I have data from a randomized survey experiment in which each respondent was assigned to one of 4 groups, one of which can be considered a "control" or "no treatment" group. The key question asked in the survey was a binary one: i.e. each respondent was faced with a choice between two products given some stimulus based on the assigned group. Of course, there are several other questions to be controlled for (demographics, pre-existing preferences, etc.).
I want to know what effect, if any, being in a particular group had on the respondent's choice for that key question, controlling for the other factors. Since my response variable is categorical I can't use ANOVA (at least R doesn't appear willing to let me have a non-numeric response variable). I have tried to do a logistic regression but it seems like the structure of my data means that this would result in the respondents in each group being compared to the rest of the respondents which seems like it would be incorrect.
My data resembles the following in structure:
| Id | Group | Product Chosen | ... (other variables) | 1 | 1 | A | ... | 2 | 4 | B | ... | 3 | 3 | B | ... | 4 | 2 | B | ... | 5 | 1 | A | ... | 5 | 2 | B | ... | 5 | 4 | A | ... | 5 | 3 | B | ...
In case it is relevant, I have been using R for my analysis.
Update: Just so it's clear, my working hypothesis is that respondents in non-control groups were more likely to choose product A than B (and less importantly, but similarly, that respondents in group 2 were more likely than those in group 3, and those in group 3 were more likely than those in group 4).