ANCOVA methods for heteroscedastic nonparametric regression models

We consider an ANCOVA design in which the relationship between the response Y_i and the covariate X_i in cell (factor-level combination) i satisfies the model Y_i = m_i(X_i)+σ_i(X_i)ϵ_i, where the error term ϵ_i is assumed to be independent of X_i, and m_i and σ_i are respectively a smooth (but unknown) regression and scale function. This model can be viewed as a generalization of the nonparametric ANCOVA model of Young and Bowman. As such it is a useful alternative for parametric or semiparametric ANCOVA models, whenever modeling assumptions such as proportional odds, normality of the error terms, linearity or homoscedasticity appear suspect. We develop test statistics for the hypotheses of no main effects, no interaction effects, and no simple effects, which adjust for the covariate values, as defined by Akritas, Arnold, and Du. The asymptotic distribution of the test statistics is obtained, its small sample behavior is studied by means of simulations and a real dataset is analyzed.