An Analytical Investigation of the Effects of Heterogeneous Regression Slopes in Analysis of Covariance

Abstract

The effects of violation of the assumption of homogeneity of regression on the Type I error rate and on the power of analysis of covariance (ANCOVA) were investigated. The study shows that the test statistic used to test the hypothesis of equal treatment effects has a non-central F-distribution with the non-centrality parameter equal to zero if and only if the regression slopes are homogeneous and/or the means of the covariables, computed on the entire sample, are all zero. Consequently, if the heterogeneous regression slopes are present, the likelihood of committing a Type I error is greater than what the researcher prespecified. The power function of the test of hypothesis was also determined. Results indicated that the power of ANCOVA was effected by heterogeneous regression slopes and furthermore it was found that the test does not maximize the power for all possible differences of the treatment effects as described by the alternative hypothesis. The method of investigation was analytical and the results are general for any number of groups, any combination of group sizes and any number of covariables, provided the number of observations in each group is larger than the number of covariables.