How do I summarise a fit for a logistic regression model?
Menard (2000) compares various R-squared measures for binary logistic regressions and concludes that the log-likelihood ratio chi-square is the most appropriate.
\[ R-squared (Likelihood ratio) = 1 - \frac{ln(L[m])}{ln(L[0] } \]
where L(m) and L(0) are the log likelihoods for the model with predictors and intercept only models respectively.
Ths statistical significance of the predictors may be jointly assessed using twice the change in log-likelihoods equal to 2 (L(m) - L(0)) which is approximately chi-square(p) if the predictors jointly have no influence on group membership. This term is routinely computed for binary logistic regressions and in SPSS is the chi-square statistic produced immediately after the predictors are added to the model under 'Block 1 Method=Enter'. For example running a logistic regression to assess the joint importance of two predictors p1 and p2
LOGISTIC REGRESSION y /METHOD = ENTER p1 p2 /CRITERIA = PIN(.05) POUT(.10) ITERATE(20) CUT(.5) .
we obtain in the output
BLOCK 1: METHOD-ENTER
Omnibus Tests of Model Coefficients
Chi-square df Sig.
Step 1 Step 3.958 2 .138
Block 3.958 2 .138
Model 3.958 2 .138which is a chi-square(2) = 3.96, p=0.14 indicating that together the two predictors, p1 and p2, do not have a statistically significant association with group, y.
Reference
Menard, S. (2000) Coefficients of determination for multiple logistic regression analysis. American Statistician, 54, 17-24.