Finite Mixture Model Specifications Accommodating Treatment Nonresponse in Experimental Research

Abstract

For researchers exploring causal inferences with simple two group experimental designs, results are confounded when using common statistical methods and further are unsuitable in cases of treatment nonresponse. In signal processing, researchers have successfully extracted multiple signals from data streams with Gaussian mixture models, where their use is well matched to accommodate researchers in this predicament. While the mathematics underpinning models in either application remains unchanged, there are stark differences. In signal processing, results are definitively evaluated assessing whether extracted signals are interpretable. Such obvious feedback is unavailable to researchers seeking causal inference who instead rely on empirical evidence from inferential statements regarding mean differences, as done in analysis of variance (ANOVA). Two group experimental designs do provide added benefit by anchoring treatment nonrespondents' distributional response properties from the control group.

Obtaining empirical evidence supporting treatment nonresponse, however, can be extremely challenging. First, if indeed nonresponse exists, then basic population means, ANOVA or repeated measures tests cannot be used because of a violation of the identical distribution property required for each method. Secondly, the mixing parameter or proportion of nonresponse is bounded between 0 and 1, so does not subscribe to normal distribution theory to enable inference by common methods.

This dissertation introduces and evaluates the performance of an information-based methodology as a more extensible and informative alternative to statistical tests of population means while addressing treatment nonresponse. Gaussian distributions are not required under this methodology which simultaneously provides empirical evidence through model selection regarding treatment nonresponse, equality of population means, and equality of variance hypotheses. The use of information criteria measures as an omnibus assessment of a set of mixture and non-mixture models within a maximum likelihood framework eliminates the need for a Newton-Pearson framework of probabilistic inferences on individual parameter estimates. This dissertation assesses performance in recapturing population conditions for hypotheses' conclusions, parameter accuracy, and class membership. More complex extensions addressing multiple treatments, multiple responses within a treatment, a priori consideration of covariates, and multivariate responses within a latent framework are also introduced.