I'm working with experimental data. Participants were assigned to one of three conditions and made a single choice (A, B, C, or D).
So the data look something like this, where the values are counts (made up numbers because I don't have them in front of me):
Condition 1: 35 A's, 23 B's, 17 C's, 2 D's
Condition 2: 17 A's, 73 B's, 45 C's, 23 D's
Condition 3: 71 A's, 15 B's, 61 C's, 17 D's
A possibility I would like to rule out is that some portion, p, of the participants in Condition 1 treat it the same way they would condition 2, while the rest, 1-p, treat it the same way they would condition 3, thus making the distribution of choices in condition 1 just a mixture of the distribution of the choices in the other two.
What I would like to do is:
1) Find p* that maximizes the likelihood of some mixture of the distribution in Condition 2 and that in Condition 3 producing the observed choices in Condition 1.
2) A goodness of fit test to show (hopefully) that even with the best fitting p*, the distribution observed in Condition 1 is unlikely to be a mixture of that in the other conditions.
I have found some similar problems solved online, BUT they all involved THEORETICAL distributions as the constituent elements of the mixture. Is there a way to do this given that I only have empirical distributions?
I am considering the following approach (which I'm guessing is probably wrong):
Because I only care about the estimate p*, not its distributional characteristics, and because the sample proportions are the MLEs for the multinomial probabilities, I will ignore the uncertainty in the sample proportions for the first step. That is, I will calculate the likelihood of a given value of p as:
(p*(17 / (17 + 73 + 45 + 23)) + (1-p)*(71 / (71 + 15 + 61 + 17))^35 + ... (for each choice option A through D)
and maximize to obtain p*.
Then for the next step I will calculate a goodness-of-fit measure for the hypothesis that my sample proportions in condition 1 and the proportions from the mixture model come from the same distribution. This will serve as the test statistic, but I cannot use standard critical values because I have ignored the uncertainty in the same proportions. Rather I would use permutation to simulate the distribution of that same test statistic under the null.