Nonparametric models and methods for nonlinear analysis of covariance

A fully nonparametric model for nonlinear analysis of covariance is proposed. The term nonlinear means that the covariate influences the response in a possibly nonlinear and nonpolynomial fashion, while the term fully nonparametric implies that the distributions for each factor level combination and covariate value are not restricted to comply with any parametric or semiparametric model. The possibility of different shapes of covariate effect in different factor level combinations is also allowed. This generality is useful whenever modelling assumptions such as proportional odds, or linearity and homoscedasticity appear suspect. In the context of this nonparametric model hypotheses, of no main effect, no interaction and no simple effect, which adjust for the covariate values are defined and test statistics are developed. Both the response and the covariate are allowed to be ordinal. The test statistics are based on averages over the covariate values of certain Nadaraya-Watson-type nonparametric regression quantities and asymptotically they have, under their respective null hypotheses, a central χ²-distribution. Simulation results show that the statistics have good power properties. The procedures are demonstrated on two real datasets.