Logistic regression when the number of options differs by group?

Question

I have data from a laboratory experiment in which I look at the effect of expanding the choice set on a particular option being chosen. The idea is that introducing an Option "B" reduces demand for an Option "A."

Participants in the Control condition had the choice between A and Nothing. Participants in the Treatment condition had the choice between A, B, Both, and Nothing.

I'm interested in the likelihood of choosing A, which includes either just A or both A and B in the Treatment condition. (In this case, we can consider the choice set ordered such that Neither < B < A < Both.)

My first approach was to just collapse the decision for the Treatment group to the same binary measure: i.e. I treat "Both" the same way as "choose A only," and "B only" the same as "Nothing." For the purpose of the effect I'm primarily interested in (the likelihood of choosing A), this seemed sufficient.

However, a reviewer asked for a more flexible model, and suggested either a nested logit or a multinomial logit with a collapsed number of options in Control. The idea would still be to look at the effect on Option A being chosen, but without the assumptions of collapsing the responses to binary in the way I had done.

I am, however, a bit stumped on how to do that. The mlogit specification I tried below doesn't "look" right and, maybe predictably, gives an error that the system is exactly singular. I'm pretty sure this is because "Both" and "B" are, of course, only options in one condition.

Below is sample code to generate the data (and the analysis that leads to a singularity error).

df <- data.frame(id = 1:1000, condition = c(rep("Control", 500), rep("Treatment", 500)), choice = c(sample(c("A", "Nothing"), 500, replace = T), sample(c("A", "B", "Both", "Nothing"), 500, replace = T))) df_mlogit <- mlogit::mlogit.data(df, shape = 'wide', chid.var = 'id', choice = 'choice') mlogit::mlogit(choice ~ condition, data = df_mlogit, nests = list(A = c("A", "Both")))

Answer 1

I don't think this is a good idea to combine data coming from different data generation processes (DGP). In short, it is very likely that the modelling errors would differ between the two types of data, and then you would need to account for this source of heteroscedasticity when pooling the data together. Collapsing choice options seems to be a bad idea as it is very likely to bias your estimates (Even if one option was never selected, it was still there when people made their choices and one could argue that this undesirable option still provided a meaningful information to make a choice - so omitting to specify this info in your model would lead to an omitted variable bias).