2. A Methodological Comparison of Age-Period-Cohort Models: The Intrinsic Estimator and Conventional Generalized Linear Models

Abstract

Age-period-cohort (APC) accounting models have long been objects of attention in statistical studies of human populations. It is well known that the identification problem created by the linear dependency of age, period, and cohort (Period ¼ Age þ Cohort or P ¼ A þ C) presents a major methodological challenge to APC analysis, a problem that has been widely addressed in demography, epidemiology, and statistics. This paper compares 75 parameter estimates and model fit statistics produced by two solutions to the identification problem in age-period-cohort models-namely, the conventional demographic approach of constrained generalized linear models Mason 1978, 1985;Mason and Smith 1985) and the intrinsic estimator method recently developed by Fu (2000;Knight and Fu 2000;Fu, Hall, and Rohan 2004). We report empirical analyses of applications of these two methods to population data on U.S. female mortality rates. Comparisons of parameter estimates suggest that both constrained generalized linear models and the intrinsic estimator method can yield similar estimates of age, period, and cohort effects, but estimates obtained by the intrinsic estimator are more direct and do not require prior information to select appropriate model identifying constraints. We also describe three statistical properties of the estimators: (1) finite-time-period bias, (2) relative statistical efficiency, and (3) consistency as the number of periods of observed data increases. These empirical analyses and theoretical results suggest that the intrinsic estimator may well provide a useful alternative to conventional methods for the APC analysis of demographic rates.

The Non-uniqueness Property of the Intrinsic Estimator in APC Models

“…Further, 0 β denotes the intercept and e represents the unit's error term. To estimate the parameters i α , j β , and k γ we follow Yang (2004) and apply the following constraints: …”

“…The IE estimates however depend on a) the choice of omitted categories and b) the type of parameterization applied. As an example of the dependence on omitted categories, we derive the IE estimates when the first categories of age, period, and cohort are omitted instead of the last ones, the latter being the default in Yang (2004).…”

“…This is illustrated in Fig. 2 where we use the same dataset on U.S. female mortality as was used by Yang et al (2004Yang et al ( , 2008. These data are from the Berkeley Human Mortality…”

“…We took all possible 19 x 8 x 26=3952 triplets of omitted categories and for each triplet we calculated the corresponding set of effect coded IE estimates, using the Poisson regression model also employed by Yang et al (2004Yang et al ( , 2008. In Fig.…”

Mapping Mortality: Using Lexis surfaces to identify complex patterns in age-year specific mortality rates within and between populations

“…For example, people born around the time of the 'Spanish Flu' of 1918 appear to have somewhat higher mortality risks at any given age than might be expected from trends observed in earlier and later cohorts, (Almond, 2006;J Minton, Vanderbloemen, & Dorling, 2013) and people born in England and Wales in the 1950s to have a somewhat lower mortality risk as they age than might be expected from broader trends. (Willets, 2003) There have been various attempts to uniquely partition away cohort effects from age effects and period effects in statistical models (for example (Yang, Fu, & Land, 2004;Yang, Schulhofer-Wohl, Fu, & Land, 2008)), but doing so is logically impossible, because each of the three effects cannot be uniquely identified (Wilmoth, 2006), leading to effects to do so being branded 'futile'. (Bell & Jones, 2014) 1.2.…”