## Compound Symmetry vs Random Intercept

Frequent Contributor
Posts: 75

# Compound Symmetry vs Random Intercept

I have been reading Robert Weiss's Modeling Longitudinal Data for help on how to correlated data and my question centers around the two proc mixed commands

repeated / type = CS
random intercept / type = UN

now when I run both models I get the same answers for the covariance parameters. According to Weiss the CS model has two parameters: the variance of the data (t^2) and correlation p. Now the RI model also has two parameters: the variance the data around the random intercept (s^2) and variance of the random intercepts themselves D.

Now Weiss says that t^2=s^2+D and that p=D/(s^2+D). SO my question is whether the two procs give us the (t^2,p) parameters or the (s^2,D) parameters????

Given that both procs give the same answers I am led to believe repeated / type=CS is actually giving me the (s^2,D) parameters and I actually have to solve for t^2 and p to get the true CS parameters. What is everyone's thoughts on this?

Thanks!!!

Thanks
Regular Contributor
Posts: 171

## Re: Compound Symmetry vs Random Intercept

You are correct that both models are returning the parameter set (s^2,D) and that if you want t^2 and p, you would have to solve for them.

I don't know why you call the parameter set (t^2,p) the "true CS parameters". The set (t^2,p) conforms to one parameterization of a compound symmetric residual covariance structure if you project t^2 on to the diagonal and p*(t^2) onto the off-diagonal terms of the covariance matrix..

You might note in this phrase that we are dealing with a covariance matrix. The parameter p is not directly observed in the variance/covariance matrix. Rather, the value of p for a particular row and column is the ratio of an off-diagonal covariance term to the square root of the product of the two variances indexed by the pairs (row,row) and (column,column). In a compound symmetric matrix, the diagonal terms are all the same (t^2) and the off-diagonal terms are all the same. Because the diagonal terms are a constant (the sum of the variance of the means and the variance about the means) and the off-diagonal terms are a constant (the variance of the means), the correlation of any two observations from a given subject is constant. But you observe that we are defining p in terms of variances. I would say that the parameterization which SAS employs for a compound symmetric covariance structure is the natural parameterization. (Note the avoidance of the term "true parameterization.")

By the way, it is probably quite rare that one would encounter a compound symmetric error structure when modeling longitudinal data. It is possible - but not likely. Observations which are adjacent in time would typically have a stronger correlation than observations which are distant in time.
Frequent Contributor
Posts: 75

## Re: Compound Symmetry vs Random Intercept

Thanks Dale
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