A simultaneous confidence-bounded true discovery proportion perspective on localizing differences in smooth terms in regression models

A method is demonstrated for localizing where two spline terms, or smooths, differ using a true discovery proportion (TDP)-based interpretation. The procedure yields a statement on the proportion of some region where true differences exist between two smooths. The methodology avoids ad hoc approaches to making such statements, like subsetting the data and performing hypothesis tests on the truncated spline terms. TDP estimates are 1-α confidence-bounded simultaneously, which means that a region's TDP estimate is a lower bound on the proportion of actual differences, or true discoveries, in that region, with high confidence regardless of the number of estimates made. The procedure is based on closed-testing using Simes local test. This local test requires that the multivariate ${\chi }^2$ test statistics of generalized Wishart type underlying the method be positive regression dependent on subsets (PRDS), a result for which evidence is presented suggesting that the condition holds. Consistency of the procedure is demonstrated for generalized additive models with the tuning parameter chosen by REML or GCV, and the achievement of confidence-bounded TDP is shown in simulation as is an analysis of walking gait.