Age-Period-Cohort models

Content:
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/...) observed for a broad age range over a reasonably long time period, and classified by age and date of follow-up and date of birth.

This type of follow-up can be shown in a Lexis-diagram. Individual life-lines can be shown with colouring according to states, or the diagram can just be shown to indicate what ages and periods are covered, and what units of analysis are used.

The Age-Period-Cohort model describes the (log)rates by a sum of 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 terms used to describe rates are linearly related, and the model can therefore be parametrized in different ways, but still produce the same estimated rates.

Parametrization of APC-models

An APC-model should be reported graphically as three functions:
The age-effect, the period effect and the cohort effect.
The following things 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.
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

Having decided this one will have 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 model for the three effects, that be either a factor model (constant rates in 1- or 5-year intervals), splines, fractional polynomials, ...

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 have been taught by Bendix Carstensen. Each of the sites contain slides, practicals and solutions to the practicals.

Last updated: 23rd May 2011, BxC