Covariance Structure
Study guide: Exam MAS-II
The covariance structure of a Linear Mixed Model is the pattern imposed on \mathrm{Var}(\mathbf{y}) = \mathbf{Z}\mathbf{D}\mathbf{Z}^\top + \mathbf{R}. Choosing it is how the model says which observations are correlated and how the correlation decays.
\mathbf{V}_i = \mathbf{Z}_i\mathbf{D}\mathbf{Z}_i^{\top} + \mathbf{R}_i
Common residual structures for a group of n repeated measures:
| Structure | Form of \mathrm{Corr}(y_{ij}, y_{ik}) |
Parameters | Fits |
|---|---|---|---|
| Independent (VC) | 0 |
1 | no within-group correlation |
| Compound symmetry (CS) | \rho for all j \neq k |
2 | exchangeable measures — the random-intercept model |
| First-order autoregressive AR(1) | \rho^{\,\lvert j-k\rvert} |
2 | equally spaced measures over time |
| Toeplitz | \rho_{\lvert j-k\rvert} |
n |
banded, decay not forced to be geometric |
| Unstructured (UN) | free | n(n+1)/2 |
anything, at maximum cost |
- A random intercept already implies compound symmetry — the two are the same model written differently
- More structure means fewer parameters and more power; the wrong structure biases the standard errors of the fixed effects even when
\hat{\boldsymbol\beta}is fine - Compare nested structures with a Likelihood Ratio Test fitted by REML (the fixed effects must be identical), or non-nested ones with AIC
- Under-specifying (assuming independence when data are correlated) understates standard errors and finds effects that are not there
Choosing Between CS and AR(1) Quarterly loss ratios for 40 accident years are modelled with a random year effect. The empirical correlations between quarters 1 apart, 2 apart and 3 apart are 0.62, 0.38 and 0.24. Which structure fits?
Answer Compound symmetry predicts one common correlation — the observed values decay, so it does not fit. AR(1) predicts
\rho, \rho^2, \rho^3. With\rho = 0.62:0.62,\ 0.384,\ 0.238— almost exactly the observed pattern. AR(1), at the same two-parameter cost as compound symmetry.