In answering this question John Christie suggested that the fit of logistic regression models should be assessed by evaluating the residuals.  I'm familiar with how to interpret residuals in OLS, they are in the same scale as the DV and very clearly the difference between y and the y predicted by the model.  However for logistic regression, in the past I've typically just examined estimates of model fit, e.g. AIC, because I wasn't sure what a residual would mean for a logistic regression.  After looking into R's help files a little bit I see that in R there are five types of glm residuals available, c("deviance", "pearson", "working","response", "partial").  The help file refers to:  


Davison, A. C. and Snell, E. J. (1991) Residuals and diagnostics. In: Statistical Theory and Modelling. In Honour of Sir David Cox, FRS, eds. Hinkley, D. V., Reid, N. and Snell, E. J., Chapman & Hall.  


I do not have a copy of that.  Is there a short way to describe how to interpret each of these types?  In a logistic context will sum of squared residuals provide a meaningful measure of model fit or is one better off with an Information Criterion?

The easiest residuals to understand are the deviance residuals as when squared these sum to -2 times the log-likelihood. In its simplest terms logistic regression can be understood in terms of fitting the function $p = \text{logit}^{-1}(X\beta)$ for known $X$ in such a way as to minimise the total deviance, which is the sum of squared deviance residuals of all the data points.

The (squared) deviance of each data point is equal to (-2 times) the logarithm of the difference between its predicted probability $\text{logit}^{-1}(X\beta)$  and the complement of its actual value (1 for a control; a 0 for a case) in absolute terms. A perfect fit of a point (which never occurs) gives a deviance of zero as log(1) is zero. A poorly fitting point has a large residual deviance as -2 times the log of a very small value is a large number. 

Doing logistic regression is akin to finding a beta value such that the sum of squared deviance residuals is minimised. 

This can be illustrated with a plot, but I don't know how to upload one.
