Test Statistics for Mixture Models

Abstract

Regression models of the forms proposed by Scheffé and by Becker have been widely and usefully applied to describe the response surfaces of mixture systems. These models do not contain a constant term. It has been common practice to test the statistical significance of these mixture models by the same statistical procedures used for other regression models whose constant term is absent (e.g., because the regression must pass through the origin). In this paper we show that the common practice produces misleading reslllts for mixtures. The mixture models require a different set of F, R ², and R _A ² statistics. The correct mixture statistics correspond to a physically consistent null hypothesis and are also consistent with the expression of the mixture model in the older “slack-variable” form. An illustrative example is included.

KEY WORDS:

• Mixture Experiments
• Mixture Models
• Mixture Null Hypothesis
• Regression Analysis
• Test Statistics