Editing Fisher transformation (section)

==Derivation==
{{cleanup|reason=the steps of the derivation are not laid out completely.|date=July 2021}}
[[File:Fisher Transformation.png|thumb|Fisher Transformation with <math>\rho =0.9</math> and <math>N=30</math>. Illustrated is the exact probability density function of <math>r</math> (in black), together with the probability density functions of the usual Fisher transformation (blue) and that obtained by including extra terms that depend on <math>N</math> (red). The latter approximation is visually indistinguishable from the exact answer (its maximum error is 0.3%, compared to 3.4% of basic Fisher).]]

Hotelling gives a concise derivation of the Fisher transformation.<ref>{{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|url-access=subscription}}</ref>

To derive the Fisher transformation, one starts by considering an arbitrary increasing, twice-differentiable function of <math>r</math>, say <math>G(r)</math>. Finding the first term in the large-<math>N</math> expansion of the corresponding [[skewness]] <math>\kappa_3</math> results<ref>{{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|url-access=subscription}}</ref> in
:<math>\kappa_3=\frac{6\rho -3(1-\rho ^{2})G^{\prime \prime }(\rho )/G^{\prime }(\rho )}{\sqrt{N}}+O(N^{-3/2}).</math>
Setting <math>\kappa_3=0</math> and solving the corresponding differential equation for <math>G</math> yields the inverse hyperbolic tangent <math>G(\rho)=\operatorname{artanh}(\rho)</math> function.

Similarly expanding the mean ''m'' and variance ''v'' of <math>\operatorname{artanh}(r)</math>, one gets
:m = <math>\operatorname{artanh}(\rho )+\frac{\rho }{2N}+O(N^{-2}) </math>
and
:v = <math>\frac{1}{N}+\frac{6-\rho ^{2}}{2N^{2}}+O(N^{-3}) </math>
respectively.

The extra terms are not part of the usual Fisher transformation. For large values of <math>\rho </math> and small values of <math>N</math> 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<sup>1/2</sup>, yields:
:<math>\frac{z-\operatorname{artanh}(\rho )-\frac{\rho }{2N}}{\sqrt{\frac{1}{N}+\frac{6-\rho ^{2}}{2N^{2}}}}</math>
which has, to an excellent approximation, a [[standard normal distribution]].<ref>{{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 }}</ref>
[[File:rsquared.png|thumb|300 px| Calculator for the confidence belt of r-squared values (or coefficient of determination/explanation or goodness of fit).<ref>[https://www.waterlog.info/r-squared.htm r-squared calculator]</ref>]]