Linear projections of joint symmetry and independence applied to exact testing treatment effects based on multidimensional outcomes

Abstract

The growing need for analyzing multivariate aspects of joint data distributions is reinforced by a diversity of experiments based on dependent outcomes. In this sense, different contexts of joint symmetry of data distributions have been dealt with extensively in both theory and practice. Univariate characterizations of properties of multivariate distributions can allow the reduction of the original problem to a substantially simpler one. We focus on research scenarios when vectors x and Ax are identically distributed, where A is a diagonal matrix and absolute values of A’s elements equal to one. It is shown that these scenarios are attractive in new characterizations of joint or mutual independence between random variables. We establish projections of the joint symmetry and independence via the one-dimensional symmetry of linear combinations of x’s components and their interactions. These projections are the most revealing of the multivariate data distribution. The usefulness of the linear projections is exemplified by constructing an efficient nonparametric exact test for joint treatment effects. In this framework, an algorithm for implementing linear projection-based tests is proven. Numerical studies based on generated vectors and a real dataset show that the proposed test can exhibit high and stable power characteristics. The present method can be also used for testing independence between symmetric random vectors.