In linear regression, I've seen (granted, not many) situations where basic transformations to some of the predictors can significantly improve the fit and stability of the model, and often a scatterplot of the response variable vs each individual predictor can provide useful clues as to whether a transformation can help.
Is there a similar approach that would work for logistic regression - i.e., what is an intuitive way to determine whether any predictor transformations can help?
If it's feasible that you can bin in such a way that the bins aren't too wide but all of them contain some 0s and some 1s, it's possible to do a logit-transformation on the proportions and see if logit(p) is reasonably linear. However, the presence of other (meaningful/important) IVs can make the impression from such marginal relationships meaningless
Another possibility is to fit a nonparametric relationship and see if it's clearly showing signs that it's not logistic in shape; this can be done in the presence of other covariates, since one can fit a GAM term in an otherwise linear (in the GLM sense) model.
This is only a partial answer: Transformation in linear regression can be used to improve model fit if residuals are not normally distributed (e.g. skewed), or if the relationship between predictor and dependent variable is not linear. However, for the latter other modeling strategies exist (quadratic terms, or using non-linear regression)
In logistic regression residuals are not assumed to be normally distributed but to follow a binomial distribution. Furthermore, the relationships between predictor and dependent variable is not assumed to be linear, yet the relationship between predictor and the predicted logits is assumed to be linear. Logits are: ln(p(y=1)/(1-p(y=1)))
Thus, I would assume that if those assumptions are violated, a transformation might improve model fit the same way as in linear regression. However, that is only my statistical intuition, and I have no prove of that. Furthermore, you can only transform the predictors (the dependent variable has to be binary).
A more modern approach would be to use generalized linear models. By this, you can choose the distribution of the residuals freely. Non-linearity can be modeled by including non-linear terms into the model. However, I am not an expert on that topic, but hopefully some of my answers will give you some first hints into the correct direction. ;)
