Analysis of covariance (ANCOVA) is intended to provide a way to exclude the influence of an extraneous variable on the comparison of means on a response measure between groups defined by the levels of an independent variable (e.g., treatment vs. control, different educational levels, different occupations, etc.). This method was first presented in print by Day and Fisher in 1938 (Day & Fisher, 1938). It has survived to the present, along with its regression-based incremental F-test version, as the only method available for excluding the influence of extraneous covariates from group means comparisons. It is widely used, especially in the analysis of data from classic experimental designs consisting of pre- and post-treatment assessments of treatment and control groups. In this situation, the pre-test is usually correlated with the post-test, and it is entirely appropriate to seek to exclude the post-test variance that is due to the pre-test.
Unfortunately, ANCOVA is only able to achieve its intended purpose in the very rare case where the slopes of the regressions of the dependent variable on the covariate are identical in all of the groups being compared. This restriction on its applicability is conventionally stretched to allow slope differences up to the point where they become statistically significant. In fact, even such nonsignificant slope differences will result in the retention of extraneous covariate variance within each group that will reduce the power of the test of group differences by enlarging the error term. In a large proportion of studies (25 - 50% in my experience), especially those involving 50 or fewer subjects per group, the assumption of homogeneity of within-group regression slopes is violated to significant degree, rendering ANCOVA formally inapplicable. Until now, the conclusion that data fails to satisfy this "homogeneity of regression slopes" assumption has left analysts with no alternative methodology for excluding covariate variance from the comparison of group means.
I have recently submitted an article for review in a major journal that proposes a new method which can be used in place of ANCOVA to exclude variance due to one or more extraneous covariates from the dependent variable on which group means are to be compared. The proposed method is called Analysis of Covariate Residuals, or ANCOVRES. The applicability of this method is unaffected by any degree of differences between the slopes of within-group regressions of the dependent variable on the covariate (i.e., lack of homogeneity of regression). It achieves complete exclusion of covariate influence on the dependent variable within each group being compared, thereby maximizing the power of the comparison for the given sample. The adjusted data is quite easily computed, especially in the case of one covariate, and this adjusted data is subsequently analyzed through ordinary ANOVA or t-test.
An abbreviated version of the article proposing the new method is available for download from the "Articles" section of this website, or by clicking this link:
https://www.prostatservices.com/articles/beyond-ancova-a-new-method-for-excluding-the-influence-of-covariates-in-comparing-group-means