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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?

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    $\begingroup$ Descriptions of the input and the output are necessary in order for this question to fit within our format. Please see the help center for more information. $\endgroup$ – whuber Jul 15 '14 at 0:08
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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!

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"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.

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