Nonparametric scanning tests of homogeneity for hierarchical models with continuous covariates

In many applications of hierarchical models, there is often interest in evaluating the inherent heterogeneity in view of observed data. When the underlying hypothesis involves parameters resting on the boundary of their support space such as variances and mixture proportions, it is a usual practice to entertain testing procedures that rely on common heterogeneity assumptions. Such procedures, albeit omnibus for general alternatives, may entail a substantial loss of power for specific alternatives such as heterogeneity varying with covariates. We introduce a novel and flexible approach that uses covariate information to improve the power to detect heterogeneity, without imposing unnecessary restrictions. With continuous covariates, the approach does not impose a regression model relating heterogeneity parameters to covariates or rely on arbitrary discretizations. Instead, a scanning approach requiring continuous dichotomizations of the covariates is proposed. Empirical processes resulting from these dichotomizations are then used to construct the test statistics, with limiting null distributions shown to be functionals of tight random processes. We illustrate our proposals and results on a popular class of two-component mixture models, followed by simulation studies and applications to two real datasets in cancer and caries research.