For a sample correlation that uses a sample from a bivariate normal distribution with correlation
, the statistic
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has a Student’s distribution with (
) degrees of freedom.
With the monotone transformation of the correlation (Fisher, 1921)
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the statistic has an approximate normal distribution with mean and variance
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where .
For the transformed , the approximate variance
is independent of the correlation
. Furthermore, even the distribution of
is not strictly normal, it tends to normality rapidly as the sample size increases for any values of
(Fisher, 1973, pp. 200–201).
For the null hypothesis , the
-values are computed by treating
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as a normal random variable with mean zero and variance , where
(Fisher 1973, p. 207; Anderson 1984, p. 123).
Note that the bias adjustment, , is always used when computing
-values under the null hypothesis
in the CORR procedure.
The ALPHA= option in the FISHER option specifies the value for the confidence level
, the RHO0= option specifies the value
in the hypothesis
, and the BIASADJ= option specifies whether the bias adjustment is to be used for the confidence limits.
The TYPE= option specifies the type of confidence limits. The TYPE=TWOSIDED option requests two-sided confidence limits and
a -value under the hypothesis
. For a one-sided confidence limit, the TYPE=LOWER option requests a lower confidence limit and a
-value under the hypothesis
, and the TYPE=UPPER option requests an upper confidence limit and a
-value under the hypothesis
.
The confidence limits for the correlation are derived through the confidence limits for the parameter
, with or without the bias adjustment.
Without a bias adjustment, confidence limits for are computed by treating
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as having a normal distribution with mean zero and variance .
That is, the two-sided confidence limits for are computed as
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where is the
percentage point of the standard normal distribution.
With a bias adjustment, confidence limits for are computed by treating
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as having a normal distribution with mean zero and variance , where the bias adjustment function (Keeping, 1962, p. 308) is
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That is, the two-sided confidence limits for are computed as
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These computed confidence limits of and
are then transformed back to derive the confidence limits for the correlation
:
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Note that with a bias adjustment, the CORR procedure also displays the following correlation estimate:
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Fisher (1973, p. 199) describes the following practical applications of the 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
To test if a population correlation from a sample of
observations with sample correlation
is equal to a given
, first apply the
transformation to
and
:
and
.
The -value is then computed by treating
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as a normal random variable with mean zero and variance .
Assume that sample correlations and
are computed from two independent samples of
and
observations, respectively. To test whether the two corresponding population correlations,
and
, are equal, first apply the
transformation to the two sample correlations:
and
.
The -value is derived under the null hypothesis of equal correlation. That is, the difference
is distributed as a normal random variable with mean zero and variance
.
Assuming further that the two samples are from populations with identical correlation, a combined correlation estimate can
be computed. The weighted average of the corresponding values is
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where the weights are inversely proportional to their variances.
Thus, a combined correlation estimate is and
. See Example 2.4 for further illustrations of these applications.
Note that this approach can be extended to include more than two samples.