I have a question about random effect in repeated measures design with 4 within-subjects factor ( x1
, x2
, x3
, x4
)
What the differences between these 2 ways of defining random effects:
(1 + x1 + x2 + x3 + x4 | Subject)
(1 + x1 | Subject) + (1 + x2 | Subject) + (1 + x3 | Subject) + (1 + x4 | Subject)
The first option fits a model where the within-subject deviations of the effects of all the variables are correlated with each other. Assuming that each of the factors has two levels, this will lead to estimating 5 variances (one for the intercept, one each for the effects of the factors) and (5*4/2 = 10) covariances/correlations.
The second option is wrong/problematic because it ends up repeating the intercept-within-Subject component four times (these will be redundant/unidentifiable). If you want the within-subject effects of the factors to be treated as independent, you can do
(1|Subject) + (0 + x1 | Subject) + (0 + x2 | Subject) +...
or glmmTMB
and diag(1 + x1 + x2 + x3 + x4 | Subject)
orafex::mixed()
and (1 + x1 + x2 + x3 + x4 || Subject)
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