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Age-Period-Cohort Models in Cancer Surveillance Research:

Ready for Prime Time?

Philip S. Rosenberg, PhD1 and William F. Anderson, MD, MPH2,*

1Division of Cancer, Epidemiology & Genetics, National Cancer Institute

2Division of Cancer Epidemiology and Genetics, National Cancer Institute

Abstract

Standard descriptive methods for the analysis of cancer surveillance data include canonical plots

based on the lexis diagram, directly age-standardized rates (ASR), estimated annual percentage

change (EAPC), and joinpoint regression. The age-period-cohort (APC) model has been used less

often. Here, we argue that it merits much broader use. Firstly, we describe close connections

between estimable functions of the model parameters and standard quantities such as the ASR,

EAPC, and joinpoints. Estimable functions have the added value of being fully adjusted for period

and cohort effects, and generally more precise. Secondly, the APC model provides the descriptive

epidemiologist with powerful new tools, including rigorous statistical methods for comparative

analyses and the ability to project the future burden of cancer. We illustrate these principles using

invasive female breast cancer incidence in the United States, but these concepts apply equally well

to other cancer sites for incidence or mortality.

Keywords

Cancer Surveillance Research; age standardized rates (ASR); estimated annual percentage change

(EAPC); joinpoint regression; Age-period-cohort (APC) model; APC estimable parameters; APC

linear trends; APC deviations; APC drifts; APC fitted age at onset curve

Introduction

Cancer incidence and mortality rates are closely monitored to track the burden of cancer and

its evolution in populations (1-4), provide etiological clues (5-11), reveal disparity (12-14),

and gauge the dissemination of screening modalities (15-17) and therapeutic innovations

(18, 19). A standard “toolbox” of graphical and quantitative methods has evolved to handle

the needs of cancer surveillance researchers. Perhaps the most widely used methods include

classical descriptive plots based on the lexis diagram (20-22), directly age-standardized rates

(ASR) (23), estimated annual percentage change (EAPC) (24), and the joinpoint regression

method (25). The underlying philosophy is agnostic and empirical; hence standard tools are

particularly well suited to descriptive, exploratory, and hypothesis-generating studies.

At the same time, the age-period-cohort (APC) model has been developed in the statistics

literature as a mathematical counter-point to purely descriptive approaches (20, 26-33). The

APC model is based on fundamental generalized linear model theory (34); in principle, it

allows the descriptive epidemiologist to both generate and test hypotheses. However,

*Corresponding Author: William F Anderson, Division of Cancer Epidemiology and Genetics, National Cancer Institute, EPS 8036,

6120 Executive Blvd, Rockville, MD, 20852-7244, United States wanderso@mail.nih.gov.

DISCLAIMER: None of the co-authors has a financial conflict of interest that would have affected this research.

NIH Public Access

Author Manuscript

Cancer Epidemiol Biomarkers Prev. Author manuscript; available in PMC 2012 July 1.

Published in final edited form as:

Cancer Epidemiol Biomarkers Prev. 2011 July ; 20(7): 1263–1268. doi:10.1158/1055-9965.EPI-11-0421.

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although the APC model is generally accepted, our sense is it remains more of a niche

methodology than an integral part of mainstream practice.

We believe two misunderstandings have slowed the uptake of the APC approach. Firstly,

there are concerns about the “identifiability problem” of the APC model (27, 28). Secondly,

close connections between the classical toolbox and the APC model have not been clearly

spelled out in the literature. In this commentary, we will attempt to clarify both

misunderstandings and thereby make the case that the APC model merits much wider use.

Data, Methods, and Results

Example: Breast cancer incidence data

We will develop this commentary using as a concrete example the incidence of invasive

female breast cancers in the United States. For this purpose, we obtained age-specific case

and population data from the National Cancer Institute’s Surveillance, Epidemiology, and

End Results 9 Registries Database (SEER9) for the 36-year time period from 1973 through

2008 (November 2010 submission) (35).

In general, for any given cancer and population group, the matrix Y = [Ypa, p = 1, …, P, a =

1,…A] contains the number of cancer diagnoses in calendar period p and age group a, and

the matrix O = [Opa, p = 1, …P, a = 1, …, A] contains the corresponding person-years. The

observed incidence rates per 100,000 person-years are λpa = 105 Ypa/Opa, and the expected

log rates are ρpa = log(E(Ypa)/Opa).

