Why does SAS random and repeated both produce the same result?
Can someone explain this in detail?
For example:
proc mixed data=test;
class variable1 ..... variableN;
model outcome=variable1+...+variableN;
random intercept/ subject=cluster type=cs;
run;
proc mixed data=test;
class variable1 ..... variableN;
model outcome=variable1+...+variableN;
repeated/ subject=cluster type=cs;
run;
Why do both program produce the same result?
The mixed effects model can be written as $Y=X\beta+Zu+\epsilon$, where $X$ and $Z$ are matrices of known constants, $\beta$ is an unknown parameter vector, $u$ is a random vector, and $\epsilon$ is a vector of random errors, all of which are appropriately conformable. The elements of $\beta$ are considered to be non-random "fixed effects" and the elements of $u$ are "random effects." $Var(\epsilon)=R$ and $Var(u)=G$. Then the variance of $Y$, $Var(Y)=ZGZ^\prime+R$ as we assume $Cov(\epsilon, u)=0$.
SAS differentiates between "G-side" and "R-side" random effects in your model. G-side effects are random effects that enter through the $G$ matrix above and R-side effects enter through the $R$ matrix above. When you use the repeated statement in proc mixed with a compound symmetric covariance structure as you have specified in your post, you are fitting a marginal model ($G=0$) and placing all the random effects on $R$, which involves the variances and covariances for the error components in $\epsilon$. In this case, you have told SAS that $G=0$ and $R$ is compound symmetric, where each element of $R$ is equal to $\sigma_{ij} = \sigma_1^2+\sigma^2$ for ($i=j$) and $\sigma_{ij} = \sigma_1^2$ for ($i\ne j$). When you use the random intercept statement in proc mixed, you are inducing correlations through both the $G$ and $R$ matrices. In this case you are setting the elements of $G$ to $\sigma_1^2$, and you are setting $R=\sigma^2I$, so that $R$ is the identity matrix with $\sigma^2$ terms on the diagonal.
Therefore, when you use the repeated statement in SAS:
$Var(Y)=ZGZ^\prime+R = 0 + R = 0 + CS = CS$,
where $CS$ is the compound symmetric matrix. When you use the random statement, you get:
$Var(Y)=ZGZ^\prime+R = \sigma_1^2ZZ^\prime + \sigma^2 I = CS$,
since $ZZ^\prime$ is a block diagonal matrix with 1's on the diagonal blocks. As you can see, in either case the end result is the same: a compound symmetric covariance matrix.
I hope this helps. Best of luck!
"Random effects" and "repeated measures" are conceptually the same thing under most formulations of the underling linear model. In R, there is no "repeated" statement; a random effect is specified the same whether it represents, eg, a student whose performance is assessed repeatedly or a classroom in which multiple students are assessed. Finney's (1990) "Repeated measurements: what is measured and what repeats?" discusses this and points out the confusion with the terminology.
