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Abstract 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. 

In many applications of zero-inflated models, score tests are often used to evaluate whether the population heterogeneity as implied by these models is consistent with the data. The most frequently cited justification for using score tests is that they only require estimation under the null hypothesis. Because this estimation involves specifying a plausible model consistent with the null hypothesis, the testing procedure could lead to unreliable inferences under model misspecification. In this paper, we propose a score test of homogeneity for zero-inflated models that is robust against certain model misspecifications. Due to the true model being unknown in practical settings, our proposal is developed under a general framework of mixture models for which a layer of randomness is imposed on the model to account for uncertainty in the model specification. We exemplify this approach on the class of zero-inflated Poisson models, where a random term is imposed on the Poisson mean to adjust for relevant covariates missing from the mean model or a misspecified functional form. For this example, we show through simulations that the resulting score test of zero inflation maintains its empirical size at all levels, albeit a loss of power for the well-specified non-random mean model under the null. Frequencies of health promotion activities among young Girl Scouts and dental caries indices among inner-city children are used to illustrate the robustness of the proposed testing procedure. 

Summary In many applications of two‐component mixture models such as the popular zero‐inflated model for discrete‐valued data, it is customary for the data analyst to evaluate the inherent heterogeneity in view of observed data. To this end, the score test, acclaimed for its simplicity, is routinely performed. It has long been recognised that this test may behave erratically under model misspecification, but the implications of this behaviour remain poorly understood for popular two‐component mixture models. For the special case of zero‐inflated count models, we use data simulations and theoretical arguments to evaluate this behaviour and discuss its implications in settings where the working model is restrictive with regard to the true data‐generating mechanism. We enrich this discussion with an analysis of count data in HIV research, where a one‐component model is shown to fit the data reasonably well despite apparent extra zeros. These results suggest that a rejection of homogeneity does not imply that the underlying mixture model is appropriate. Rather, such a rejection simply implies that the mixture model should be carefully interpreted in the light of potential model misspecifications, and further evaluated against other competing models.