The Interval Testing Procedure: Inference for Functional Data Controlling the Family Wise Error Rate on Intervals.
Thursday 14th March 2013
Pini, A.; Vantini, S.
We propose a novel inferential technique based on permutation tests that enables the statistical comparison between two functional populations. The procedure (i.e., Interval Testing Procedure) involves three steps: (i) representing functional data on a suitable high-dimensional ordered functional basis; (ii) jointly performing univariate permutation tests on the coefficients of the expansion; (iii) combining the results obtaining a suitable family of multivariate tests and a p-value heat-map to be used to correct the univariate p-values. The procedure is provided with an interval-wise control of the Family Wise Error Rate. For instance this control, which lies in between the weak and the strong control of the Family Wise Error Rate, can imply that, given any interval of the domain in which there is no difference between the two functional populations, the probability that at least a part of the domain is wrongly detected as significant is always controlled. Moreover, we prove that the statistical power of the Interval Testing Procedure is always higher than the one provided by the Closed Testing Procedure (which provides a strong control of the Family Wise Error Rate but it is computationally unfeasible in the functional framework). On the contrary, we prove that the power of the Interval Testing Procedure is always lower than the Global Testing Procedure one (which however provides only a weak control of the Family Wise Error Rate and does not provide any guide to the interpretation of the test result). The Interval Testing Procedure is also extended to the comparison of several functional populations and to the estimation of the central function of a symmetric functional population. Finally, we apply the Interval Testing Procedure to two case studies: Fourier-based inference for the mean function of yearly recorded daily temperature profiles in Milan, Italy; and B-spline-based inference for the difference between curvature, radius and wall shear stress profiles along the Internal Carotid Artery of two pathologically-different groups of subjects. In the supplementary materials we report the results of a simulation study aiming at comparing the novel procedure with other possible approaches. An R-package implementing the Interval Testing Procedure is available as supplementary material.