# MATS: Inference for potentially Singular and Heteroscedastic MANOVA

## TL;DR

This paper develops a new inference method for multivariate analysis of variance that handles singular, heteroscedastic, and high-dimensional data without strict distributional assumptions, using resampling techniques.

## Contribution

It extends existing MANOVA tests to accommodate singular and heteroscedastic covariance matrices in factorial designs, with inference based on resampling quantiles.

## Key findings

- Method performs well in simulations under various conditions.
- Successfully applied to real election data with complex covariance structures.
- Provides confidence regions and ellipsoids for multivariate parameters.

## Abstract

In many experiments in the life sciences, several endpoints are recorded per subject. The analysis of such multivariate data is usually based on MANOVA models assuming multivariate normality and covariance homogeneity. These assumptions, however, are often not met in practice. Furthermore, test statistics should be invariant under scale transformations of the data, since the endpoints may be measured on different scales. In the context of high-dimensional data, Srivastava and Kubokawa (2013) proposed such a test statistic for a specific one-way model, which, however, relies on the assumption of a common non-singular covariance matrix. We modify and extend this test statistic to factorial MANOVA designs, incorporating general heteroscedastic models. In particular, our only distributional assumption is the existence of the group-wise covariance matrices, which may even be singular. We base inference on quantiles of resampling distributions, and derive confidence regions and ellipsoids based on these quantiles. In a simulation study, we extensively analyze the behavior of these procedures. Finally, the methods are applied to a data set containing information on the 2016 presidential elections in the USA with unequal and singular empirical covariance matrices.