The Behrens-Fisher Problem with Covariates and Baseline Adjustments

TL;DR

This paper extends the Welch-Satterthwaite t-test to include covariate adjustments and heteroscedastic variances, providing unbiased variance estimators and improved error rate control in small samples.

Contribution

It introduces a generalized Welch-Satterthwaite t-test with unbiased variance estimators for covariate-adjusted analysis under heteroscedasticity.

Findings

Accurately controls type-1 error rate in simulations

Performs well with small sample sizes and skewed data

Demonstrated effectiveness on real data

Abstract

The Welch-Satterthwaite t-test is one of the most prominent and often used statistical inference method in applications. The method is, however, not flexible with respect to adjustments for baseline values or other covariates, which may impact the response variable. Existing analysis of covariance methods are typically based on the assumption of equal variances across the groups. This assumption is hard to justify in real data applications and the methods tend to not control the type-1 error rate satisfactorily under variance heteroscedasticity. In the present paper, we tackle this problem and develop unbiased variance estimators of group specific variances, and especially of the variance of the estimated adjusted treatment effect in a general analysis of covariance model. These results are used to generalize the Welch-Satterthwaite t-test to covariates adjustments. Extensive simulation studies show that the method accurately controls the nominal type-1 error rate, even for very small sample sizes, moderately skewed distributions and under variance heteroscedasticity. A real data set motivates and illustrates the application of the proposed methods.