Variable selection - A review and recommendations for the practicing statistician

Abstract

Statistical models support medical research by facilitating individualized outcome prognostication conditional on independent variables or by estimating effects of risk factors adjusted for covariates. Theory of statistical models is well-established if the set of independent variables to consider is fixed and small. Hence, we can assume that effect estimates are unbiased and the usual methods for confidence interval estimation are valid. In routine work, however, it is not known a priori which covariates should be included in a model, and often we are confronted with the number of candidate variables in the range 10-30. This number is often too large to be considered in a statistical model. We provide an overview of various available variable selection methods that are based on significance or information criteria, penalized likelihood, the change-in-estimate criterion, background knowledge, or combinations thereof. These methods were usually developed in the context of a linear regression model and then transferred to more generalized linear models or models for censored survival data. Variable selection, in particular if used in explanatory modeling where effect estimates are of central interest, can compromise stability of a final model, unbiasedness of regression coefficients, and validity of p-values or confidence intervals. Therefore, we give pragmatic recommendations for the practicing statistician on application of variable selection methods in general (low-dimensional) modeling problems and on performing stability investigations and inference. We also propose some quantities based on resampling the entire variable selection process to be routinely reported by software packages offering automated variable selection algorithms.

Keywords: change-in-estimate criterion; penalized likelihood; resampling; statistical model; stepwise selection.

Figure 1

Simulation study to illustrate possible differential effects of variable selection. Graphs show scatterplots of estimated regression coefficients ${\widehat{\beta }}_1$ and ${\widehat{\beta }}_2$ in 50 simulated datasets of size $N=50$ with two standard normal IVs with correlation $\rho =0.5$. Circles and dots indicate simulated datasets where a test of the null hypothesis ${\beta }_2=0$ yields p‐values greater or lower than 0.157, respectively. The dashed lines are regression lines of β₁ on β₂; thus they indicate how β₁ would change if β₂ is set to 0

Figure 2

A schematic network of dependencies arising from variable selection. β, regression coefficient; IV, independent variable; RMSE, root mean squared error