It is instructive to think of the rate matrix in terms of its corresponding Lexis diagram

(Figure 1), which makes visually clear how the diagonals of matrices Y and O, from upper

right to lower left, represent successive birth cohorts indexed by c = p − a + A, from the

oldest (c = 1) to the youngest (c = C ≡ P + A − 1). From this perspective, it becomes clear

that a new cohort enters prospective follow-up with each consecutive calendar period. For

this reason, one can think of a registry as a “cohort of cohorts.” Because cancer registries are

operated in perpetuity, over time, a substantial number of birth cohorts are followed. Our

example includes C = 24 nominal 8-year cohorts born from 1892 through 1984 (referred to

by mid-year of birth).

The APC model: formulation

APC analysis is based on a log-linear model for the expected rates with additive effects for

age, period, and cohort:

(1)

The generic additive effects in equation (1) can be partitioned into linear and non-linear

components (28). There are number of equivalent ways to make this partition while

incorporating the fundamental constraint that c ≡ p − a. Two of the most useful (36) are the

age-period form

(2)

and the age-cohort form

(3)

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Notation and parameters are summarized in Table 1. Importantly, all the parameters in

equations (2) and (3) can be estimated from the data without imposing additional

constraints, and fitted rates from both forms are identical.

There is a close correspondence between APC parameters and estimable functions in Table

1 and fundamental aspects of the data investigated using the standard descriptive toolbox.

Before highlighting some of these connections below, we hopefully can shed further light on

the much discussed identifiability problem.

Identifiability: “problem” or uncertainty principle?

The aspect of identifiability in question concerns whether log-linear trends in rates can

uniquely be attributed to the influences of age, period, or cohort, quantified by parameters

αL, πL, and γL. Mathematically, it has been shown by Holford (28) that one cannot do this

without imposing additional unverifiable assumptions, because the three time scales are co-

linear (cohort equals period minus age, c = p − a). This issue has often implicitly been held

out as a unique and unfortunate limitation of the APC model. In fact, the same issue affects

time-to-event analysis of any cohort study.

To see this, consider the following thought experiment. Suppose one enrolls a cohort of

exchangeable persons of identical age (e.g., the 1956 birth cohort in Figure 1) and follows

them longitudinally over a decade for cancer. At the end of the study, one observes that the

log incidence rate increases linearly with age. It is natural to attribute this trend entirely to

the effects of ageing, and equate the age-associated slope to the value of a parameter αL.

However, suppose one had also assembled an identical cohort of persons of the same age,

but this study had been conducted ten years earlier. It is possible that the age-associated

slopes of the two studies would be very different, if disease-causing exposures out of

experimental control had been increasing or decreasing in prevalence over time. Hence, the

observed age-associated slope actually estimates parameter (αL + πL) or longitudinal age

trend (LAT in Figure 1) (32), where αL is the component of the trend that is attributable to

aging and πL is the component of the trend due to the net impact of unknown and

uncontrollable exposures over successive calendar-periods.

A similar issue affects any cross-sectional analysis. To “control” for the effects of ageing,

suppose one studied in succession over time an event rate in persons of the same age (e.g.,

age group 65-69 years in Figure 1), to estimate the slope of the time-trend πL. By definition,

each successive group in this cross-sectional study was born a year later. Hence, both

unknown factors and factors out of experimental control associated with birth cohort could

also play a role. Therefore, the observed slope over time actually estimates a parameter (πL +

γL) or net drift in Figure 1 (29, 30), where πL is the component of the trend that is

attributable to calendar time and γL is the component of the trend attributable to the

successive cohorts enrolled in the study.

These simple thought experiments, Figure 1, and Table 1 illustrate an important ‘uncertainty

principle’ regarding the measurement of absolute rates in cohorts. Interestingly, this

principle is seldom considered in the context of most epidemiological cohort and case-

control studies, perhaps because these studies have a fairly narrow accrual window and often

focus on relative rates rather than absolute rates. In contrast, this issue is often centralin the

analysis of registry data, because the follow-up has sufficient breadth and depth to reveal

long-term secular trends in the population associated with age, period, and cohort. Indeed, a

unique role of registry studies is to identify and quantify such trends, thereby providing

direction and guidance regarding the needs for targeted analytical studies.

