My question is conceptual. Suppose $n$ patients, where each one is measured at 4 different time points. The outcome is continuous. The patients are randomly assigned to two groups, intervention yes/no. The first time measurement is the baseline outcome before the intervention, 2nd till 4th time measurements are after the intervention.
Two possibilities (in my view):
- Something like:
Outcome ~ Intervention, random = ~ patient id | time.
Intervention as fixed effect, random intercepts for each patient, random slopes allowed at different time measurements.
Am I right that the random intercepts already imply a correlation BETWEEN patients whereas random slopes imply a correlation WITHIN patients (even though not explicitly)? In other words, if I didn't assume a correlation within patients over the different time measurements, I wouldn't allow random slopes and I would just estimate a random intercept model. So there is no need in additionally specifying an autoregressive covariance matrix, since an autoregressive pattern is already implied and captured in the model by allowing random slopes to vary over the different time measurements. Am I right about this point?
Another question: Should I include Baseline (= outcome before intervention) as a covariate, e.g.,
Outcome ~ Baseline + Intervention, random = ~ patient id|time?
Outcome ~ Baseline, random = ~ Intervention | time
Baseline as fixed effect, Intervention as grouping variable, random slopes allowed over different time measurements.
Let's again assume an autoregressive pattern of the error terms (correlation within each patient) over the different time measurements. Here I would actually specify an autoregressive correlation matrix, since a correlation of error terms is not yet captured in the model.
Under the assumption that the variance among groups varies over the different time measurements, should I maybe even consider an unstructured covariance matrix to additionally account for the different variance within groups?
Which model would u prefer and why?