# Accounting for nuisance confounding variable in multi level model for crossed or multiple membership study

I am trying to use a multi level model for a study where participants take part in 3 separate sessions where they complete a cognitive task and I measure reaction time. During each session, they complete 3 blocks of the task, so I have 3 measurements per session. Participants receive a different treatment condition each session: Placebo A, Drug B, and Drug C. It is a Latin Square design, and order is counter balanced to 6 orders: ABC, ACB, BAC, BCA, CAB, CBA. There are several participants that received each order, but it isn't perfectly balanced (i.e. some orders have 9 participants, others have 12, others have somewhere in between). We had a washout period that was selected so there wouldn't be a carryover effect.

I want to know whether there is an effect of treatment condition on reaction time. One problem that I am having trouble accounting for is the fact that there is a large session effect. The version of the task they completed during session 2 has a reaction time that is much lower than the reaction time during sessions 1 and 3. I initially had planned to analyze using RM ANOVA, but when I was unblinded, I became aware of the session effect which has made it difficult to isolate whether a treatment effect is present because it instead picks up on the within-subjects session*treatment interaction.

I have been trying to learn about multi level modeling as I have heard it is better able to handle this. However, with my level of understanding, I am unsure whether I am on the right track. I have read many helpful answers on the site, but am concerned my limited understanding that I am making an obvious mistake or misunderstanding what the models I made are doing. I do not care about session order effect--only want to account for it so I can isolate the within-subjects effect of treatment.

Currently, I believe the following is capturing the simplest version what I want, which based on my understanding should be what I go with unless I have good reason to change to something more complex. Is this correct?
model1 <- lmer(formula=RT ~ Condition + (1|participant) + (1|session) + (1|Block), data=data)

I also considered whether this may be the right approach--if block should be nested in participant. model2 <- lmer(formula=RT ~ Condition + (1|Block/participant) + (1|session), data=data, na.action=na.exclude)

Based on some reading, I wondered if I need to include session as a fixed factor. But again, I'm really not interested in it and am just trying to account for that session 2 issue. One thing that confuses me about this model is that I have convergence issues if I try to add condition or session as a random effect here. This has been one clue that I have a fundamental misunderstanding.

model3 <- lmer(formula=RT ~ Condition*session + (1|Block/participant), data=data, na.action=na.exclude)

I also considered whether this was the proper way. Thus, I keep going back and forth on the structure the random effect is capturing, and whether I need sometime like this instead. But, if I need to specify that blocks are nested in person, then this doesn't converge. model4_fit <- lmer(formula= RT ~ Condition + (1|participant/Condition) + (1|session) + (1|Block), data=data)

I'm also considering whether all of these are wrong and I need to fit a multiple membership model as in here https://github.com/jvparidon/lmerMultiMember as each participant completes 3 sessions and 3 treatment conditions.

Thank you so much for any help anyone is willing to provide.

Note that:

model1 <- lmer(formula=RT ~ Condition + (1|participant) + (1|session) + (1|Block), data=data)


and

model2 <- lmer(formula=RT ~ Condition + (1|Block/participant) + (1|session), data=data, na.action=na.exclude)


are the same model, depending on how the data are coded, as explained in the first link in the OP.

I'm also considering whether all of these are wrong and I need to fit a multiple membership model

I don't see any multiple membership with this design, unless I have not understood the OP