Fisher transformation

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

:$r\; =\; \backslash frac\{\backslash operatorname\{cov\}(X,Y)\}\{\backslash sigma\_X\; \backslash sigma\_Y\}\; =\; \backslash frac\{\backslash sum\; ^N\; \_\{i=1\}(X\_i\; -\; \backslash bar\{X\})(Y\_i\; -\; \backslash bar\{Y\})\}\{\backslash sqrt\{\backslash sum\; ^N\; \_\{i=1\}(X\_i\; -\; \backslash bar\{X\})^2\}\; \backslash sqrt\{\backslash sum\; ^N\; \_\{i=1\}(Y\_i\; -\; \backslash bar\{Y\})^2\}\}.$

Here $\backslash operatorname\{cov\}(X,Y)$ stands for the covariance between the variables $X$ and $Y$ and $\backslash sigma$ stands for the standard deviation of the respective variable. Fisher's z-transformation of r is defined as

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

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

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

and a standard deviation which does not depend on the value of the correlation rho (i.e., a Variance-stabilizing transformation)

:$\{1\; \backslash over\; \backslash sqrt\{N-3\}\},$

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

This transformation, and its inverse

:$r\; =\; \backslash frac\{\backslash exp(2z)-1\}\{\backslash exp(2z)+1\}\; =\; \backslash operatorname\{tanh\}(z),$

can be used to construct a large-sample confidence interval for r using standard normal theory and derivations. See also application to partial correlation.

{{cleanup|reason=the steps of the derivation are not laid out completely.|date=July 2021}}

File:Fisher Transformation.png

Hotelling gives a concise derivation of the Fisher transformation.{{Cite journal|last=Hotelling|first=Harold|date=1953|title=New Light on the Correlation Coefficient and its Transforms|url=http://dx.doi.org/10.1111/j.2517-6161.1953.tb00135.x|journal=Journal of the Royal Statistical Society, Series B (Methodological)|volume=15|issue=2|pages=193–225|doi=10.1111/j.2517-6161.1953.tb00135.x|issn=0035-9246}}

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 $\backslash kappa\_3$ results{{Cite journal|last=Winterbottom|first=Alan|date=1979|title=A Note on the Derivation of Fisher's Transformation of the Correlation Coefficient|url=http://dx.doi.org/10.2307/2683819|journal=The American Statistician|volume=33|issue=3|pages=142–143|doi=10.2307/2683819|jstor=2683819 |issn=0003-1305}} in

:$\backslash kappa\_3=\backslash frac\{6\backslash rho\; -3(1-\backslash rho\; ^\{2\})G^\{\backslash prime\; \backslash prime\; \}(\backslash rho\; )/G^\{\backslash prime\; \}(\backslash rho\; )\}\{\backslash sqrt\{N\}\}+O(N^\{-3/2\}).$

Setting $\backslash kappa\_3=0$ and solving the corresponding differential equation for $G$ yields the inverse hyperbolic tangent $G(\backslash rho)=\backslash operatorname\{artanh\}(\backslash rho)$ function.

Similarly expanding the mean m and variance v of $\backslash operatorname\{artanh\}(r)$, one gets

:m = $\backslash operatorname\{artanh\}(\backslash rho\; )+\backslash frac\{\backslash rho\; \}\{2N\}+O(N^\{-2\})$

and

:v = $\backslash frac\{1\}\{N\}+\backslash frac\{6-\backslash rho\; ^\{2\}\}\{2N^\{2\}\}+O(N^\{-3\})$

respectively.

The extra terms are not part of the usual Fisher transformation. For large values of $\backslash rho$ 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:

:$\backslash frac\{z-\backslash operatorname\{artanh\}(\backslash rho\; )-\backslash frac\{\backslash rho\; \}\{2N\}\}\{\backslash sqrt\{\backslash frac\{1\}\{N\}+\backslash frac\{6-\backslash rho\; ^\{2\}\}\{2N^\{2\}\}\}\}$

which has, to an excellent approximation, a standard normal distribution.{{cite journal |last1=Vrbik |first1=Jan |title=Population moments of sampling distributions |journal=Computational Statistics |date=December 2005 |volume=20 |issue=4 |pages=611–621 |doi=10.1007/BF02741318|s2cid=120592303 }}

File:rsquared.png

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 {{Clarify|date=May 2022}}, and accepting a 90% confidence interval, the r-squared value in another random sample from the same population may range from 0.656 to 0.888. When r-squared is outside this range, the population is considered to be different.

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{{cite journal | last=Gayen | first=A. K. |title=The Frequency Distribution of the Product-Moment Correlation Coefficient in Random Samples of Any Size Drawn from Non-Normal Universes | volume=38 | year=1951 | pages=219–247 | journal=Biometrika | jstor=2332329 | issue=1/2 | doi=10.1093/biomet/38.1-2.219}}

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{{cite journal |authorlink=Harold Hotelling | last=Hotelling | first=H | year=1953 | title=New light on the correlation coefficient and its transforms | journal=Journal of the Royal Statistical Society, Series B | volume=15 | pages=193–225 | jstor=2983768 |issue=2 }} and Hawkins in 1989 discovered the asymptotic distribution of z for data from a distribution with bounded fourth moments.{{cite journal | last=Hawkins | first=D. L. | year=1989 | title=Using U statistics to derive the asymptotic distribution of Fisher's Z statistic | journal=The American Statistician | volume=43 | pages=235–237 | doi=10.2307/2685369 | issue=4 | jstor=2685369| title-link=u-statistic }}

An alternative to the Fisher transformation is to use the exact confidence distribution density for ρ given by{{Cite journal|last=Taraldsen|first=Gunnar|date=2021|title=The Confidence Density for Correlation|url=https://doi.org/10.1007/s13171-021-00267-y|journal=Sankhya A|language=en|doi=10.1007/s13171-021-00267-y|s2cid=244594067 |issn=0976-8378|doi-access=free|hdl=11250/3133125|hdl-access=free}}{{Cite journal|last=Taraldsen|first=Gunnar|date=2020|title=Confidence in Correlation|url=http://rgdoi.net/10.13140/RG.2.2.23673.49769| language=en|doi=10.13140/RG.2.2.23673.49769}}

$$\backslash pi\; (\backslash rho\; |\; r)\; =$$

\frac{\Gamma(\nu+1)}{\sqrt{2\pi}\Gamma(\nu + \frac{1}{2})}

(1 - r^2)^{\frac{\nu - 1}{2}} \cdot

(1 - \rho^2)^{\frac{\nu - 2}{2}} \cdot

(1 - r \rho )^{\frac{1-2\nu}{2}} F\!\left(\frac{3}{2},-\frac{1}{2}; \nu + \frac{1}{2}; \frac{1 + r \rho}{2}\right)

where $F$ is the Gaussian hypergeometric function and $\backslash nu\; =\; N-1\; >\; 1$ .

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.{{cite encyclopedia|last=Zar | first= Jerrold H. | year=2005| title=Spearman Rank Correlation: Overview | encyclopedia=Encyclopedia of Biostatistics| doi= 10.1002/9781118445112.stat05964 | isbn= 9781118445112 }} A similar result for the asymptotic distribution applies, but with a minor adjustment factor: see the cited article for details.

• Data transformation (statistics)
• Meta-analysis (this transformation is used in meta analysis for stabilizing the variance)
• Partial correlation
• {{slink|Pearson correlation coefficient#Inference}}

• R [http://personality-project.org/r/html/fisherz.html implementation]

{{statistics}}

{{DEFAULTSORT:Fisher Transformation}}

Category:Covariance and correlation

Category:Transforms

Category:Ronald Fisher