# What's the correct analysis to use for a proportional, multinomial, repeated measures DV?

I'm looking at data from an experiment that examines how three different types of cognitive load (two levels each: present or not present) affects self-reports of thought category. Every subject performs a task eight times, each time under a different configuration of the three types of load until all possible configurations are covered. The order of the conditions is randomized for each subject. Essentially: three categorical IVs, two levels each, within-subjects design. (An example block would be: load type 1 present, load type 2 not present, load type 3 present.) Sample size is N = 32.

During a single run of the task (we'll call this a "block"), the subject is "probed" either three or four times during the course of the task. A probe consists of choosing between four different types of thought (let's call them T1, T2, T3, and T4.) So by the time the entire experiment (8 blocks) has run its course, the subject has been probed between 24 and 32 times.

It's expected that the vast majority of responses will be either T1 or T2, T3 is infrequent, and T4 is particularly rare.

I want to analyze the effect of the IVs (factorial design) on two separate but highly related dependent variables - DV1 is the frequency of responses that are either T3 or T4 within a given block, and DV2 is the ratio of T4 responses to T3 responses within a given block, somehow taking into account DV1. The effect of the IVs on DV1 is fairly easy to analyze, but I'm not sure what the correct way to approach the DV2 analysis is, due to the complicated nature of my data (3 IVs, within-subjects design, and 3 or 4 measurements per block) and also the fact that T3 is infrequent and T4 is rare.

I considered running a multinomial logistic regression, but I don't think it's the correct analysis since I'm really looking at two different dependent variables extracted from the categorical measurement, as opposed to one. I cannot run a factorial ANOVA due to the large amount of missing data. Does anyone what would be the most statistically reliable and practical way to run the second analysis?

Any help would be appreciated, and I am happy to provide additional clarification. Thank you!