Is there anything like a two-way ANOVA but for continuous independent variables and a nominal dependent? Also, alternatives to logistic regression? I want to determine the interaction between two of my continuous (scale) independent variables on the one dependent variable, which is dichotomous (cases are coded as 0 and 1). I was going to use a two-way ANOVA, but you need categorical or continuous independent variables as well as a continuous dependent. Is there any similar test that would work for my variable types?
I was also going to do a logistic regression to determine how my independent variables predict my dependent. However, upon performing the Box-Tidwell test in SPSS, I learned that two of my independent variables do not have a linear relationship with the logit transformation of the dependent. This violates an assumption of performing logistic regression. My options are to either take out the offending variables (I'd prefer not to) or not do the regressions. What is recommended when a logistic regression cannot work?

People have been routinely incorporting nonlinear relationships in binary, ordinal, and polytomous logistic regression since the early 1980s using regression splines.  It is not a good idea to use some statistical test to screen the model (which distorts operating characteristics).  It's a better idea to pre-specify which continuous predictors are not known to act linearly, and to allow them to be nonlinear.  This will result in valid compatibility intervals and p-values.  I cover this in detail in RMS.
