Overall Model Fit in Logistic Regression

Abstract

The social sciences have seen a marked increase in the use of logistic regression for the analysis of dichotomous dependent variables. Walsh (1987) provides a very good non-technical introduction to the use and interpretation of logit models, but he has incorrectly reported on the method for calculating a pseudo R-squared which is an important statistic for evaluating an overall model fit. An example is provided below, using SPSSx logistic regression, to show a method for the correct calcultion of pseudo R-squared. The Pearson goodness of fit chi-square reported in SPSSX varies so that a large value of chi-square indicates a poor fit to the data. Walsh (1987, p.182) has used this chi-square to calculate a pseudo R-squared and has suggested that the fit will vary from 0 to 1, approaching 1 as the quality of fit improves. Table 1 shows the results of an example available in the SPSSX User's Guide (1988, p. 815). This example is compared with results where the values for the independent variable have been replaced using a random numbers table. The purpose of this substitution is to produce a poor fitting model with the data. The Pearson chi-square in the original model is not significant and indicates an acceptable fit of the model to the data. As we would expect, the chi-square in the random numbers model is significant indicating a poor overall fit. Calculations of pseudo R-squared as described by Walsh shows that there is a higher value where the quality of fit is in fact lower. Unfortunately, Walsh has not heeded the warning of Aldrich and Nelson (1984, p. 56) that canned programs may not directly provide the likelihood ratio statistic for goodness of fit. The Pearson chi-square reported by SPSSx is a variant of the likelihood ratio but its interpretation is exactly the opposite to Walsh's suggestion. The pseudo R-squared (Aldrich and Nelson 1984, p. 57) may be calculated using the Pearson chi-square statistic reported by SPSSx: Pseudo R-squared = 1 C/(N + C). This may be verified by comparing the algorithm for Pearson chi-square in SPSSX (Norusis 1985, p. 304) to the formulas provided in Aldrich and Nelson (1984, pp. 55-56). A recalculation of pseudo R-squared for the examples in Table 1 suggests an acceptable overall fit for the original