Focused Relative Risk Information Criterion for Variable Selection in Linear Regression

TL;DR

This paper introduces a new variable selection method for linear regression called Focused Relative Risk Information Criterion (FRIC), which assesses models based on relative risks and confidence distributions, improving model selection accuracy.

Contribution

The paper develops FRIC scores and plots for variable selection, extending the theory to multiple focus parameters and connecting to existing criteria like Mallows.

Findings

FRIC provides accurate relative risk assessments for submodels.

The method balances model complexity and fit effectively.

Extensions handle multiple focus parameters simultaneously.

Abstract

This paper motivates and develops a novel and focused approach to variable selection in linear regression models. For estimating the regression mean $\mu=\E\,(Y\midd x_0)$, for the covariate vector of a given individual, there is a list of competing estimators, say $\hatt\mu_S$ for each submodel $S$. Exact expressions are found for the relative mean squared error risks, when compared to the widest model available, say $\mse_S/\mse_\wide$. The theory of confidence distributions is used for accurate assessments of these relative risks. This leads to certain Focused Relative Risk Information Criterion scores, and associated FRIC plots and FRIC tables, as well as to Confidence plots to exhibit the confidence the data give in the submodels. The machinery is extended to handle many focus parameters at the same time, with appropriate averaged FRIC scores. The particular case where all available covariate vectors have equal importance yields a new overall criterion for variable selection, balancing complexity and fit in a natural fashion. A connection to the Mallows criterion is demonstrated, leading also to natural modifications of the latter. The FRIC and AFRIC strategies are illustrated for real data.