# Structuring residual (R) covariance matrix to account for dependencies in a mixed effects model of repeated measures data

I have repeated measures data measured at randomly chosen time points from different individuals. I am trying to analyze using a mixed effects model. The observations made at different sites were measured at the same (randomly chosen) time point.

I think of the model as multi-level, in that I am considering the repeated measures as at Level 1, and the different time points (and their characteristics) as at Level 2.

My problem and question is how to account for the dependencies within observations made from the same individual, since:

• all of the observations at the same individual were measured at the same time point, so, a cross-classified model will not work
• I chose to account for temporal dependencies among observations made at the same time point, rather than dependencies by individual
• there is not a three-level hierarchy such as observations (at Level 1) nested within time points (Level 2) nested within site (Level 3)
• the data are repeated measures but are not longitudinal (no change in the outcome is expected)

After reading about structuring the residual covariance matrix (i.e., Wolfinger, 1993 and Wolfinger, 1996), I tried to pursue the strategy for this problem by fixing a homogeneous residual variance by individual.

My initial strategy was to consider structuring the residual (R) matrix by fixing a homogeneous residual variance for the same individuals As an example, for the first three observations for one time point, here is the R matrix, with a fixed residual variance based on from whom the observation was made:

$$R\quad =\quad \left( \begin{matrix} var({ \varepsilon }_{ 1iindividual1 }) & 0 & 0 \\ 0 & var({ \varepsilon }_{ 2iindividual2 }) & 0 \\ 0 & 0 & var({ \varepsilon }_{ 3iindividual3 }) \end{matrix} \right)$$

Would such a strategy address the issue with a cross-classified model - that all of the observations at the same individual were measured at the same time point?