# Which mixed model is closest to GLS with unstructured covariance? Random intercept + slopes or random slopes only?

I would like to closely reproduce a model fit via generalized least square method with unstructured residual covariance by using a mixed model.

Which mixed model is closest to GLS with unstructured covariance? Random intercept + slopes or random slopes only? If if both - correlated or not?

Let's assume it's about repeated observations over time across two groups: Treatment A and Treatment B, where Time is categorical: T1, T2, T3, T4

Treatment is a fixed effect. Time is a fixed effect. Subject ID is the random effect.

We can consider the following options and I am curious which one is the best:

1. random intercept only: NO, because I expect different correlations over time for each subject, not the same. This rather refers to a compound symmetry, which is not of my interest.

2. random slope only: ? I expect different correlations over time for each subject, but they start from a similar conditions. In randomized trial it may be pretty sensible approach, isn't it?

3. random slope and random intercept uncorrelated: ? I expect different correlations but subjects may start from different conditions. The initial conditions do not have to be linked to the further correlations. I have a randomization, so I'm wondering about the random intercept here, but maybe this approach is anyway worthwhile?

4. random slope and random intercept correlated: ? As above, I expect different correlations and subjects start from different conditions and maybe the initial conditions are somehow related to the correlations. I am not sure. This seems to be the most advanced approach, but maybe too complex?

I expect there may be no clear translation from the G (random effects) correlations to R (residual) correlations, but I think the random slopes or random slopes + random intercepts seem closest to unstructured covariance (no constraints). But which one?

Suppose we have the linear mixed model $$y_i = X_i\beta + Z_i b_i + \varepsilon_i,$$ with $$b_i \sim \mathcal N(0, D)$$, and $$\varepsilon_i \sim \mathcal N(0, \sigma^2 I)$$. By using random effects, you imply the following structure for the marginal covariance matrix: $$V_i = Z_i D Z_i^\top + \sigma^2 I.$$ Hence, if you would like $$V_i$$ to become an unstructured matrix, you would need to consider a suitable version for the random-effects design matrix $$Z_i$$. An option is to include a separate random effect per time point (though there could be a small identifiability issue with $$\sigma^2$$).