A note on Left-Spherically Distributed test with covariates

Abstract

In this note we extend the Left-Spherically Distributed linear scores test (LSD) of Läuter, Glimm and Kropf test (1998, The Annals of Statistics). The LSD test is a method for multivariate testing also applicable to the p >> n (much more variables than observations). As a key feature, the score coefficients are chosen such that a left-spherical distribution of the scores is reached under the null hypothesis. Here the test is extended to account for nuisance parameters, particularly for covariates that are assumed to explain (part of) the response variables but are not under test. Moreover it is shown how to deal with the random matrix D which is a Borel function of the residuals under the tested null hypothesis instead of a Borel function of total residuals. This provides a D matrix -and a test as well -which is more focused on the effects of the predictors under test. A R code is available on the someMTP package in CRAN.

FDR Control with Pseudo‐Gatekeeping Based on a Possibly Data Driven Order of the Hypotheses

“…One can adjust the p values for confounders and non-parametric approaches can be accommodated via the rank transformation (Conover and Iman, 1982). Similar results on data driven ordering are shown for instance in Läuter, Glimm, and Kropf (1998), later extended to the more general case of covariates in Finos (2011). Here we propose a much simpler proof based on the Basu theorem, and further generalize previous results to a wider class of test statistics.…”

Rotation‐based multiple testing in the multivariate linear model

“…In addition, it is known that the class of distributions satisfying when is the orthogonal group is the class of left‐spherical distributions (Dawid, ; Finos, ). This means that to have rotatability of , we need to assume that is left‐spherically distributed.…”