying; this is an

assumption of homogeneity. Under the homogeneity

assumption, the variance of an age-specific rate on a unit

basis is given by

Variance ( m i ) =

m i * (1 – m i )

pi

NOTE: If the rates are per 100,000, then the (1– m i ) term

becomes (100,000– m i ).

The variance of an ADR can be defined as weighted average

of the variances of the ASDRs. Under the assumption that

ASDR are independent, or that the covariance ( mi , m j ) = 0 for

i not equal to j , then the standardized rates are simply

weighted averages of the age-specific rates, and the variance is

given by

Variance (ADR) = (^) ∑ i

w i^2 * Variance ( m i )

An example of the calculation of the variances of the age-

adjusted rates for males and females computed in table III is

shown in table V. In order to obtain meaningful variances, the

number of deaths and the populations for males and females

used in table V are those of the same hypothetical medium-

sized State used in table III. The variance of an age-adjusted

death rate for the entire U.S. population is extremely small.

Confidence intervals can be formed using the variances. If

the number of deaths is large enough (again, a rough principle

is 25 or more) then a 95-percent confidence interval for the

age-adjusted rate is formed as:

(ADR – 1.96 * √ var (mi ) , ADR + 1.96 * √ var (m (^) i ) )

For the example in table III, the 95-percent confidence interval

for the ADR for strokes for males for the hypothetical State is

[33.0 – (1.96 * 1.05) , 33.0 + (1.96 * 1.05)]

or

For females, the 95-percent confidence interval is

[27.9 – (1.96 * 0.80) , 27.9 + (1.96 * 0.80)]

or

Because the 95-percent confidence intervals do not overlap,

the difference between the ADRs for males and females is

statistically significant at the 0.05 level.

Some care has to be exercised when both the rates are low

and the number of deaths is small. In this case, the above

formula can result in the lower bound being less than zero, and

death rates cannot be negative. One way to avoid this is to use

log transformations of the rates; another way is to use a

discrete distribution function. These methods are beyond the

scope of this report. The NCHS Office of Research and

Methodology can provide assistance on computing variances

for ADRs based on small frequencies.

When ASDRs are based on sufficiently small numbers, a

simple Poisson approximation may be used to compute the

variance of the ASDRs, as follows:

m i^2

d i

where m i is the ASDR on a unit basis for the i-th age group,

and di is the corresponding number of deaths

In these cases, the resultant Poisson age-specific variances can be

used in the formulae described in this section to compute the

variance of the ADR. More information on random variation

can be found in the annual vital statistics volumes (5).

Table II. Standard million age distribution used

to adjust death rates to the U.S. population

in 1940

Age

Standard million (psi)

Unit basis (psi/ 1,000,000)

All ages............. 1,000,000 1.

Under 1 year.......... 15,343 0. 1–4 years............ 64,718 0. 5–14 years............ 170,355 0. 15–24 years........... 181,677 0. 25–34 years........... 162,066 0. 35–44 years........... 139,237 0. 45–54 years........... 117,811 0. 55–64 years........... 80,294 0. 65–74 years........... 48,426 0. 75–84 years........... 17,303 0. 85 years and over....... 2,770 0.

Revised Table V

Table V. Variance calculation for the age-adjusted death rate calculation for stroke (ICD-9 430–438) for males

and females: Hypothetical medium-sized State

Age di

pi (thousands)

mi (per 100,000)

wsi (unit basis) Var (mi) w^2 i* Var(mi)

Males

Under 1 year...................... 1 38 2.6 0.015343 6.84193 0. 1–4 years........................ – 150 – 0.064718 0.00000 0. 5–14 years........................ 1 322 0.3 0.170355 0.09317 0. 15–24 years....................... 2 344 0.6 0.181677 0.17442 0. 25–34 years....................... 8 443 1.8 0.162066 0.40631 0. 35–44 years....................... 21 379 5.5 0.139237 1.45111 0. 45–54 years....................... 46 256 18.0 0.117811 7.02998 0. 55–64 years....................... 103 189 54.5 0.080294 28.82026 0. 65–74 years....................... 254 136 186.8 0.048426 137.09637 0. 75–84 years....................... 371 57 650.9 0.017303 1134.49700 0. 85 years and over................... 212 12 1,766.7 0.002770 14462.39759 0.

Variance of age-adjusted death rate = Sumw^21 * Var (mi ) = 1. Standard error of ADR = Square root of variance = 1.

Females

Under 1 year...................... 1 36 2.8 0.015343 7.77756 0. 1–4 years........................ – 143 – 0.064718 0.00000 0. 5–14 years........................ 1 309 0.3 0.170355 0.09709 0. 15–24 years....................... 2 337 0.6 0.181677 0.17804 0. 25–34 years....................... 7 458 1.5 0.162066 0.32751 0. 35–44 years....................... 21 401 5.2 0.139237 1.29669 0. 45–54 years....................... 41 267 15.4 0.117811 5.76690 0. 55–64 years....................... 83 208 39.9 0.080294 19.17504 0. 65–74 years....................... 245 178 137.6 0.048426 77.19700 0. 75–84 years....................... 553 100 553.0 0.017303 549.94191 0. 85 years and over................... 661 34 1,944.1 0.002770 5606.77868 0.

Variance of age-adjusted death rate = Sumw^21 * Var (mi ) = 0. Standard error of ADR = Square root of variance = 0.

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