The Relation Between Rao's Paradox in Discriminate Analysis and Regression Analysis

Area of Science:

• Multivariate statistics
• Statistical modeling

Background:

• A paradox exists in discriminant analysis where individual variables show group discrimination, but not when combined.
• This phenomenon has been noted by Rao and requires statistical explanation.

Purpose of the Study:

• To explain the paradox in discriminant analysis where combined variables lose discriminatory power.
• To establish the relationship between this discriminant analysis paradox and similar paradoxes in regression analysis.
• To provide a clear statistical expression for the F statistic in discriminant analysis.

Main Methods:

• Mathematical derivation to establish relationships between discriminant and regression analysis.
• Formulation of an expression for the F statistic in discriminant analysis.
• Analysis of the F statistic in terms of the average of squares of the t-value.

Main Results:

• An exact mathematical relationship is demonstrated between the discriminant analysis paradox and regression analysis paradoxes.
• A novel expression for the F statistic in discriminant analysis is derived.
• The F statistic expression clarifies the relationship by using the average of squares of the t-value.

Conclusions:

• The paradox in discriminant analysis is statistically explained and linked to regression analysis.
• The derived F statistic expression provides a clearer understanding of variable interactions in discriminant analysis.
• This work offers insights into the behavior of statistical models with multiple predictor variables.