I have a MBC study, where a dependent variable has several choice categories plus "None" in it (off-state).
In order to handle "None" response estimation I brought ASC into the analysis. But I am getting ASC values for all response options in Dependent variable, not just for None category in it (which is constrained to Zero due to its off-state coding).
As I understand, these ASC values represent observed but unexplained residue or predisposition to select each of choice options shown.
I am thinking that negative average of these ASCs might be a good proxy to "None" utility.
Share=exp(A+ASC1) / (exp(A+ASC1)+exp(B+ASC2)+exp(C+ASC3)+exp(None+ASC4))
Since ASC4=0 and None has total zero utility, also assuming there is not that much difference between ASC1, ASC2 and ASC3 ~= ASC
Share=exp(A+ASC) / (exp(A+ASC)+exp(B+ASC)+exp(C+ASC)+exp(0+0))
Share=exp(A)*exp(ASC) / (exp(A)*exp(ASC)+exp(B)*exp(ASC)+exp(C)*exp(ASC)+exp(0+0))
dividing both nominator and denominator by exp(ASC) we would get
Share=exp(A) / (exp(A)+exp(B)+exp(C)+exp(0)/exp(ASC))
But exp(0)/exp(ASC) = exp(0-ASC), so
Share=exp(A) / (exp(A)+exp(B)+exp(C)+exp(-ASC))
In order words my reversed sign average ASC took place of "None" response option utility.
Is it conceptually right?
NOTE: I understand I would get exactly the same share without all these tricks if I used straight ASC1, ASC2 and ASC3. But I am trying to model choice tasks with different number of options, not just 3.
So I am not that interested these specific ASC1, ASC2 and ASC3 linked to order of options or whatever unidentified factors or biases as much as I am interested in proper None estimate.
I didn't code and structure data that way that would allow my dependent variable to be 0-1 only. This would have accommodated straightforward "None" estimation for sure. But I felt that merging other predictors in the model the way they were actually shown in choice task would allow better estimate of these merged predictors. This required having data and dependent variable structured the way I have.