Categorical Predictors and categorical responses

Question

I have a dataset consisting of 5 categorical predictors and a categorical response (class). I want to find out which predictor has an effect on the response. Additionally, I can't guarantee whether the predictors are not correlated to each other.

I know that in the case of:
continuous predictors and continuous response -> I'd use Multiple Linear Regression
continuous predictors and categorical response -> LDA or Logistic Regression
categorical predictors and continuous response -> ANOVA

But what do I do in the case of:
categorical predictors and categorical response -> ???

What approach should I use and what are the assumptions of this approach?

Answer 1

You use logistic regression. All these forms of regression/ANOVA depend only on the nature of the dependent variable. ANOVA is the same thing as linear regression. So, here are starting places for various types of DV

Continuous, unbounded response with normal errors: Linear regression/ANOVA

Binary, categorical or ordinal response: Logistic regression of one type or another

Count response: Poisson or negative binomial regression

Time to event response: Survival methods, probably Cox model to start

Answer 2

First: independent of the predictors, if your response has

• 2 classes (binary), we would usually use a logistic regression

• 2 classes, we are speaking about a multinomial regression.

Second: regardless of what regression model you use (linear, logistic, multinomial), if you have categorical predictors, most software packages offer the choice between showing

• Regression tables (which include tests for the individual predictors, i.e. n-1 tests for a categorical predictor with n classes), and
• ANOVA, which tests for the overall significance of a predictor (i.e. test for an influence of all classes of a predictor at once).

Hence, regression tables and ANOVA have the same model underlying, but apply a different test. You can always do both. It's your choice what you want.

In R, if you are doing a logistic regression, you can do an ANOVA as you would do for the linear model (with the catch discussed here Choice between Type-I, Type-II, or Type-III ANOVA). You have to check to what extent ANOVA functions are implemented for the various multinomial regressions options in R. If ANOVA is not available, a simple substitute for a specific hypothesis would be a likelihood ratio test.