Alternatives to ANOVA to find differences between three groups? So my data is looking at the incidence of emotions within therapy sessions. I am looking at three groups that I suspect have different pattern across several emotions
Group 1 n=36
Group 2 n=12
Group 3 n=6
The variables are measured in terms of time spent in that emotion (e.g., 10 minutes of anger, 5 minutes of sadness). I have also constructed a proportion for each emotion based on how much time spent in a given emotion divided by the total time spent experiencing any emotion (e.g., 10 minutes of anger divided by 20 minutes in emotion = 50% anger in that session).
I am aware that the group sizes are uneven, and that one group is quite small - these are exploratory analyses so I would just like to find a way to compare them statistically as a starting point.
Given that the groups are different across emotions - there is a low incidence of certain emotions in some groups, thus violating normality and homogeneity of variance.
What I want to know is whether these groups are different across each emotion type. Any ideas of what would be appropriate or other considerations before making a decision?
 A: If at each evaluation time there is either no emotion or exactly 1 emotion, then multinomial regression should be a good choice. That models the log-odds of each of a defined set of categorical outcomes against a reference category, in a situation where there is exactly one outcome per observation. This is effectively a set of binomial regressions modeled in a way that ensures the probabilities over all outcomes sum up to 1. Normality isn't an issue. Unequal group sizes will limit the ability to detect true differences but shouldn't matter otherwise.
You should include "emotion-free" as one of the outcomes, at least to start your model, as you want there to be exactly 1 outcome at each observation. "Emotion-free" might be a reasonable choice for the outcome reference category. Group membership would be the predictor/independent variable. The above link illustrates the procedure for single observations, but the help page for the multinom function used there shows how to set up the outcome as a set of columns of counts for each of the outcomes. Those values should be integers. For example, with evaluation of emotion done once per minute, they would be the number of minutes in each state.
After you've built that model, you can interrogate the results in terms of probabilities to perform the comparisons the you want. That could include comparisons among groups of proportion of any emotion time in each emotion. Post-modeling tools like those in the emmeans package can make that task easier.
