Identifiability Issue Sample Clauses

Identifiability Issue. Despite the popular application of APC model, equivalently log-linear model, it is well known that there is a non-identifiability problem associated with all three factors because of the exact linear relationship among them [Xxxxxxx, 1983, 1991]. For any c ∈ IR, the linear predictor for log(λij) is not changed between the parameter set (µ, xx, xx, xx) xxx (x∗, xx∗, xx∗, γk∗), where µ∗ = µ − c · I ; αi∗ = αi + c · i ; βj∗ = βj − c · j ; γk∗ = γk + c · k. Since k = I + j − i, we can have log(λi∗j ) = µ∗ + αi∗ + βj∗ + γk∗ = µ − c · I + αi + c · i + βj − c · j + γk + c · k = µ + αi + βj + γk − (I + j − i) + k = µ + αi + βj + γk = log(λij). Therefore, the age, period, and cohort effects cannot be identified and interpreted uniquely. Figure 2.1 demonstrates the problem of nonidentifibility in APC models. Here we choose the arbitrarily c value as 1 and make the transformation on the age-period-cohort estimates as shown above. As we can see from Figure 2.1, the period effect is increasing in the top row but the new period effects in the bottom row has a decreasing trend over period j. Significant changes are also observed on age and cohort effects. In the top row, we can see the trends of age effects changing at age group of 5 but in the bottom row, a straight linear pattern is shown in the age effects. The cohort effects are orginially decreasing over the cohorts k, however, after transformation, the decreasing trend diminishes around cohort 6. To overcome this non-identifiability problem, several parameter constraints or assumptions were proposed for the APC model [Xxxxxx and Xxxxxxx, 1982, Xxxxxxx and Xxxxxxxxxx, 1987]. However, those constraints lack a sound biological explanation. To explain the