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# Regression model for data where limited budget is assigned to different items

I have a task where participants are given a limited number of tokens which they can assign acorss different items. The number of tokens given to an item represents the importance of one item over the others so they have to make trade-offs the fewer tokens they get. The participants get a random number of tokens they can allocate.

Since I think the data violates the assumptions of a standard linear regression just looking at total tokens assigned to each item (e.g. since the total tokens assigned to each item aren't independent because if you assign some tokens to an item you have fewer tokens for the other items) I was wondering what kind of model people would recommend for this.

I have looked at using a mutinomial logistic model where the number of tokens a participant has is represented as a the number of trails the participant does. Each trial the participant selects one of the items to assign a token to. Then I would have random intercepts for participant and random slopes for amount of trails.

How does this sound and how else might you model this?

• That sounds reasonable to me!

E.J.

• I have just realized this may also be problematic since your choice at trial 3, for example, isn't independent of your choice at trails 1 and 2.

That is, if you have 5 items, if you chose to put a token on item 2 in trial 1, that would effect whether you put a token on item 2 in the second trail (you are possibly more likely to choose to put a token on a different item).

I guess I need some kind of analysis for predicting proportions on items where choice isn't independent. Any ideas? 😅

• I wonder if I need to add a five predictors (one for each trait) for "total-given-so-far" to that trait for each trial? Is there a more sane analysis for this that would be recommended?

• If you want to model the process by which people distribute the tokens one-by-one you may need a more complicated model. What do you wish to learn from these data?