A nonparametric alternative to analysis of covariance

The Analysis of Covariance (ANCOVA) is designed for the many practical situations in which factor effects are obscured by concomitant variables, or the main purpose of the investigation lies in assessing the effect of the concomitant variables. Not taking covariates into account may cause unprecise or biased results. However, if the response variable or the covariate are only measured on an ordinal scale (like typically psychological and other scores or grading scales), or if they show distinct nonnormal distributions, one would be reluctant to use parametric ANCOVA methods. In this paper, we consider a nonparametric model with covariates. The information contained in the covariates is used to minimize the variance of certain nonparametric estimators for the response variable. This model combines the power gain through introduction of covariates into a factorial design with the robustness of nonparametric procedures. We discuss asymptotic test procedures for inference about factor effects as well as for testing the effect of a covariate. To apply the suggested methods to real data, a SAS macro is provided and available for download. The use of this macro is briefly explained. The tests can be used for data with ties, and even for purely ordinal data, including ordinal covariates. The number of covariates that can be included into the model is not restricted. Simulations show extremely good small-sample performance. In many situations, the proposed tests only require sample sizes around 10.