A hot topic and a futile quest? The recent discussion on age-period-cohort analysis

Last year, Yang & Land released their book on age-period-cohort (APC) analysis. A large section of the December issue of Demography is dedicated to age-period-cohort analysis. And just a week or two ago a paper by Bell & Jones on the Yang & Land APC model was published in Demographic Research. APC analysis appears to be a hot topic these days!

What is APC analysis?

APC analysis comes from to the desire of demographers, epidemiologists and sociologists to want to break down phenomena of interest into constituent effects caused by (or associated with) age, calendar time and time of birth (or some other event, but birth is most common). For example, we know that health deteriorates with increasing age, and hence mortality rates are tied to age. However, they are also affected by calendar time; populations in Western Europe living in the 20th century are certainly more healthy than populations in the 19th century. But it is less well-known that time of birth is also related to health; for example individuals born in the Netherlands in 1945-1946 have worse health due to a famine during that period. Of course, this is a strong effect, but more subtle cohort effects may also exist. Being able to break down some topic into age, period and cohort is of interest to demographers, for example to come to better projections of future mortality, fertility, population size and composition. For other scientists, it may be of interest for explanatory reasons (e.g. changes in voting behavior).

The futility of this quest

APC analysis is sometimes considered to be a ‘futile’ quest or even an ‘unholy’ quest. The reason for this is the identification problem: age = period – cohort. If you know two of these three variables, then you automatically know the third. Basically, this means that if you build a linear model (e.g. outcome = constant + age + period + cohort), the model won’t run because there are an infinite number of solutions to this equation. While there may be an infinite number of solutions, if we constrain models in a certain way, we can make sure only one solution is found. We can do this by making certain assumptions (and constraining our models to reflect those assumptions). In the past 40 years there have been many technical papers which discuss ways to constrain the APC model. These technical papers, alongside papers studying APC effects in real-world phenomena, have kept APC analysis in focus and have recently made it a hot topic in demography!

The recent discussion

The recent fuss in Demography and Demographic Research has been about APC models by Yang & Land. They developed two methods, one is called the Intrinsic Estimator, and the other the Hierarchical Age-Period-Cohort (HAPC) model. Yang & Land claimed that their model produced estimates that are unbiased and consistent. This statement made many substantive scientists very happy; no longer would model constraints have to be justified: a few clicks in some statistical program would produce outcomes that could immediately be interpreted!

Alas, it was not to be: in the recent issue of Demography, Liying Luo demonstrated quite eloquently why these claims are not true, or at least misleading. Without going into the technical details, Luo demonstrated that the Yang & Land model still uses constraints to estimate the age-period-cohort model. In fact, a critique is that the constraints placed by the Intrinsic Estimator are far less transparent than those of other methods. Only after these constraints have been applied is the model unbiased and consistent; however, this is also true for most other popular APC models! In other words, the Intrinsic Estimator is not an improvement over previous methods.

Artistic rendering of demographers fighting over methods.

Artistic rendering of demographers fighting over methods.

…In fact, it may even be worse. A further critique by Luo and various distinguished commenters is that since the constraints are less transparent in the Yang & Land approach, it is more difficult for substantive researchers to justify these constraints. For example, a popular constraint is to say that two birth cohorts have the same coefficient. These are often consecutive cohorts, which might be justified by saying that individuals born in time periods close together will have more similar formative experiences than cohorts born further apart. We can keep this constraint regardless of the length of the time series, so that we can even compare time series of different lengths. On the other hand, constraints in the Intrinsic Estimator will change if the researcher changes the length of the time series under investigation. Finally, Luo writes that the intrinsic estimator is remarkably similar to a model developed in by Kupper et al. known as the principal component estimator.

A lesson for the future

In the discussion, criticism is not only aimed at Yang & Land, but also at applied APC researchers. This is especially clear in a comment made by Stephen Fienberg in response to Luo’s paper. After decades of technical statistical research, it appeared that researchers had to be reminded of the true consequences of the well-known identification problem. It is quite simply naïve to assume that a solution can be found to the identification problem; it is not a problem to be solved by statistical modelling constraints, it is a mathematical problem that is unsolvable. The problem with APC analysis therefore lies in the ‘APC accounting framework’ (outcome = constant + age + period + cohort). Therefore, Luo writes that her methodological efforts are now aimed at models for APC analysis that work from a different framework and are more informed by substantive (social) theory. This is in line with, and inspired by, work by O’Brien. These developments fit in with the recent ‘paradigm shift’ from doing demographic analysis using macro-level data (e.g. the analysis of mortality levels in a country) towards analysis using micro-level data (e.g. studying events or trajectories in the life course). For this reason, I foresee that ‘old’ APC analysis (using the APC accounting framework and finding some way to avoid the identification problem) will stick around for a while longer. However, with the development of new methodological approaches in the field of APC analysis, I foresee an exciting time ahead for APC analysts.

Maarten J. Bijlsma is an editor of, and regular contributor to, Demotrends. He has a special interest in APC analysis through his research on birth cohort effects in drug utilization and effectiveness.


  1. very good text, thank you Maarten

  2. Thank you for re-blogging, eldemographo!

  3. Thanks for reblogging, vcjha!

  4. Very nice summary! A familiar topic is the decomposition of mortality improvement in a age-dependent component bx and age-invariant component kt in the Lee-Carter model, where a solution could only be found if constraints where imposed on the parameters – already in the AP Lee-Carter!

    In the spirit of Bens note on causality , we might want to step back from the APC model to a more descriptive approach nicely illustrated in Roland Raus rates of mortality improvement . Even without any model you clearly see period and cohort effects in these figures. Maybe we have to live with the fact that the precise stochastic quantification of these visual impressions will remain impossible due to the “demographic uncertainty principle” inherent in APC analysis.

  5. […] (where retrospective information is not provided), and therefore, inability to fully account for age-period-cohort effects, the issue of spurious relations between research variables, the problem of identifying causality, […]

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.

%d bloggers like this: