Fisher transformation explained

In statistics, the Fisher transformation (or Fisher z-transformation) of a Pearson correlation coefficient is its inverse hyperbolic tangent (artanh). When the sample correlation coefficient r is near 1 or -1, its distribution is highly skewed, which makes it difficult to estimate confidence intervals and apply tests of significance for the population correlation coefficient ρ.^{[1] [2] [3]} The Fisher transformation solves this problem by yielding a variable whose distribution is approximately normally distributed, with a variance that is stable over different values of r.

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 = (\operatorname{cov)/((X,Y)}{\sigma) _{X} \sigma _{Y}} = (\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}}.}

Here

\operatorname{cov}(X,Y) stands for the covariance between the variables X and Y and \sigma stands for the standard deviation of the respective variable. Fisher's z-transformation of r is defined asz = { 1 \over 2 } ln \left({ 1+ r \over 1- r}\right) = \operatorname{artanh}(r), where "ln" is the natural logarithm function and "artanh" 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

{ 1 \over 2 } ln \left({{ 1+ \rho} \over { 1- \rho}}\right), and a standard deviation which does not depend on the value of the correlation rho (i.e., a Variance-stabilizing transformation){ 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( 2 z) -1)/(\exp( 2 z) +1) = \operatorname{tanh}(z), 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

Hotelling gives a concise derivation of the Fisher transformation.^[4]

To derive the Fisher transformation, one starts by considering an arbitrary increasing, twice-differentiable function of

r , say G(r) . Finding the first term in the large-N expansion of the corresponding skewness \kappa _{3} results^[5] in\kappa _{3= (6 \rho -3 ( 1- \rho ^{2} )G ^{\prime} (\rho )/G ^{\prime} (\rho ))/(\sqrt{N)} }+O(N^).Setting \kappa _{3=0} and solving the corresponding differential equation for G yields the inverse hyperbolic tangent G(\rho) = \operatorname{artanh}(\rho) function.

Similarly expanding the mean m and variance v of

\operatorname{artanh}(r) , one gets

m =

\operatorname{artanh}(\rho ) + (\rho)/(2 N) + O(N ^{-2} ) and

v =

(1)/(+ N) (6- \rho ^{2})/(2 N ^{2}) + O(N ^{-3} ) respectively.

The extra terms are not part of the usual Fisher transformation. For large values of

\rho and small values of N 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, i.e., computing (z-m)/v^1/2, yields:(z - \operatorname{artanh)/() - (\rho) (\rho)/(2 N) }which has, to an excellent approximation, a standard normal distribution.^[6]

Application

The application of Fisher's transformation can be enhanced using a software calculator as shown in the figure. Assuming that the r-squared value found is 0.80, that there are 30 data, and accepting a 90% confidence interval, the r-squared value in another random sample from the same population may range from 0.588 to 0.921. When r-squared is outside this range, the population is considered to be different.

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^[7] 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^[8] and Hawkins in 1989 discovered the asymptotic distribution of z for data from a distribution with bounded fourth moments.^[9]

An alternative to the Fisher transformation is to use the exact confidence distribution density for ρ given by^{[10] [11]} $$\backslash pi\; (\backslash rho\; |\; r)\; =\backslash frac(1\; -\; r^2)^\; \backslash cdot(1\; -\; \backslash rho^2)^\; \backslash cdot(1\; -\; r\; \backslash rho)^\; F\backslash !\backslash left(\backslash frac,-\backslash frac;\; \backslash nu\; +\; \backslash frac;\; \backslash frac\backslash right)$$where

F is the Gaussian hypergeometric function and \nu = N -1 > 1 .

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.^[12] A similar result for the asymptotic distribution applies, but with a minor adjustment factor: see the cited article for details.

See also

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

References

1. Fisher . R. A. . 1915 . Frequency distribution of the values of the correlation coefficient in samples of an indefinitely large population . Biometrika . 10 . 507–521 . 2331838. 4. 10.2307/2331838. 2440/15166 . free .
2. Ronald Fisher . Fisher . R. A. . 1921 . On the 'probable error' of a coefficient of correlation deduced from a small sample . Metron . 1 . 3–32.
3. Rick Wicklin. Fisher's transformation of the correlation coefficient. September 20, 2017. https://blogs.sas.com/content/iml/2017/09/20/fishers-transformation-correlation.html. Accessed Feb 15,2022.
4. Hotelling. Harold. 1953. New Light on the Correlation Coefficient and its Transforms. Journal of the Royal Statistical Society, Series B (Methodological). 15. 2. 193–225. 10.1111/j.2517-6161.1953.tb00135.x. 0035-9246.
5. Winterbottom. Alan. 1979. A Note on the Derivation of Fisher's Transformation of the Correlation Coefficient. The American Statistician. 33. 3. 142–143. 10.2307/2683819. 2683819 . 0003-1305.
6. Vrbik . Jan . Population moments of sampling distributions . Computational Statistics . December 2005 . 20 . 4 . 611–621 . 10.1007/BF02741318. 120592303 .
7. Gayen . A. K. . The Frequency Distribution of the Product-Moment Correlation Coefficient in Random Samples of Any Size Drawn from Non-Normal Universes . 38 . 1951 . 219–247 . Biometrika . 2332329 . 1/2 . 10.1093/biomet/38.1-2.219.
8. Harold Hotelling . Hotelling . H . 1953 . New light on the correlation coefficient and its transforms . Journal of the Royal Statistical Society, Series B . 15 . 193–225 . 2983768 . 2 .
9. Hawkins . D. L. . 1989 . Using U statistics to derive the asymptotic distribution of Fisher's Z statistic . . 43 . 235–237 . 10.2307/2685369 . 4 . 2685369. u-statistic .
10. Taraldsen. Gunnar. 2021. The Confidence Density for Correlation. Sankhya A. en. 10.1007/s13171-021-00267-y. 244594067 . 0976-8378. free.
11. Taraldsen. Gunnar. 2020. Confidence in Correlation. en. 10.13140/RG.2.2.23673.49769.
12. Encyclopedia: Zar . Jerrold H. . 2005. Spearman Rank Correlation: Overview . Encyclopedia of Biostatistics. 10.1002/9781118445112.stat05964 . 9781118445112 .

External links

• R implementation