Date: Wed, 10 Sep 2003 07:15:55 -0700 Dale McLerran "SAS(r) Discussion" Dale McLerran Re: repeated measures, correlation To: Akin Pala text/plain; charset=us-ascii

Akin Pala [mailto:akin@COMU.EDU.TR] wrote (originally): > I have two repeated Y variables and I want to measure the correlation > between them. I am guessing just use proc corr? Do I need to do any > corrections because the errors are correlated in these two dependent > factors? >

Akin,

As David Cassell points out, the simple correlation coefficient obtained through PROC CORR assumes independent samples, an assumption which your data do not meet due to the repeated measurement structure. I would note that the procedure MIXED offers the ability to estimate the covariance structure in a multivariate repeated measures model. One must employ the REPEATED statement with appropriate TYPE statement. The appropriate TYPE statements all employ a specification of UN@??? where ??? is either CS, AR(1), or UN. If you have Y1,Y2 at only two time points, then it does not matter which of the three TYPE specifications you employ. If you have more repeated measures on each subject then you will want to choose the TYPE which maximizes AIC and/or BIC statistics. It should also be observed that in order to employ the MIXED procedure with its capability of fitting multivariate repeated measures models, you must construct your data with a single response vector and an indicator variable for the response. That is, rather than constructing the data as

subject time Y1 Y2 1 1 3 4 1 2 8 7 2 1 6 8 ...

your data would need to be constructed as

subject time indic value 1 1 1 3 1 1 2 4 1 2 1 8 1 2 2 7 2 1 1 6 2 1 2 8 ...

When you fit a multivariate repeated measures model, the covariance will be partitioned into two component sets: 1) the multivariate response covariance structure, and 2) the covariance structure for the repeated measurements. The multivariate response covariance structure is reported first in the covariance parameters table, followed by the covariance structure for the repeated measurement model.

For any of the multivariate repeated measures specifications, the correlation between Y1 and Y2 is obtained from the variance/covariance matrix for the multivariate response. Given that your multivariate response has only two variables, these will be the first three components reported by the MIXED procedure in the covariance parameters table. The first parameter in the table will be the variance for variable 1, the second parameter will be the covariance between the two variables, while the third parameter will be the variance for variable 2. The correlation between Y1 and Y2 can be computed as

R(Y1,Y2) = cov(Y1,Y2) / sqrt(V(Y1)*V(Y3)) = parm2 / sqrt(parm1*parm3)

Given that your data are constructed as above, specific code for fitting the multivariate repeated measures model would be

proc mixed; class subject time indic; model value =; repeated indic time / subject=subject type=un@???; run;

again substituting CS, AR(1), or UN for ???. I don't know what the standard error of the correlation would be. If there is perfect within-subject correlation over time, then the standard error would be

se = 1 / sqrt(N-3)

where N is the number of independent subjects. If the within-subject correlation over time is zero (such that every observation could be treated as though they were independent), then the standard error would be

se = 1 / sqrt(2*N-3)

But if the within-subject correlation over time is between 0 and 1, then I do not know immediately what the standard error would be. You could obtain an estimate of the standard error of the correlation through the delta method, or by bootstrapping the mixed model. Sorry, I don't have time to elaborate on either of these.

Dale

===== --------------------------------------- Dale McLerran Fred Hutchinson Cancer Research Center mailto: dmclerra@fhcrc.org Ph: (206) 667-2926 Fax: (206) 667-5977 ---------------------------------------

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