Testing for covariate effects in the fully nonparametric analysis of covariance model

Traditional inference questions in the analysis of covariance mainly focus on comparing different factor levels by adjusting for the continuous covariates, which are believed to also exert a significant effect on the outcome variable. Testing hypotheses about the covariate effects, although of substantial interest in many applications, has received relatively limited study in the semiparametric/nonparametric setting. In the context of the fully nonparametric analysis of covariance model of Akritas et al., we propose methods to test for covariate main effects and covariate-factor interaction effects. The idea underlying the proposed procedures is that covariates can be thought of as factors with many levels. The test statistics are closely related to some recent developments in the asymptotic theory for analysis of variance when the number of factor levels is large. The limiting normal distributions are established under the null hypotheses and local alternatives by asymptotically approximating a new class of quadratic forms, The test statistics bear similar forms to the classical F-test statistics and thus are convenient for computation. We demonstrate the methods and their properties on simulated and real data.