A model for extreme stacking of data at endpoints of a distribution: Illustration with W-shaped data

Analyses will only provide correct inferences when certain assumptions are met. Regrettably, researchers do not always assess the validity of their assumptions. One scenario requiring such consideration is when data is stacked at one or both ends of a distribution. When data are stacked at zero, regression models are not appropriate but zero-altered regression models (e.g., zero-inflated Poisson or negative binomial, and hurdle models), can be valid and effective models. When data are stacked at two known extremes, a different modeling structure must be used. In this paper, we propose a model in which the complete distribution of the outcome is modeled by three component distributions: the two extreme responses and a distribution for responses in between. We illustrate this modeling structure in an educational data application, where the data exhibited two extreme responses with a near normal distribution in between, resulting in a bimodal W-shape distribution. We compare our results to standard modeling approaches with respect to goodness of fit, protection against bias in parameter estimation, statistical power, and interpretation.