# 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.

• I was able to create the seven models you described. However, in this dataset, the Vision*Sound*Touch interaction seemed to dominate the M4, M5 and M6 models. That is, the negative impact on the odds of success observed in that condition results in these pair models showing a negative effect when the raw counts are higher than the individual conditions. Thank you for suggesting this analysis, I did learn something from trying the series of models. – Adam Faeth Jan 4 '13 at 1:51