The CORR Procedure

You can use the following statements to test the null hypothesis against a two-sided alternative . The test is requested with the option FISHER(RHO0=0.5).

title 'Analysis for Batch 1';
proc corr data=Sim (where=(Batch=1)) fisher(rho0=.5);
   var X Y;
run;

Output 1.4.1 displays the results based on Fisher’s transformation. The null hypothesis is rejected since the p-value is less than 0.0001.

You can use the following statements to test for equality of two population correlations, and . Here, the null hypothesis is tested against the alternative .

ods output FisherPearsonCorr=SimCorr;
title 'Testing Equality of Population Correlations';
proc corr data=Sim (where=(Batch=1 or Batch=2)) fisher;
   var X Y;
   by Batch;
run;

The ODS OUTPUT statement saves the "FisherPearsonCorr" table into an output data set in the CORR procedure. The output data set SimCorr contains Fisher’s z statistics for both batches.

The following statements display (in Output 1.4.2) the output data set SimCorr:

proc print data=SimCorr;
run;

The p-value for testing is derived by treating the difference as a normal random variable with mean zero and variance , where and are Fisher’s z transformation of the sample correlations and , respectively, and where and are the corresponding sample sizes.

The following statements compute the p-value in Output 1.4.3:

data SimTest (drop=Batch);
   merge SimCorr (where=(Batch=1) keep=Nobs ZVal Batch
                  rename=(Nobs=n1 ZVal=z1))
         SimCorr (where=(Batch=2) keep=Nobs ZVal Batch
                  rename=(Nobs=n2 ZVal=z2));
   variance = 1/(n1-3) + 1/(n2-3);
   z = (z1 - z2) / sqrt( variance );
   pval = probnorm(z);
   if (pval > 0.5) then pval = 1 - pval;
   pval = 2*pval;
run;

proc print data=SimTest noobs;
run;

In Output 1.4.3, the p-value of 0.2795 does not provide evidence to reject the null hypothesis that . The sample sizes and are not large enough to detect the difference at a significance level of .

Assume that sample correlations and are computed from two independent samples of and observations, respectively. A combined correlation estimate is given by , where is the weighted average of the z transformations of and :

The following statements compute a combined estimate of by using Batch 1 and Batch 3:

ods output FisherPearsonCorr=SimCorr2;
proc corr data=Sim (where=(Batch=1 or Batch=3)) fisher;
   var X Y;
   by Batch;
run;

data SimComb (drop=Batch);
   merge SimCorr2 (where=(Batch=1) keep=Nobs ZVal Batch
                   rename=(Nobs=n1 ZVal=z1))
         SimCorr2 (where=(Batch=3) keep=Nobs ZVal Batch
                   rename=(Nobs=n2 ZVal=z2));
   z = ((n1-3)*z1 + (n2-3)*z2) / (n1+n2-6);
   corr = tanh(z);
   var = 1/(n1+n2-6);
   zlcl = z - probit(0.975)*sqrt(var);
   zucl = z + probit(0.975)*sqrt(var);
   lcl= tanh(zlcl);
   ucl= tanh(zucl);
   pval= probnorm( z/sqrt(var));
   if (pval > .5)  then pval= 1 - pval;
   pval= 2*pval;
run;

proc print data=SimComb noobs;
   var n1 z1 n2 z2 corr lcl ucl pval;
run;

Output 1.4.4 displays the combined estimate of . The table shows that a correlation estimate from the combined samples is r=0.2264. The confidence interval is (0.10453,0.34156), using the variance of the combined estimate. Note that this interval contains the population correlation 0.3.