The probability of obtaining a significant statistic, using the parametric analysis of covariance (ANCOVA) and the rank transform ANCOVA, was estimated for three conditions defined in terms of conditional distributions for two groups. The distributions were both normal, both skewed in the same direction but to different degrees, or both skewed to the same degree but in different directions. For each type of pair of conditional distributions both equal and unequal conditional means were studied. When the conditional distributions were skewed and the conditional means were unequal, both positive and negative correlations between height of the conditional mean function and degree or direction of skew were investigated. Sample size and strength of the covariate-dependent variable relationship were manipulated. The results were interpreted in terms of Type I error probabilities and power for testing two different null hypotheses: equality of two conditional mean functions, or equality of two conditional distributions. The results indicated the parametric ANCOVA should be used to test both hypotheses provided that the conditional distributions are normal or the correlation between the height of the conditional mean functions and the degree of skew is expected to be positive. (Author/DWH)