Is it a good idea to cluster predictor variables to try an improve classification performance with logistic regression? I have trained a logistic regression model on on a selection of 10 socio-demographic predictor variables, all of which are categorical, in order to predict customer behavior on an outcome measure. Out of the 10 variables only three significantly contribute to the model, as assessed by a forward elimination model comparison procedure. The AUC on the testing set is .64 which is ok but we want to improve the model more. My question has two parts:
1) is it a good idea/good practice to cluster the remaining seven variables and enter these clusters as predictors in the model? 
2) When clustering, should I include the outcome measure as an input variable to the cluster analysis in order to make the clusters more related to the outcome measure?
 A: Ultimately it is hard to say whether this will help or not. Many preprocessing ideas like this work in specific situations despite being useless in many others. Obviously you can try it out using cross-validation on an "experimentation part" of your dataset that isn't used for the evaluation of what you finally come up with.
That said, chances are that this is not a particularly good idea, and as long as you don't do things such as double cross-validation, you need to be aware that the more different approaches you compare to arrive at something good, the more your results will be over-optimistic on your data due to model selection bias.
A general consideration here is that forward selection is not a suitable method to decide which variables are significant, and very often not a good method to do variable selection at all. If you want to make statements about the significance of individual variables, do this from the p-values that you get from the full model, at least as long as you have enough observations to fit it properly (say number of variables times 5-10). It is absolutely not mandatory to remove insignificant variables, and for prediction purposes the full model is often better than the model you get by forward selection (as could be checked once more by cross-validation). Note in particular that standard tests and p-values computed on a subset of variables arrived at by forward selection on the same data are invalid. So the first possibility that you have is to just use the full model regardless of how many variables are significant there. Another option are regularisation methods such as the Lasso.
Regarding adding information of clusters of variables, my intuition is that this doesn't have much potential to improve things. Obviously it provides you with some kind of summary of the variables that otherwise you wouldn't use (although you may well use them all, see above), and if you're lucky, with this summary you can explain your response to some amount, but as the response is not used for clustering, there is no guarantee for this, and using the clustering information instead of all the variables may well miss some relevant content of/interaction between them. It becomes a bit more promising if these variables have many categorical levels, and you don't have enough observations to fit all the parameters (which may also be a reason why you used forward selection). In any case "forward selection and then clustering of not selected variables" is, as far as I know, nowhere systematically investigated, so nobody can grant you that it's any good. Intuitivaly I don't expect it to be good, but thinking a bit more about it, I see a potential in very specific situations, as long as it's done in a way that model selection bias is controlled (e.g., with double cross-validation or using three parts of the data to fit, select, and validate).
Regarding using the response variable for clustering, this will incur model selection bias, so normally this is not a good idea, however once more, in specific situations it may help if you can control that bias. In any case I doubt you'll find any literature advising to do this.
A: Conditional logistic regression or fixed/random effect models may be useful for your purpose. It attempts to model the behaviour of sub groups within your larger dataset.
Researching them might help you avoid reinventing the wheel. 
