Fisher's exact test in more than 2 variables

I have some data with three independent categorical variables (one has a sense of order, the others do not) and one dependent categorical variable. Is there an analogue to Fisher's exact test where I can simultaneously evaluate whether the independent variables affect the dependent one?

Specific details for anyone who is curious: Subjects (37) do a series of 3 different activities. After completing all activities, we ask them to rank the activities in order of their preference (6 possible ranking orders: ABC, ACB, BAC, BCA, CAB, CBA) (dependent variable). We also collect their age range (4 levels, ordered), impairment (4 categories: cognitive, motor, motor and cognitive, none), and order they performed the activities (2 possible). So, in multiple dimensions, this creates a table of 6x4x4x2.

I can certainly run fisher's exact test for each factor with the dependent variable. But that doesn't seem complete or satisfying.

• Are you sure the ranking orders are the independent variable? Please state precisely which are the independent and with is the dependent variables. Commented Feb 17, 2022 at 6:58
• wow, I need more sleep. Thanks @frank for calling that out. Edited to fix. Commented Feb 17, 2022 at 7:28
• You might want to consider a multinomial probit model. Here you model the preference of the categories A, B, C as random draws from conditional Gaussian distributions, and these Gaussian distributions relate to the categorical probability of the different orders ABC, ACB, BAC, BCA, CAB, CBA as function of the independent variables. Commented Feb 17, 2022 at 9:44
• Hmm, that is actually really interesting. I am going to explore using linear models for this a bit more. Thanks Commented Feb 18, 2022 at 20:50