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 procedure yields a statement on the proportion of some region where true differences exist between two smooths, which results from use of hypothesis tests on collections of basis coefficients parametrizing the smooths. The methodology avoids otherwise ad hoc means of doing so such as performing hypothesis tests on entire smooths of subsetted data. TDP estimates are 1-alpha confidence bounded simultaneously, assuring that the estimate for a region is a lower bound on the proportion of actual difference, or true discoveries, in that region with high confidence regardless of the number, location, or size of regions for which TDP is estimated. Our procedure is based on closed-testing using Simes local test. We develop expressions for the covariance of quadratic forms because of the multiple regression framework in which we use closed-testing results, which are shown to be non-negative in many settings. Our procedure is well-powered because of a result on the off-diagonal decay structure of the covariance matrix of penalized B-splines of degree two or less. We demonstrate achievement of estimated TDP in simulation for different specified alpha levels and degree of difference and analyze a data set of walking gait of cerebral palsy patients. Keywords: splines; smoothing; multiple testing; closed-testing; simultaneous confidence