Age-Period-Cohort models

Contents:
What is an Age-Period-Cohort model?
Parametrizations
Software for fitting APC-models
Statistical papers discussing APC-models
Courses

What is an Age-Period-Cohort model?

Age-Period-Cohort models is a class of models for demographic rates (mortality/morbidity/fertility/...) usually observed over a broad age range over a reasonably long time period, and classified by age at follow-up (age), date of follow-up (period) and date of birth (cohort).

This type of follow-up can be shown in a Lexis-diagram; a coordinate system with date of follow-up (calendar time) along the x-axis, and age along the y-axis. A single person's life-trajectory is therefore a straight line with slope 1 (as calendar time and age advance at the same pace). Tabulated data enumerates the number of events and the risk time (sum of lengths of life-trajectories) in some subsets of the Lexis diagram, usually subsets classified by age and period in equally long intervals.

Individual life-lines can be shown with colouring according to different states (disease or other), or the diagram can just be shown to indicate what ages and periods are covered, and what subsets are used for classification of events and risk time.

The Age-Period-Cohort model describes the (log-)rates as a sum of (non-linear) age- period- and cohort-effects. The three variables age (at follow-up), a, period (i.e. date of follow-up), p, and cohort (date of birth), c, are related by a = p − c ; any one person's age is calculated by subtracting the date of birth from the current date. Hence the three variables used to describe rates are linearly related, and the model can therefore be parametrized in different ways, and still produce the same estimated rates.

In popular terms you can say that it is possible to move a linear trend around between the three terms, because the age-terms contains the linear effect of age, the period-terms contains the linear effect of period and the cohort effect contains the linear effect of cohort.
An illustration of this phenomenon is in this little "film" of APC-effects on testis cancer rates in Denmark. All sets of estimates will yield the same set of fitted rates.

Parametrization of APC-models

An APC-model can be reported graphically as three functions, that glued together form the fitted values:
The age-effect, the period effect and the cohort effect.
The following must be considered when devising these:

Which of the three terms should have the rate-dimension?
Normally one would choose the age-effect to have this.
What should be the reference points for the period and cohort effects?
Normally one would choose a reference point for either period or cohort, and constrain the other to be 0 on average. But choosing a reference point for both would work too; the choice will only influence the level of the age-specific rates.
Where should the drift (linear trend) be included?
Normally one would put this either with the cohort or the period effect, leaving the other one to have 0 slope on average. This choice will influence the slope of the age-specific rates.

Having decided on these issues will result in three curves that for example could be:

The estimated age-specific rates in the 1940-cohort.
The cohort rate-ratio relative to the 1940-cohort.
The period rate-ratio taken as a residual RR (because it is constrained to be 0 on average with 0 slope).

This sort of choice for the parametrization is unrelated to the particular choice of how to model the three effects, that be either a factor model (constant rates in 1- or 5-year intervals), splines, fractional polynomials, ...
The snag is that "constrained to be 0 on average with 0 slope" is not a mathematically well defined concept, it can be defined in many ways.

Software for fitting APC-models

In the Epi package for R is a function apc.fit that fits APC-models in various guises, and with different options for parametrizations. Moreover there are functions that are designed to plot the estimates in a nicely groomed fashion.

Statistical papers discussing APC-models

T.R. Holford: The estimation of age, period and cohort effects for vital rates. Biometrics, 39:311-324, 1983.
This is the first paper that suggests to constrain period/cohort effects to be 0 with 0 slope on average. It also derives the estimable drift parameters. It refers to only to factor parametrizations of models.
D. Clayton & E. Schifflers: Models for temporal variation in cancer rates. I: Age-period and age-cohort models; II: Age-period-cohort models. Statistics in Medicine, 6:449-481, 1987.
These two companion papers are the classical references which very carefully explain how the three effect play together and how to report models in practice. It also discusses pitfalls when tabulating data by all three factors (in Lexis triangles). Only discusses factor parametrizations of effects.
B. Carstensen: Age-Period-Cohort models for the Lexis diagram. Statistics in Medicine, 26: 3018-3045, 2007.
This is an overview paper which tries to separate data format, model structure and model parametrization. Discusses models that will work for any type of data tabulation and gives guidance on choice of parametrization.

Courses

The following courses on APC-models have been taught by Bendix Carstensen. Each of the (live) sites contain slides, practicals and solutions to the practicals. Some links are intentionally dead.

Last updated: 22 March 2023, BxC