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We demonstrate a method for localizing where two spline terms, or smooths,differ using a true discovery proportion (TDP) based interpretation. Theprocedure yields a statement on the proportion of some region where truedifferences exist between two smooths, which results from use of hypothesistests on collections of basis coefficients parameterizing the smooths. Themethodology avoids otherwise ad hoc means of making such statements likesubsetting the data and then performing hypothesis tests on the truncatedspline terms. TDP estimates are 1-alpha confidence bounded simultaneously. Thismeans that the TDP estimate for a region is a lower bound on the proportion ofactual difference, or true discoveries, in that region with high confidenceregardless of the number of regions at which TDP is estimated. Our procedure isbased on closed-testing using Simes local test. This local test requires thatthe `multivariate chi-sq test statistics of generalized Wishart type'underlying the method are positive regression dependent on subsets (PRDS),which we show. The method is well-powered because of a result on theoff-diagonal decay structure of the covariance matrix of penalized B-splines ofdegree two or fewer. We demonstrate achievement of estimated TDP in simulationand analyze a study of walking gait of cerebral palsy patients.

Localizing differences in smooths with simultaneous confidence bounds on the true discovery proportion