Multinomial logistic regression or ANOVA to predict percentage? I have several categorical predictors:

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*experimental_group [A,B] # medicated/not

*factor1 [i,j] # task conditions

*factor2 [x,y] # task conditions

And I need to predict:

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*percentage of choices of 4 categories [c1,c2,c3,c4] in N trials

My professor suggests ANOVA: % ~ category
But this feels wrong.
And I also want to check for factor1 & factor2
Google suggests Multinomial logistic regression.

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*Should it be like this then: % ~ exp_group + factor1 + factor2

*or i should dummify categories somehow? how will the formula look then?

Data is like that:

 A: Yes, multinomial regression can be a good choice when there are more than 2 mutually exclusive outcomes. ANOVA can work when the residuals around the model predictions can come close to a normal distribution, but that's typically not the case with percentage outcomes. If there's a natural ordering to the outcomes (say, each level of strategy is known to the participant to be more difficult than the lower-numbered strategies), then ordinal regression would be a good choice.
This UCLA web page has links to implementation of multinomial or ordinal logistic regression in 5 software packages. See individual pages for cautions about important considerations like the Independence of Irrelevant Alternatives (IIA) assumption that underlies the multinomial analysis.
Data formatting depends on the software and experimental design. The R software used in the implementation explained here can accept as an outcome either a matrix with columns including the numbers of each category choice for a combination of predictor values, or a categorical outcome (strategy) with timesChosen for each particular combination of predictors as case weights. You can't just use prob as the outcome as you seem to suggest in the question, as a 10% probability is a lot more reliable if it's based on 1000 cases rather than 10 cases. Somehow the software needs to have information about the number of cases.
You will need to include subject in some form in the model, to account for correlations within individuals in their choices. How best to do that would depend on details of your overall experimental design.
