Using old results to produce new solutions in age–period–cohort multiple classification models

There are other reasons for setting a new constraint such as examining the effects of different parameterizations on the solutions produced by the intrinsic estimator (Luo et al. 2016; Pelzer et al. 2015). This too can be accomplished using the \(s\)-constraint with different null vectors based on different parameterizations.

The age-group of the observation is determined by cohort and period as: \(i = I + j - k\), and the period of the observation is determined by age and cohort as: \(j = k + i - I\).

The null vector that consists of all zeros is often labelled the “trivial null vector” in this context.

Kupper et al. (1983) noted this relationship and derived it geometrically.

The number of cohorts \(\left( K \right)\) is \(K = I + J - 1.\)

Note that with effect coding, the left out variable for periods (and for ages and for cohorts) is the negative sum of the period (age or cohort) coefficients in the estimation equation. These values are calculated as such in the column for the new solution.

In many areas of research, we may have better knowledge about the age effects based on theory and substantive knowledge than about the period effects and the cohort effects (O’Brien 2019).

A reviewer suggested that one might use the \(s\)-constraint method (O’Brien 2014a) or the approach of Fosse and Winship (2016) to build bounds around the estimates to get some idea of how error in the estimates might affect the results. This could be done by making small changes in the slope of, for example, the estimated age coefficients to see how this changes the results for ages, periods, and cohorts. I do not do this here, but these techniques are available. This procedure will not give us 95% confidence intervals and will not have the sort of band that such intervals typically have, since it will only change the slopes of the estimates, but it will give some indication of how much the results might change with small changes (assuming large sample sizes) in the slopes.

Note that this is for the raw data that have not been corrected for differences in educational attainment over time [as Alwin and McCammon (1999) do]. It is an accounting of the relationships between age, period, and cohort in terms of vocabulary knowledge not controlling for any other variables. Again the focus of this example is not to settle the debate between these authors.

We can see why this is an estimable function from Eq. (7). If we add the formula for the trend in periods and the formula for the trend for cohort together we have \(t_{p }^{*} + t_{c }^{*} = t_{p} + t_{c} - k + k\). The sum of the linear trends for period and cohort from any new solution equals the sum of the trends for period and cohort for any other old solution.

One of the reviewers of this paper noted an idea from the Bayesian literature of a “community of priors,” which Bell and Jones (2015) use in the context of APC models. It is based on the choice of plausible constraints (using strong priors) and then choosing a solution from among these priors. This shares an affinity to setting bounds and sensitivity tests using plausible constraints (Fosse and Winship 2016; O’Brien 2014a, b, 2019), but using different methods.