Fisher transformation

In statistics, the Fisher transformation (aka Fisher "z"-transformation) can be used to test hypotheses about the value of the population correlation coefficient ρ between variables "X" and "Y". This is because, when the transformation is applied to the sample correlation coefficient, the sampling distribution of the resulting variable is approximately normal, with a variance that is stable over different values of the underlying true correlation.
Definition.
Given a set of "N" bivariate sample pairs ("X""i", "Y""i"), "i" = 1, ..., "N", the sample correlation coefficient "r" is given by
Here formula_2 stands for the covariance between the variables formula_3 and formula_4 and formula_5 stands for the standard deviation of the respective variable. Fisher's z-transformation of "r" is defined as
where "ln" is the natural logarithm function and "arctanh" is the inverse hyperbolic tangent function.
If ("X", "Y") has a bivariate normal distribution with correlation ρ and the pairs ("X""i", "Y""i") are independent and identically distributed, then "z" is approximately normally distributed with mean
and standard error
where "N" is the sample size, and ρ is the true correlation coefficient.
This transformation, and its inverse
can be used to construct a large-sample confidence interval for "r" using standard normal theory and derivations. See also application to partial correlation.
Derivation.
To derive the Fisher transformation, one starts by considering an arbitrary increasing function of formula_10, say formula_11. Finding the first term in the large-formula_12 expansion of the corresponding skewness results in
Making it equal to zero and solving the corresponding differential equation for formula_14 yields the formula_15 function. Similarly expanding the mean and variance of formula_16, one gets
and
respectively. The extra terms are not part of the usual Fisher transformation. For large values of formula_19 and small values of formula_12 they represent a large improvement of accuracy at minimal cost, although they greatly complicate the computation of the inverse – a closed-form expression is not available. The near-constant variance of the transformation is the result of removing its skewness – the actual improvement is achieved by the latter, not by the extra terms. Including the extra terms yields:
which has, to an excellent approximation, a standard normal distribution.
Discussion.
The Fisher transformation is an approximate variance-stabilizing transformation for "r" when "X" and "Y" follow a bivariate normal distribution. This means that the variance of "z" is approximately constant for all values of the population correlation coefficient "ρ". Without the Fisher transformation, the variance of "r" grows smaller as |"ρ"| gets closer to 1. Since the Fisher transformation is approximately the identity function when |"r"| < 1/2, it is sometimes useful to remember that the variance of "r" is well approximated by 1/"N" as long as |"ρ"| is not too large and "N" is not too small. This is related to the fact that the asymptotic variance of "r" is 1 for bivariate normal data.
The behavior of this transform has been extensively studied since Fisher introduced it in 1915. Fisher himself found the exact distribution of "z" for data from a bivariate normal distribution in 1921; Gayen in 1951
determined the exact distribution of "z" for data from a bivariate Type A Edgeworth distribution. Hotelling in 1953 calculated the Taylor series expressions for the moments of "z" and several related statistics and Hawkins in 1989 discovered the asymptotic distribution of "z" for data from a distribution with bounded fourth moments.
An alternative to the Fisher transformation is to use the exact confidence distribution density for "ρ" given by
formula_22
where formula_23 is the Gaussian hypergeometric function and formula_24 .
Other uses.
While the Fisher transformation is mainly associated with the Pearson product-moment correlation coefficient for bivariate normal observations, it can also be applied to Spearman's rank correlation coefficient in more general cases. A similar result for the asymptotic distribution applies, but with a minor adjustment factor: see the latter article for details.