Goodness of Fit in Regression Analysis – R ² and G ² Reconsidered

Abstract

There has been considerable debate on how important goodness of fit is as a tool in regression analysis, especially with regard to the controversy on R ² in linear regression. This article reviews some of the arguments of this debate and its relationship to other goodness of fit measures. It attempts to clarify the distinction between goodness of fit measures and other model evaluation tools as well as the distinction between model test statistics and descriptive measures used to make decisions on the agreement between models and data. It also argues that the utility of goodness of fit measures depends on whether the analysis focuses on explaining the outcome (model orientation) or explaining the effect(s) of some regressor(s) on the outcome (factor orientation).

In some situations a decisive goodness of fit test statistic exists and is a central tool in the analysis. In other situations, where the goodness of fit measure is not a test statistic but a descripitive measure, it can be used as a heuristic device along with other evidence whenever appropriate. The availability of goodness of fit test statistics depends on whether the variability in the observations is restricted, as in table analysis, or whether it is unrestricted, as in OLS and logistic regression on individual data. Hence, G ² is a decisive tool for measuring goodness of fit, whereas R ² and SEE are heuristic tools.