Properties of R(2) statistics for logistic regression

Area of Science:

• Statistics
• Biostatistics
• Econometrics

Background:

• R(2) statistics are crucial for evaluating logistic regression model fit.
• Variance-based R(2) measures quantify the predictive power of covariates.
• Assessing model performance requires understanding the statistical properties of these measures.

Purpose of the Study:

• To investigate the asymptotic properties of three popular variance-based R(2) statistics for logistic regression.
• To compare the asymptotic behavior of sum of squares, squared Pearson correlation, and Gini's concentration measure.
• To provide a theoretical foundation for evaluating logistic regression model predictivity.

Main Methods:

• Asymptotic analysis of variance-based R(2) statistics.
• Comparative study of statistical distributions for selected R(2) measures.
• Derivation of asymptotic confidence intervals for R(2) statistics.

Main Results:

• The sum of squares and squared Pearson correlation R(2) statistics exhibit identical asymptotic distributions.
• Gini's concentration measure demonstrates a different asymptotic behavior.
• Gini's measure may inflate the perceived predictivity of logistic regression models, particularly under mis-specification.

Conclusions:

• The study provides theoretical justification for observed discrepancies in R(2) statistic performance.
• Asymptotic confidence intervals can be constructed, enabling more robust model evaluation.
• Statistical variability must be considered when assessing the predictive accuracy of logistic regression models.