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I’m looking to build a linear mixed effects model to examine the impact of three different interventions on individuals’ ability to perform a task.

Each individual receives the three different treatments (Placebo, Treat1, Treat2) on three different days in a fully randomised order. On each day their performance was measured at three time points: [Time 1: ‘Pre-treatment’] [Time 2: ‘During-treatment’] [Time 3: ‘Post-treatment’].

Because we assume that every participant will demonstrate comparable performance at ‘Time 1’ (Pre-Treatment) on each day, we expect that the effects of a given treatment will be captured by a Treatment X Time interaction.

However, we also expect that this Treatment response will be moderated by a physiological variable that we measure at the ‘Pre-’ testing phase on each day. Therefore, using the nlme package in R I have specified the following regression (with intercept (Subject) taken as a random effect) to predict performance:

mod1 <-lme(Performance ~ Time * Treatment * Phys , random = ~ 1|Subject, data = TBRT, method = 'ML’).

We observe the hypothesised three-way interaction effect here. However, I also believe that we should model the TimeTreatment and the TimeTreatment*Phys interactions as random effects as we do not expect every individual to respond in the same way to a given Treatment, nor do we do expect that the physiological variable will moderate this relationship to the same extent for each individual.

I would greatly appreciate if someone could verify:

1) that this latter point about including these interactions as random effects seems like sound reasoning?

2) advise how to specify the random effects for just the interaction terms of TimeXTreatment and TimeXTreatmentXPhys -- as I believe the manner by which I have specified below will also treat the main effects of Time and Treatment as random (which it shouldn’t do as these are fixed categorical variables; Further, the below model, not surprisingly, does not converge).

mod2 <- lme(Performance ~ Time * Treatment * Phys, random = ~Time * Treatment * Phys | SUBJECT, data = TBRT, method = 'ML’).

Many thanks in advance,

S

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