I'm fitting a linear mixed model by SAS.
There are 596 sectors and 8489 subjects. (each sector contains 10~15 subjects).
Each subject is measured at most 6 times, so the total number of observation is 50043.
SAS code are as follows.
proc mixed data=work.dat2 covtest method=ml maxfunc=1000 ;
class group_k sectorid childuid;
model laz=group_k x1 x2 x4 x6 x1_k x2_k x4_k x6_k
/ solution cl outpm=out;
random sectorid;
repeated / subject=childuid type=cs ;
run;
One of the result tables is as follows. The first table is without the option 'type=cs' (default type=VC), and the second table is with the option 'type=cs'.
I don't understand the relation between the SAS result table and the theory presented here http://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/viewer.htm#statug_mixed_sect003.htm
The covariance matrix for the default 'variance component' is $ \sigma_k^2 1(i=j) $ and that for the type 'compound symmetry' is $ \sigma_1 + \sigma^2 1(i=j) $
In the two tables, I don't understand which corresponds to $ \sigma_k, \sigma_1, \sigma $ respectively.
Best Answer
The statement
random sectorid
stipulates that the variablesectorid
enters as a random effect ($G$-side of the model). This is the only random effect for the $G$-side ($k=1$). By default, the covariance structure is "Variance Components" (VC). The variance component is given in the row calledSECTORID
in the "Covariance Parameter Estimates" Table. In your first case, then, $ G = 0.08433 \, I_{596}, $ with $I_{n}$ the identity matrix of dimension $n$.The statement
repeated / subject=childuid
stipulates that the $R$ matrix, i.e. the covariance matrix of the residuals, is block diagonal with one block for each level ofchilduid
. By default, the common covariance structure is "Variance Components" (VC). The corresponding variance component, $\sigma^2$, is given in the row calledResidual
in the "Covariance Parameter Estimates" Table. In your first case, then, each block has a $1.1341$ on the diagonal.Specifying
type=
in therepeated
statement changes the common block structure of the $R$ matrix. Withtype=cs
, $\sigma^2$ is still in the row calledResidual
and $\sigma^2_1$ is given in the row calledCS
.