Table 2.

Methods used to estimate age, period, and cohort effect

Method	Description
Linear model60	Generalized linear model; log or logit transformation of age-period-cohort specific rates are modeled as a linear function of additive effects of age, period, and cohort Suffers from “identification problem” induced by linear dependency between age, period, and cohort Design matrix is less than full rank, leading to multiple rather than unique estimators of the three effects1,61
Coefficient-constraints approach60	Placing one or more identifying constrain on the parameter vector to just-identify or over-identify the model Model coefficients are sensitive to choice of constraint
Estimable function approach57–59	Focuses on non-linear (vs. linear) components and uses deviations, curvatures, and drift to derive unique estimates
Intrinsic estimator62,63	Estimates the unique estimable function of linear and non-linear components of the age-period-cohort model Determined by the Moore-Penrose generalized inverse function using principal component regression
Hierarchical model	Mixed-effect models estimate fixed effects of age at the individual level and random effects of period and cohort at a higher level Capture contextual effects of cohort membership and historical time relevant in disease processes Allows researchers to include additional covariates at different levels to test explanatory hypotheses about specific risk factors (e.g., obesity, smoking) contributing to observed trends
NCI web tool64	Publically available web tool for researchers, providing a panel of estimable functions and corresponding Wald test

NOTE: Coefficient-constraints and estimable function approaches are two approaches within the linear model framework; intrinsic estimator is a specific example of an estimable function