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Estimable functions: separating signal from noise

The APC model provides a unique set of best-fitting log incidence rates, ρ̂pa or equivalently

ρ̂ca, obtained by plugging in maximum likelihood estimators into equations (2) or (3),

respectively. The corresponding variances are readily calculated. In our experience the fitted

rates have an appealing amount of smoothing, and we use them routinely in our studies

(36-45), especially for rare cancer outcomes. Experience suggests that for “moderate” sized

rate matrices (in terms of A and P), the APC model smoothes the data conservatively, about

as much as a 3-point moving average, yielding around a 40-60% reduction in the width of

the confidence intervals. Of course, the precise amount of noise reduction depends on a

number of technical details including whether over-dispersion is present or accounted for.

This application of the APC model is illustrated in Figure 2 for the breast cancer data. The

age-standardized rates (ASRs) over time calculated using the observed rates are nearly

identical to the ASRs calculated using the APC fitted rates. However, the point-wise

confidence intervals for the fitted rates are substantially narrower, by around 40% averaged

over the 10-year time period.

Estimable functions: connections to the classical approaches

The APC parameter called the net drift (Table 1 and equations (2) and (3)) estimates the

same quantity as the EAPC of the ASR, i.e. the overall long-term secular trend. The point

estimates for these quantities are almost identical for the breast cancer data in Figure 2; net

drift = 0.83% per year (95% CI: 0.78 to 0.85%/yr) and EAPC = 0.78% percent per year

(0.18 to 1.39%/yr). However, for this example, the estimated confidence bands are much

narrower for the net drift.

We introduced a novel estimable function called the fitted age-at-onset curve to summarize

the longitudinal (i.e. cohort-specific) age-associated natural history (Table 1 and figure 3)

(46). By construction, the fitted curve extrapolates from observed age-specific rates over the

full range of birth cohorts to estimate past, current, and future rates for the referent cohort,

e.g., the 1932 cohort in this example. The fitted age-at-onset curve provides a longitudinal

age-specific rate curve that is adjusted for both calendar-period and birth-cohort effects. We

view it as an improved version of the cross-sectional age-specific rate curve, improved

because the cross-sectional curve is not adjusted for period and cohort effects (47). The

fitted curve has proven very useful in practice (38-40, 42-44, 46, 48).

Finally, period deviations in the APC model (Table 1) identify changes over time; such

change points are often analyzed non-parametrically using joinpoint regression methods

(25). Similarly, cohort deviations can provide an explanation for joinpoint patterns in age-

specific rates over time.

APC analysis: beyond the basics

There are many useful extensions to the basic APC model. Estimable functions are

amenable to formal hypothesis tests (29, 30). Parameters associated with age, period, and

cohort can be smoothed (49). Parametric assumptions about the shape of the age incidence

curve derived from mathematical models of carcinogenesis can be incorporated (50). Other

extensions have included parametric (33) and nonparametric (51, 52) assessments of

changes in period and cohort deviations, and simultaneous modeling of a moderate or large

number of strata, such as geographic areas, using Bayes and Empirical Bayes methods (53).

Recently, we developed novel methods to compare age-related natural histories and time-

trends between distinct event rates assuming that separate APC models hold for each (36).

Using this approach one can formally contrast the incidence of a given tumor such as breast

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cancer in two populations, say Black versus White women (46), or the incidence of two

tumor subtypes in the same population, say, ER positive versus ER negative breast cancers

((46), supplemental Figure). We demonstrated that two event rates are proportional over age,

period, or cohort if and only if certain sets of APC parameters are all equal across the

respective event-specific models (36). We also developed corresponding tests of

proportionality and estimators of rate ratios.

A number of authors have forecast future cancer rates using the APC model (54-58).

Projections quantify the future implications of current trends, for example, the impact of a

net drift of 1% versus 2% over time, or the future impact of recent changes in birth cohort

patterns.

Discussion

Successful technological evolution builds on effective design. This is just as true for

statistical methods as for computers and cellular phones. We have argued here that the APC

model provides a useful evolutionary extension to the standard armamentarium of methods

available to the descriptive epidemiologist. The APC model is not a replacement for existing

methods, which are popular and successful. Rather, it provides a refined means of estimating

the same quantities, while also adding useful new capabilities, such as formal methods for

comparing two sets of rates or projecting the future cancer burden.

