# Multivariate analysis of covariance when standard assumptions are violated

## TL;DR

This paper develops a flexible multivariate analysis of covariance method that remains valid under violations of classical assumptions, including heteroskedasticity and singular covariance matrices, using bootstrap techniques and simulations.

## Contribution

It introduces a modified MANCOVA ANOVA-type statistic (MANCATS) combined with bootstrap methods, extending applicability to non-normal, heteroskedastic, and singular covariance scenarios.

## Key findings

- Parametric bootstrap MANCATS controls type I error well.
- Bootstrap MANCATS shows higher power in simulations.
- Method applied successfully to real achievement test data.

## Abstract

In applied research, it is often sensible to account for one or several covariates when testing for differences between multivariate means of several groups. However, the "classical" parametric multivariate analysis of covariance (MANCOVA) tests (e.g., Wilks' Lambda) are based on quite restrictive assumptions (homoscedasticity and normality of the errors), which might be difficult to justify in small sample size settings. Furthermore, existing potential remedies (e.g., heteroskedasticity-robust approaches) become inappropriate in cases where the covariance matrices are singular. Nevertheless, such scenarios are frequently encountered in the life sciences and other fields, when for example, in the context of standardized assessments, a summary performance measure as well as its corresponding subscales are analyzed. In the present manuscript, we consider a general MANCOVA model, allowing for potentially heteroskedastic and even singular covariance matrices as well as non-normal errors. We combine heteroskedasticity-consistent covariance matrix estimation methods with our proposed modified MANCOVA ANOVA-type statistic (MANCATS) and apply two different bootstrap approaches. We provide the proofs of the asymptotic validity of the respective testing procedures as well as the results from an extensive simulation study, which indicate that especially the parametric bootstrap version of the MANCATS outperforms its competitors in most scenarios, both in terms of type I error rates and power. These considerations are further illustrated and substantiated by examining real-life data from standardized achievement tests.