How should I handle categorical variables with multiple levels when doing backward elimination? I'm doing a simple AIC-based backward elimination model where some variables are categorical variables with multiple levels. These variables are modeled as a set of dummy variables. When doing backward elimination, should I be removing all the levels of a variable together? Or should I treat each dummy variable separately? And why?
As a related question, step in R handles each dummy variable separately when doing backward elimination. If I wanted to remove an entire categorical variable at once, can I do that using step? Or are there alternatives to step which can handle this? 
 A: I think you'd have to remove the entire categorical variable.  Imagine a logistic regression in which you're trying to predict if a person has a disease or not.  Country of birth might have a major impact on that, so you include it in your model.  If the specific USAmerican origin didn't have any impact on AIC and you dropped it, how would you calculate $\hat{y}$ for an American?  R uses reference contrasts for factors by default, so I think they'd just be calculated at the reference level (say, Botswana), if at all.  That's probably not going to end well...
A better option would be to sort out sensible encodings of country of birth beforehand - collapsing into region, continent, etc. and finding which of those is most suitable for your model.
Of course, there are many ways to misuse stepwise variable selection, so make sure that you're doing it properly.  There's plenty about that on this site, though; searching for "stepwise" brings up some good results.  This is particularly pertinent, with lots of good advice in the answers.
A: As for the example of countries, I think if the dummy variable for a specific country is selected, then it means this country is a predictor in comparison with all other countries combined (no need to create a new binary variable). The problem I have very often is dummy variables that reflects, for example, the severity of a disease (such as -, +, ++, +++). Sometimes the dummy variable for ++ is selected but the dummy variable for +++ is not. In this case, reclassification might be useful.
