Model Selection Issues

Many authors caution against the use of "automatic variable selection" methods and describe pitfalls that plague many such methods. For example, Harrell (2001) states that "stepwise variable selection has been a very popular technique for many years, but if this procedure had just been proposed as a statistical method, it would most likely be rejected because it violates every principle of statistical estimation and hypothesis testing." He lists and discusses several of these issues and cites a variety of studies that highlight these problems. He also notes that many of these issues are not restricted to stepwise selection, but affect forward selection and backward elimination, as well as methods based on all-subset selection.

In their introductory chapter, Burnham and Anderson (2002) discuss many issues involved in model selection. They also strongly warn against "data dredging," which they describe as "the process of analyzing data with few or no a priori questions, by subjectively and iteratively searching the data for patterns and 'significance'." However, Burnham and Anderson also discuss the desirability of finding parsimonious models. They note that using "full models" that contain many insignificant predictors might avoid some of the inferential problems arising in models with automatically selected variables but will lead to overfitting the particular sample data and produce a model that performs poorly in predicting data not used in training the model.

One problem in the traditional implementations of forward, backward, and stepwise selection methods is that they are based on sequential testing with specified entry (SLE) and stay (SLS) significance levels. However, it is known that the "-to-enter" and "-to-delete" statistics do not follow an distribution (Draper, Guttman, and Kanemasu 1971). Hence the SLE and SLS values cannot reliably be viewed as probabilities. One way to address this difficulty is to replace hypothesis testing as a means of selecting a model with information criteria or out-of-sample prediction criteria. While Harrell (2001) points out that information criteria were developed for comparing only prespecified models, Burnham and Anderson (2002) note that AIC criteria have routinely been used for several decades for performing model selection in time series analysis.

Problems also arise when the selected model is interpreted as if it were prespecified. There is a "selection bias" in the parameter estimates that is discussed in detail in Miller (2002). This bias occurs because a parameter is more likely to be selected if it is above its expected value than if it is below its expected value. Furthermore, because multiple comparisons are made in obtaining the selected model, the -values obtained for the selected model are not valid. When a single best model is selected, inference is conditional on that model.

Model averaging approaches provide a way to make more stable inferences based on a set of models. PROC GLMSELECT provides support for model averaging by averaging models that are selected on resampled data. Other approaches for performing model averaging are presented in Burnham and Anderson (2002), and Bayesian approaches are discussed in Raftery, Madigan, and Hoeting (1997).

Despite these difficulties, careful and informed use of variable selection methods still has its place in modern data analysis. For example, Foster and Stine (2004) use a modified version of stepwise selection to build a predictive model for bankruptcy from over 67,000 possible predictors and show that this yields a model whose predictions compare favorably with other recently developed data mining tools. In particular, when the goal is prediction rather than estimation or hypothesis testing, variable selection with careful use of validation to limit both under and over fitting is often a useful starting point of model development.