Nonlinear Logistic Regression Mixture Experiment Modeling for Binary Data Using Dimensionally Reduced Components

This article motivates, presents, and illustrates an approach using nonlinear logistic regression (NLR) for modeling binary response data from a mixture experiment when the components can be partitioned into groups used to form dimensionally reduced components (DRCs). A DRC is formed from a linear combination of the components in a group having similar roles and/or effects of the same sign, where the linear combinations over all groups are normalized so that the DRC proportions sum to one. Linear combinations of a particular form provide for quantifying the effects of the remaining components in a group relative to a chosen component. This reason, plus dimensional reduction, are the primary motivations for the proposed DRC mixture experiment modeling approach. NLR is required because models expressed in terms of the DRCs are nonlinear in the parameters that specify the linear combinations. A method for obtaining nonparametric tolerance limits on the probability of a “success” for the binary response variable using a bootstrap approach is also presented. Finally, the article shows how DRCs provide for visualizing data and modeling results that otherwise would be impossible. A real database on the presence or absence of nepheline crystals in simulated nuclear waste glass is used to illustrate the NLR modeling and nonparametric tolerance limit approaches. The methodology is general and can be applied to other applications.