Multivariate tests based on interpoint distances with application to magnetic resonance imaging

Abstract

The multivariate location problem is addressed. The most familiar method to address the problem is the Hotelling test. When the hypothesis of normal distributions holds, the Hotelling test is optimal. Unfortunately, in practice the distributions underlying the samples are generally unknown and without assuming normality the finite sample unbiasedness of the Hotelling test is not guaranteed. Moreover, high-dimensional data are increasingly encountered when analyzing medical and biological problems, and in these situations the Hotelling test performs poorly or cannot be computed. A test that is unbiased for non-normal data, for small sample sizes as well as for two-sided alternatives and that can be computed for high-dimensional data has been recently proposed and is based on the ranks of the interpoint Euclidean distances between observations. Five modifications of this test are proposed and compared to the original test and the Hotelling test. Unbiasedness and consistency of the tests are proven and the problem of power computation is addressed. It is shown that two of the modified interpoint distance-based tests are always more powerful than the original test. Particularly, the modified test based on the Tippett criterium is suggested when the assumption of normality is not tenable and/or in case of high-dimensional data with complex dependence structure which are typical in molecular biology and medical imaging. A practical application to a case-control study where functional magnetic resonance imaging is used is discussed.

Permutation tests of scale using deviances

“…Type I error rate and power were estimated based on 1,000 randomly selected data sets from each distribution, for each setting of sample sizes and scale parameter patterns. Marozzi (2016) suggested that only 253 random permutations are necessary with 1,000 random data sets if the goal of the simulation is to estimate the power of a test and only a "rough" estimate of the permutation p-value is required, while Keller-McNulty and Higgins (1987) recommended a random sample of at least 1,600 permutations to estimate the exact p-value for a permutation test. Since precise estimation of the permutation test p-values was considered important, a conservative 2,000 random permutations was utilized.…”

Bootstrap for dependent Hilbert space-valued random variables with application to von Mises statistics

“…, we obtain a test that has asymptotically level α. This gives an alternative to the classical Hotelling test, which is not well suited for the high dimensional setup, see Marozzi [28].…”

Interpoint Distance Test of Homogeneity for Multivariate Mixture Models

“…Berrendero, Cuevas, and Pateiro-López (2016) identify a shape with the corresponding IPD distribution. Marozzi (2015Marozzi ( , 2016 discusses multivariate tests based on IPDs for high dimensional low sample size data and applies them to magnetic resonance images. Baringhaus & Franz (2004), Rosenbaum (2005) and Jurecková & Kalina (2012) utilise IPDs to construct tests for the general two-sample problem.…”