In statistics, hypotheses about the value of the population correlation coefficient ρ between variables X and Y can be tested using the Fisher transformation (aka Fisher z-transformation) applied to the sample correlation coefficient.
Given a set of N bivariate sample pairs (Xi, Yi), i = 1, ..., N, the sample correlation coefficient r is given by
and standard error
where N is the sample size, and ρ is the true correlation coefficient.
This transformation, and its inverse
To derive the Fisher transformation, one starts by considering an arbitrary increasing function of , say . Finding the first term in the large- expansion of the corresponding skewness results in
Making it equal to zero and solving the corresponding differential equation for yields the function. Similarly expanding the mean and variance of , one gets
respectively. Note that the extra terms are not part of the usual Fisher transformation. For large values of , they represent a large improvement of accuracy at minimal cost, although they greatly complicate the computation of the inverse. (A simple closed-form inverse is not available.) Also note that 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. Now, it is
which has, to an excellent approximation, the standardized Normal distribution.
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.
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[clarification needed] for details.
- Data transformation (statistics)
- Meta-analysis (this transformation is used in meta analysis for stabilizing the variance)
- Partial correlation
- R implementation
- 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.
- Fisher, R. A. (1921). "On the 'probable error' of a coefficient of correlation deduced from a small sample" (PDF). Metron. 1: 3–32.
- Vrbik, Jan (December 2005). "Population moments of sampling distributions". Computational Statistics. 20 (4): 611--621.
- 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.
- 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.
- Hawkins, D. L. (1989). "U statistics". The American Statistician. American Statistical Association. 43 (4): 235–237. doi:10.2307/2685369. JSTOR 2685369.
- Zar, Jerrold H. (2005). "Spearman Rank Correlation: Overview". Encyclopedia of Biostatistics. John Wiley & Sons.