Q:

How do I model interaction of three boolean variables for logistic regression when there are empty cells?

I collected data to find whether the presence or absence of vision, sound, and touch during a task affected the successful completion of that task. However, there were no samples collected where all three senses were absent. So the dependent variable is boolean success but I have a question about how to model the independent variables in a logistic regression.
My initial analysis used a single categorical variable with seven levels representing each combination of senses (seven because there were no cases where all three senses were absent).
summary( glmer( Success ~ Condition + ( 1 | Participant ), family=binomial, data=trials))

When I tried to build a model with the Vision, Sound, and Touch as separate variables, the analysis fails. I believe this is because I have empty cells when including the vision*sound*touch interaction because we did not collect results where all senses were absent.
summary( glmer( Success ~ Vision + Sound + Touch + Vision*Sound + Vision*Touch + 
                Sound*Touch + Vision*Sound*Touch + ( 1 | Participant ), 
                family=binomial, data=trials))

I followed the suggestion linked above to use the interaction function to drop the unused factor (all three senses absent). However, this seems to create a variable that looks like my original single categorical variable.
senses <- interaction( trials$Vision, trials$Sound, trials$Touch, drop=TRUE )
summary( glmer( Success ~ senses + ( 1 | Participant ), family=binomial, data=trials))

As I try to refine this analysis, is there a way to model the senses as separate variables to make the interaction between these variables clearer? That is, to appropriately model the contribution of vision in the vision, vision*sound, vision*touch and vision*sound*touch conditions. From the initial analysis, the vision*sound*touch interaction is the most interesting.

A:

You could run a series of analyses, with different combinations of the levels. This would, I think, essentially duplicate what a typical analysis would do. You can then compare the results of the different models. So
M1, M2, M3 - each have only two levels for the IV, corresponding to the three dichotomous IVs.
M4, M5, M6 - each have four levels of the IV corresponding to pairs of variables
M7 is your original model.

