Are you familiar with the game theory research on repeated prisoner's dilemma problems? Look into TF2T vs. TFT research on the repeated game and the implied reputation correction functions. The reason this is important is that it speaks to the true nature of your problem, and why a multilevel model may not work. @SteveDenham is probably not going to like my response to this, but you should not attempt to use multilevel modeling when there are likely to be lag Y and X dependencies (unless you can rule them out), the mathematics and interpretations can get very bogged down, because the lag information will capture random effect information moreso when both sides are dichotomous. Behavioral models such as this are virtually impossible to estimate accurately due to the number of potential dependencies (is Y1,2 dependent on Y1,1?, is Y1,2 dependent on X1,1? Is Y1,2 dependent on X1,1*X1,2 or Y1,1,*X1,2?). Additionally, you may have a misspecification problem. Is doing/complying different than doing/not-complying and not-doing/complying different than not-doing/not-complying? This may not actually be a truly dichotomous outcome variable, but you can control for this by using additional predictors. You have a multiclass problem that you need to control for. You are correct in searching for a multilevel model (but explicit use of one may ignore the true nature of your problem), even if you have each subject over many trials. You simply need to observe all of the possible dependencies. You have a frequency of instruction issue, a frequency of response (repeated event) issue, a consecutive event issue etc.. Consider that I have a subject that I request to do something 2 times, and not to do it the next 3. This subject never does the thing. However, presume on that same subject, I had requested them to do it 3 consecutive times then not to do it the next two times. This time, on the third instruction, they do as requested. Not only is frequency of instruction a potential concern, but so is consecutive frequency, since you have a behavioral bias problem. Additionally, not complying last time may impact my decision to comply this time. From your one IV and observed outcomes, you now have a broad set of additional dichotomous IVs that must be ruled out first. Once you have ruled out the significance of frequency and consecutive frequency concerns and prior outcome dependence, then you can apply multilevel modeling if you wish (providing that lag Y information and interactions of lag Y are not significantly predictive). My guess is, that once you control for all of these potential dependencies, you will already have a multilevel model (in essence) without using a multilevel approach. Using lags of y and x and their interactions as dichotomous predictors will pick up most of the random effects. This is because purely boolean LHS/RHS equations also have boolean lags and interactions. Hence, when both the DV and the predictive variables are dichotomous, random effects can often be picked up (more accurately, though less explicitly and it may be more difficult to interpret) within the model via additional dichotomous variables. Otherwise, simply use dichotomous regression to rule out some of these potential dependencies, then structure a multilevel model with the retained information. It seems by my reading that you are overcomplicating and oversimplifying your problem at the same time.
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