Suessmilch Lecture | October 1, 2018

Age-Period-Cohort Analysis: What Is It Good For?

© Marcus T. Wright

On October 9th, 2018, Herbert L. Smith from the University of Pennsylvania, USA, will give a talk at the MPIDR about Age-Period-Cohort Analysis.


Data arrayed by age against time are of recurrent fascination because they represent social and demographic evolution.  One observes change associated with the aging process occurring alongside cohort succession, as well as temporal drift and the shocks associated with historical events.  One sees these things all at once, because they are occurring simultaneously.  Our desire to distinguish among them derives from our internal conceptual apparatus.  There are no data generating functions in the social world.  Our models are our conceit.

Conceited or not, models arise and persist to the extent that they are useful, and thus the abstract philosophy of the situation is typically of little account.  In the case of so-called age-period-cohort models, however, perhaps it is time for a dollop of philosophy.  The charm of the age×time data array has generated many efforts at a statistical decomposition of factors associated with the temporal dimensions age, period, and cohort.  They are efforts because models that are additive with respect to age, period, and cohort founder in the first instance due to the identity among them.  Chronological age is the time elapsed between birth (cohort) and observation (period).  The statistical consequence is that there is no way to identify simultaneously the linear elements of age, period, and cohort, absent some restriction on the model.  The methodological literature has subsequently organized itself around observations of the sensitivity of estimates of linear effects to the choice of identifying restrictions; under what circumstances various restrictions are justified; statistical properties of restrictions; and ways to re-conceptualize the data array.

Generalized linear models in A, P, and C, however statistically high-powered they appear from the standpoint of identification and estimation, are very crude.  They do not reflect the sociological concept of a generation, as per Mannheim and Ryder.  From a statistical and demographic standpoint, they are almost certainly mis-specified when populations are heterogeneous, and hence selectivity is at work within cohorts.  They conjure under-identification issues long before the degrees of freedom in the data arrays are exhausted.  Still, they draw us to them!

The association of the data structure with a set of challenges and strictures from statistics and mathematics has had the unfortunate consequences of (a) turning attention from substantive issues, first assuming that, then making us look as though, we are less knowledgeable than we are; and (b) causing us to overlook what the basic identity A≡P-C is telling us about how we should be thinking about age, period, and cohort as explanatory concepts.  Age, period, and cohort might be exchangeable algebraically and geometrically, but conceptually they are distinct.  For the two constructs associated with historical time, period and cohort, it is hard to think of when and why you would want to imagine linear trends in both.

This paper offers no solutions to the identification problem that do not already exist in the literature, but does find a preference among them:  primary identification via a zero linear trend, probably in period, perhaps in cohort. This preference derives from a series of interpenetrating arguments and observations.  First, the theory of age effects is almost always developmental and longitudinal, hence within cohorts, not cross-sectional.  Second, Occam’s razor: Trends that are purportedly the combination of offsetting period and cohort effects may be represented parsimoniously by one or the other.  Third, many trends in historical time--whatever their purported source--may not be of substantial interest.  They may be assumed and/or taken for granted, whereas the oscillations around this trend may be of primary interest, especially with respect to cohorts.  Fourth, there may be theoretical reasons to posit long-run stationarity.  Fifth, the empirical evidence may be against the existence of a linear trend in the effects of period-specific phenomena.  Thus age-cohort models are the canonical framework for the study of age-differentiated outcomes over time.  Sixth, period-specific events are not necessarily period effects.  They may bear on change over time in a way that is just as easily interpreted in terms of differences between cohorts.  Analysts may disagree on what is a period effect and what is a cohort effect.  Insofar as trends are concerned, it may not matter.

Finally, there is the matter of “seeing the future”: using models with terms for age, period, and cohort to extrapolate beyond the data.  The nonlinear terms associated with cohort have a value that the nonlinear terms associated with cohort do not.  Even in situations where observed temporal variation is dominated by period effects, the “structure” of the phenomenon--at least from the standpoint of extrapolation/projection/forecasting--may be better represented by trends in cohort.

Viewed from a mathematical standpoint, a brief that seems to privilege one temporal dimension over another is not compelling: a bit of shift in notation, some sleight-of-hand with respect to the data array, and any argument for one factor can be mapped onto another.  Which is in essence the point:  There is nothing in the planes of mathematics or statistics that is going to make an “age-period-cohort analysis” intrinsically useful.  Only when we start with our own understandings of how the world works, and measure these models against them, will we discover what they are good for.

About the speaker

Herbert L. Smith is Professor of Sociology and Director of the Population Studies Center (2005-09, 2011-19) at the University of Pennsylvania. Formerly he was Associate Dean for the Social Sciences, School of Arts and Sciences (2002-05). He is an elected member (1991) of the Sociological Research Association and in 2002 received the Clifford C. Clogg Award from the Population Association of America for early career achievement. From 2009-13 he served as Président, Commission d’Évaluation, Institut National d’Études Démographiques (France). Current research concerns bias due to non-response in survey research, organizational change in nursing and health outcomes, and the practice of cohort analysis.  Other research interests include the design of social and demographic studies, the experimental model, causation, representation, measurement, and matching. Previous work by Professor Smith involved social stratification, including the demography of educational attainment and race differences in non-marital childbearing; the family planning system in four counties of North China; and the relationship between the status of women and fertility in five South and Southeast Asian nations (Pakistan, India, Thailand, Malaysia, Philippines). He is or has been a Principal Investigator on two NICHD R01 awards, a T32 award, and R24 and P2C awards; as well as on grants from the Rockefeller, Hewlett, Mellon, and Compton foundations.

Time and Venue

Tuesday, October 9, 2018, 3 p.m., in the Institute's Auditorium.