  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.
