Fisher transformation

A graph of the transformation. The untransformed sample correlation coefficient is plotted on the horizontal axis, and the transformed coefficient is plotted on the vertical axis. The identity function is also shown for comparison.

In statistics, hypotheses about the value of the population correlation coefficient ρ between variables X and Y can be tested using the Fisher transformation^[1][2] (aka Fisher z-transformation) applied to the sample correlation coefficient.

Definition

Given a set of N bivariate sample pairs (X_i, Y_i), i = 1, ..., N, the sample correlation coefficient r is given by

$r = \frac{\sum ^N _{i=1}(X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum ^N _{i=1}(X_i - \bar{X})^2} \sqrt{\sum ^N _{i=1}(Y_i - \bar{Y})^2}}$

Fisher's z-transformation of r is defined as

$z := {1 \over 2}\ln\left({1+r \over 1-r}\right) = \operatorname{arctanh}(r),$

where "ln" is the natural logarithm function and "arctanh" is the inverse hyperbolic tangent function.

If (X, Y) has a bivariate normal distribution, and if the pairs (X_i, Y_i) are independent, then z is approximately normally distributed with mean

${1 \over 2}\ln\left({{1+\rho} \over {1-\rho}}\right),$

and standard error

${1 \over \sqrt{N-3}},$

where N is the sample size, and ρ is the true correlation coefficient.

This transformation, and its inverse

$r = {{\exp(2z)-1} \over {\exp(2z)+1}} = \operatorname{tanh}(z),$

can be used to construct a large-sample confidence interval for ρ using standard normal theory and derivations.

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^[3] 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^[4] and Hawkins in 1989 discovered the asymptotic distribution of z for data from a distribution with bounded fourth moments.^[5]

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.

See also

• Data transformation (statistics)
• Meta-analysis (this transformation is used in meta analysis for stabilizing the variance)

References

1. Fisher, R. A. (1915). "Frequency distribution of the values of the correlation coefficient in samples of an indefinitely large population". Biometrika (Biometrika Trust) 10 (4): 507–521. doi:10.2307/2331838. JSTOR 2331838.
2. Fisher, R. A. (1921). "On the 'probable error' of a coefficient of correlation deduced from a small sample" (PDF). Metron 1: 3–32.
3. Gayen, A. K. (1951). "The Frequency Distribution of the Product-Moment Correlation Coefficient in Random Samples of Any Size Drawn from Non-Normal Universes". Biometrika (Biometrika Trust) 38 (1/2): 219–247. doi:10.1093/biomet/38.1-2.219. JSTOR 2332329.
4. Hotelling, H (1953). "New light on the correlation coefficient and its transforms". Journal of the Royal Statistical Society, Series B (Blackwell Publishing) 15 (2): 193–225. JSTOR 2983768.
5. Hawkins, D. L. (1989). "Using U statistics to derive the asymptotic distribution of Fisher's Z statistic". The American Statistician (American Statistical Association) 43 (4): 235–237. doi:10.2307/2685369. JSTOR 2685369.