Localizing differences in smooths with simultaneous confidence bounds on the true discovery proportion
We demonstrate a method for localizing where two smooths differ using a true discovery proportion (TDP) based interpretation. The methodology avoids the otherwise ad hoc means of doing so, which performs more standard hypothesis tests on smooths of subsetted data. TDP estimates are 1-α confidence bounded simultaneously, assuring the proportion of actual difference in the region with a TDP estimate is at least that with high confidence regardless of the number or location of regions estimated. Our procedure is based in closed-testing Hommel (1986) and recent results of Goeman and Solari (2011) and Goeman et al (2019). We develop expressions for the covariance of quadratic forms because of the multiple regression framework in which we use these authors' foundation, which are shown to be non-negative in many settings. The procedure is well-powered because of a given result on the off-diagonal decay structure of the covariance matrix of penalized B-splines of degree 2 or less. We demonstrate achievement of actual TDP and type 1 error rates in simulation and analyze a data set of walking gate of cerebral palsy patients.
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