Editing Fisher transformation (section)

== 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''|&nbsp;<&nbsp;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 [[Ronald Fisher|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<ref>{{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}}</ref>
determined the exact distribution of ''z'' for data from a bivariate Type A [[Edgeworth distribution]]. [[Harold Hotelling|Hotelling]] in 1953 calculated the Taylor series expressions for the moments of ''z'' and several related statistics<ref>{{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 }}</ref> and Hawkins in 1989 discovered the asymptotic distribution of ''z'' for data from a distribution with bounded fourth moments.<ref>{{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 }}</ref>

An alternative to the Fisher transformation is to use the exact [[confidence distribution]] density for ''ρ'' given by<ref>{{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}}</ref><ref>{{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}}</ref>
<math display="block">\pi (\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)</math>
where <math>F</math> is the Gaussian hypergeometric function and <math>\nu = N-1 > 1</math> .