= 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:

$$ \mbox{R-squared (Likelihood ratio)} =  1 -   \frac{ln(L[m])}{ln(L[0] } $$

where L(m) and L(0) are the log likelihoods for the model with predictors and the model containing only the intercept respectively.

Ths statistical significance of the predictors may be ''jointly'' assessed using twice the change in the log-likelihoods in the above expression. This equals 2 (L(m) - L(0)) which is distributed as chi-square(p) if the predictors ''jointly'' have no influence on group membership. This chi-square is computed and outputted by most software which performs binary logistic regressions. In SPSS, for example, this term is denoted by the chi-square statistic produced immediately after the predictors are added to the model under the heading 'Block 1 Method=Enter'. For example running a logistic regression in SPSS to assess the joint importance of two predictors p1 and p2 with the syntax below

{{{
LOGISTIC REGRESSION  y
  /METHOD = ENTER p1 p2
  /CRITERIA = PIN(.05) POUT(.10) ITERATE(20) CUT(.5) .
}}}

we obtain the likelihood ratio chi-square in the output which is of form:

{{{
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	.138
}}}

This may be expressed as 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.