This example illustrates some applications of Fisher’s z transformation. For details, see the section Fisher’s z Transformation.
The following statements simulate independent samples of variables X
and Y
from a bivariate normal distribution. The first batch of 150 observations is sampled using a known correlation of 0.3, the
second batch of 150 observations is sampled using a known correlation of 0.25, and the third batch of 100 observations is
sampled using a known correlation of 0.3.
data Sim (drop=i); do i=1 to 400; X = rannor(135791); Batch = 1 + (i>150) + (i>300); if Batch = 1 then Y = 0.3*X + 0.9*rannor(246791); if Batch = 2 then Y = 0.25*X + sqrt(.8375)*rannor(246791); if Batch = 3 then Y = 0.3*X + 0.9*rannor(246791); output; end; run;
This data set will be used to illustrate the following applications of Fisher’s z transformation:
testing whether a population correlation is equal to a given value
testing for equality of two population correlations
combining correlation estimates from different samples
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 2.4.1 displays the results based on Fisher’s transformation. The null hypothesis is rejected since the p-value is less than 0.0001.
Output 2.4.1: Fisher’s Test for
Analysis for Batch 1 |
Pearson Correlation Statistics (Fisher's z Transformation) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Variable | With Variable | N | Sample Correlation | Fisher's z | Bias Adjustment | Correlation Estimate | H0:Rho=Rho0 | |||
95% Confidence Limits | Rho0 | p Value | ||||||||
X | Y | 150 | 0.22081 | 0.22451 | 0.0007410 | 0.22011 | 0.062034 | 0.367409 | 0.50000 | <.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 2.4.2) the output data set SimCorr
:
proc print data=SimCorr; run;
Output 2.4.2: Fisher’s Correlation Statistics
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 2.4.3:
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;
Output 2.4.3: Test of Equality of Observed Correlations
In Output 2.4.3, 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 2.4.4 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.
Output 2.4.4: Combined Correlation Estimate