Using the APC model, cancer registry data can be analyzed in the same spirit as any other

epidemiological cohort using the same concepts, such as proportional hazards, confounding,

and effect modification/interaction. Importantly, because cancer registries follows a cohort

of cohorts, analysis of registry data can reveal fundamental changes in population rates that

are not usually discernable in standard cohort or case-control studies.

Currently, software for APC analysis is available only through fairly specialized packages

(SAS, R, Matlab). Development of good stand-alone software, in addition to education and

training, are needed if the full potential of the APC model is to be exploited by descriptive

epidemiologists.

Acknowledgments

This research was supported by the Intramural Research Program of the National Institutes of Health, National

Cancer Institute. All of the authors had full access to all of the data in the study and take responsibility for the

integrity of the data and the accuracy of the data analysis.

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Figure 1.

Rate matrix or Lexis diagram (20-22) for invasive female breast cancer. Data from the

National Cancer Institute’s Surveillance, Epidemiology, and End Results 9 Registries

Database (SEER 9) for cases diagnosed between 1973 through 2008 (35). Sixteen 4-year age

groups (21-24, 25-29, …, 81-84 years) and nine 4-year time periods (1973-1976, 1977-1980,

…, 2005-2008) span 24 partially overlapping 8-yr birth cohorts, with age groups in the

columns, time-periods in the rows, and birth-cohorts along the diagonals. Conditioned upon

time period (e.g., 1973-1977), the cross sectional age trend (CAT) cuts across increasing age

groups and decreasing birth cohorts, CAT = αL − γL). Conditioned upon birth cohort (e.g.,

1956), the longitudinal age trend (LAT) cuts across increasing age groups and time periods,

LAT = αL + πL). Conditioned upon age group (e.g., 65-69 years), the net drift cuts across

increasing time periods and birth cohorts, Net drif t = (πL + γL).

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Figure 2.

Age standardized rates (ASRs, 2000 standard US population) for invasive female breast

cancer. Data from the National Cancer Institute’s SEER 9 Database. ASRs calculated using

observed rates (grey) and age-period-cohort fitted rates (red). Point estimates (solid circles)

and 95% confidence limits (shaded areas) are shown.

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Figure 3.

Cohort-specific age-specific incidence rates for invasive female breast cancer. Data from the

National Cancer Institute’s SEER 9 Database, stratified by 8-year birth cohorts. The age-

period-cohort (APC) fitted age at onset curve (red line and shaded 95% confidence intervals)

‘stitches together’ the experience of 11 cohorts observed over staggered age intervals. By

construction, the fitted (or longitudinal) curve is centered on the referent-cohort; e.g., the

1932 birth-cohort. Because these breast cancer incidence rates are increasing over time with

a net drift of 0.83% per year (figure 2 and text), the fitted rates for the referent-cohort lie

below the observed rates for the younger cohorts and above the rates for the older cohorts. In

other words, with rising incidence rates from older to younger generations (figure 2 and

text), the referent-cohort rates on average are lower than rates for the younger cohorts and

higher than rates for the older cohorts, respectively. See the text and Table 1 for additional

details.

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Table 1

Some estimable parameters and functions in the APC model*

Quantity*

Nomenclature

μ

Grand mean

ãa, π̃p, and γ̃c

Age, period, and cohort deviations

(αL + πL)Longitudinal age trend

(αL − γL)Cross-sectional age trend

(πL + γL)

Net drift ≈ EAPC of the ASR**

μ + (αL + πL)(a − ā) + α̃a

Fitted longitudinal age-at-event curve

μ + (αL − γL)(a − ā) + α̃a

Fitted cross-sectional age-at-event curve

μ + (πL + γL)(p − p̄) + π̃p

Fitted temporal trends

*The APC model is defined over a P × A event matrix Y and corresponding matrix of person-years O. The referent age, period, and cohort are ā =

[(A + 1)/2], p̄ = [(P + 1)/2], and c̄ = p̄ − ā + A, respectively, where P and A are the total numbers of period and age groups and [.] is the greatest

integer function.

**EAPC, estimated annual percentage change; ASR, direct age standardized incidence rate (ASR) (23).

Cancer Epidemiol Biomarkers Prev. Author manuscript; available in PMC 2012 July 1.