Logistic Regression and the Chi-Square Test Statistic

• In Event: 46.063 - Multiple Regression: Principles and Practices

Abstract

This presentation has three objectives. The first objective is to demonstrate how logistic regression predicts the probability of a dichotomous outcome with respect to admission/non-admission to Graduate School. The second objective is to set forth seven logistic regression equations to determine which three independent variables can improve prediction of admission to Graduate School. The third objective is to establish the chi-square statistic as a test of significance for the actual versus predicted probabilities from the logistic regression equation.

Logistic regression, unlike ordinary least squares regression has a different set of assumptions.
The assumptions of logistic regression, in contrast to multiple regression, are as follows: (1) A linear relationship between the dependent and independent variables is not assumed. (2) There is no assumption that the dependent variable is normally distributed. (3) There is no assumption of homogeneity of variance among or within the categories. (4) There is no assumption that error terms are normally distributed. (5) Interval data is not required of independent variables.

Test(s) of model fit are also different in logistic regression. A -2LL chi-square test is used to determine model fit rather than the traditional R-squared value. Logistic regression predicts a probability, that is, as the values in X increase the probability that Y will equal 1 (a successful outcome) will increase. Regression weights are interpreted as an odds ratio (OR) or risk ratio (RR), which again is different from ordinary least squares regression. For example, if exp(B) equals 2, the probability that Y will equal 1 is twice as likely as X increases by one unit. If exp(B) equals .5 the probability that Y will equal 1 is half as likely as X increases by one unit. If exp(B) equals 1 there is no relationship between X and Y.

The data source consisted of four hundred participants (N=400) with the following independent variables: GRE score, GPA, and rank of the undergraduate institution attended. The dichotomous dependent variable was admission status (1=yes, the student was admitted to a Graduate School, 0=no, not admitted to a Graduate School). The logistic regression model indicated that the independent variables GRE, GPA, and Rank were significant predictors of whether or not a student was admitted to a Graduate School.

The presentation will discuss two practical issues related to logistic regression. First, the order of independent variable entry into the equation can effect the predictability of the model. The second issue relates to testing model fit. The traditional -2LL chi-square reduction when adding predictors provides an omnibus model fit measure. A practical outcome is the classification accuracy table, that is, percent correctly classified. A chi-square test can be computed based on the actual versus predicted values from a given logistic regression equation. To our knowledge this is a test of significance that has not been previously reported.

Authors

• Alan Luther Webb, The University of Alabama

• Robin Harvey, The University of Alabama