AgePeriodCohort models
Contents:
What is an AgePeriodCohort model?
Parametrizations
Software for fitting APCmodels
Statistical papers discussing APCmodels
Courses
What is an AgePeriodCohort model?
AgePeriodCohort 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 and date of
followup (period) and date of birth (cohort).
This type of followup can be shown in a Lexisdiagram; a coordinate
system with data of followup along the xaxis, and age along
the yaxis. A single person's lifetrajectory is therefore a straight
line with slope 1 (as calender time and age advance at the same
pace). Tabulated data enumerates the number of events and the risk
time (sum of lengths of lifetrajectories) in some subsets of the
Lexis diagram, usually subsets classified by age and period in
equally long intervals.
Individual lifelines 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 AgePeriodCohort model describes the (log)rates as a sum of
(nonlinear) age period and cohorteffects. The three variables age
(at followup), a, period (i.e. date of followup), p,
and cohort (date of birth), c, are related by a=pc
— 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 ageterms contains
the linear effect of age, the periodterms 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 APCeffects on
testis cancer rates in Denmark. All sets
of estimates will yield the same set of fitted rates.
Parametrization of APCmodels
An APCmodel can be reported graphically as three functions:
The ageeffect, the period effect and the cohort effect.
The following must be considered when devising these:
 Which of the three terms should have the ratedimension?
Normally one would choose the ageeffect 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 agespecific 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
agespecific rates.
Having decided on these issues will result in three curves that for example could be:
 The estimated agespecific rates in the 1940cohort.
 The cohort rateratio relative to the 1940cohort.
 The period rateratio 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 5year intervals), splines,
fractional polynomials, ...
Software for fitting APCmodels
In the Epi package
for R is a
function apc.fit that fits APCmodels 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 APCmodels

T.R. Holford: The estimation of age, period and cohort effects for vital
rates. Biometrics, 39:311324, 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: Ageperiod and agecohort models; II: Ageperiodcohort models.
Statistics in Medicine, 6:449481, 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: AgePeriodCohort models for the Lexis diagram.
Statistics in Medicine, 26: 30183045, 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 APCmodels have been taught by Bendix
Carstensen. Each of the (live) sites contain slides, practicals and
solutions to the practicals. Some links are intentionally dead.

Max Planck Institute for Demographic Research, Rostock, May 2016

Lisboa, 1921 September, 2011

Institutionen för Medicinsk Epidemiologi och Biostatistik,
Karolinska Institutet, Stockholm, May 2010.

Max Planck Institute for Demographic Research, Rostock, March 2009

Department of Biostatistics, University of Copenhagen, 2004
Last updated: 12 April 2016, BxC