Diagnostics in Linear Discriminant Analysis

Abstract

Some new diagnostic measures in discriminant analysis are proposed. They can be expressed in terms of the two fundamental influence statistics in discriminant analysis: d _i ² and ψ _i . A theorem on the asymptotic distributions of the fundamental statistics is derived. Based on the theorem, the proposed measures can be shown to be asymptotically distributed as functions of independent chi-squared and standard normal random variables. Critical values and expected quantiles of the measures can then be constructed. Hence influential observations are detected using Q-Q plots and significance tests. Two measures have analogous forms in regression. The theorem is also useful for getting the asymptotic distributions of existing measures that are functions of d _i ² and ψ _i . A comparison of the diagnostics in linear discriminant analysis, linear regression, and linear logistic regression (discriminant) analysis is made. Although discriminant coefficients can be determined under a regression model, regression diagnostic measures are shown to be inappropriate for detecting influential observations in linear discriminant analysis. The temptation of applying regression diagnostic measures in linear discriminant analysis must be resisted.

Key Words:

• Asymptotic distributions
• Influential observations
• Q-Q plots
• Regression diagnostics
• Significance tests