Age-Period-Cohort Analysis

Because the three time scales are exactly linearly dependent: cohort equals period minus age. This makes the model's design matrix rank-deficient by one, so infinitely many combinations of linear age, period, and cohort slopes fit the data identically. Holford showed that only nonlinear features - curvatures and deviations from each effect's own linear trend - are estimable without further assumptions. The individual linear slopes can be recovered only by imposing an external constraint, and that constraint, not the data, determines them.

Not in the sense of recovering the truth from data alone; nothing can. The intrinsic estimator yields a unique, coding-invariant minimum-norm solution by projecting out the unidentifiable linear direction, which is more principled than an arbitrary equality constraint. But it still assumes that the special linear trend lying in the design matrix's null space is zero. If that assumption is wrong, the estimated slopes are biased. Best practice is to report it as one defensible solution, check sensitivity to alternatives, and emphasize the estimable curvatures.

Estimable functions - quantities that take the same value under every solution that fits the data. The most useful are second differences (curvatures) of the age, period, and cohort effects, and deviations of each category from its own fitted linear trend. These capture accelerations, slowdowns, and turning points, such as a birth cohort whose risk bends upward relative to its neighbors. Presenting curvature plots lets readers separate robust temporal structure from the parts that rest on an identifying assumption.