Model selection for penalized spline smoothing using akaike information criteria

Two different forms of Akaike's information criterion (AIC) are compared for selecting the smooth terms in penalized spline additive mixed models. The conditional AIC (cAIC) has been used traditionally as a criterion for both estimating penalty parameters and selecting covariates in smoothing, and is based on the conditional likelihood given the smooth mean and on the effective degrees of freedom for a model fit. By comparison, the marginal AIC (mAIC) is based on the marginal likelihood from the mixed-model formulation of penalized splines which has recently become popular for estimating smoothing parameters. To the best of the authors' knowledge, the use of mAIC for selecting covariates for smoothing in additive models is new. In the competing models considered for selection, covariates may have a nonlinear effect on the response, with the possibility of group-specific curves. Simulations are used to compare the performance of cAIC and mAIC in model selection settings that have correlated and hierarchical smooth terms. In moderately large samples, both formulations of AIC perform extremely well at detecting the function that generated the data. The mAIC does better for simple functions, whereas the cAIC is more sensitive to detecting a true model that has complex and hierarchical terms.