A simplified approach for establishing estimable functions in fixed effectage‐period‐cohortmultiple classification models

Abstract

Estimable functions play an important role in learning about certain aspects of the impact of ages, periods, and cohorts in age‐period‐cohort multiple classification (APCMC) models. The advantage of these estimates is that they are unbiased estimates of, for example, the deviations of age, period, and cohort effects from their linear trends, or changes in the linear trends of cohort effects within cohorts, or the residuals of fixed effect APCMC models. If the fixed effect APCMC model contains the relevant variables (is well specified), these estimable functions are unbiased estimates of functions of the parameters that generated the dependent variable data, even though the parameters that generated that data are not identified. I provide a simplified approach to establishing which functions are estimable in fixed effect APCMC models that provides an intuitive understanding of estimable functions by showing clearly and simply why they are estimable. This approach involves the partitioning of the age, period, and cohort effects into linear components and deviations from the linear components; the use of the “line of solutions”; and of the “extended null vector.”

Trend analysis and age-period-cohort effects on morbidity and mortality of liver cancer from 2010 to 2020 in Guangzhou, China

“…The limitation of the model lies in the liner independent of the above three factors. In order to obtain the unique solution of the parameter, we used the period zero linear trend method (ZLT-P) contained in the method of “Equality Restrictions” in the APC model to orthogonally decompose the time effect into an unidentifiable linear trend and a linear deviation independent of specific constraints ( 25 ). That is to say, the linear trend of the period effect is shrunk to zero in this study, and the linear degree of freedom is assigned to the other two factors, so as to achieve the division of the linear trend for the three factors.…”

“…The method of “Equality Restrictions”, proposed by HolfFord and Clayton ( 26 , 27 ), is based on the hypothesis that the effects of age, cohort, and period can be orthogonally decomposed into linear and non-linear components. The linear deviations of the time effects and specific combinations of slopes are identifiable, and the defined effects building on these identifiable functions are therefore also reliable ( 25 ). Depending on the constraints, this method contains two different sets of parameter estimates.…”

Advances in statistical methods for cancer surveillance research: an age-period-cohort perspective

“…Estimable Functions (EFs) are linear combinations of model parameters that are invariant with respect to the particular identifiability constraints imposed on the parameters to fit the model (4,5,45).…”