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{{Redirect-distinguish|Fisher z-transformation|Fisher's z-distribution}}
{{about||standard z-score in statistics|Standard score|z-transformation to complex number domain|Z-transform}}


[[Image:fisher transformation.svg|300px|right|thumb|A graph of the transformation. The untransformed sample correlation coefficient is plotted on the horizontal axis, and the transformed coefficient is plotted on the vertical axis. The identity function is also shown for comparison.]]In [[statistics]], hypotheses about the value of the population [[Pearson product-moment correlation coefficient|correlation coefficient]] ρ between variables ''X'' and ''Y'' can be tested using the '''Fisher transformation'''<ref>{{cite journal| last=Fisher | first= R.A. | year=1915 | title= Frequency distribution of the values of the correlation coefficient in samples of an indefinitely large population | journal=Biometrika | volume=10 | pages=507&ndash;521 | jstor=2331838| issue=4| publisher=Biometrika Trust}}</ref><ref>{{cite journal|authorlink=Ronald Fisher | last=Fisher | first= R.A. | year=1921 | title=On the `probable error' of a coefficient of correlation deduced from a small sample | journal=Metron | volume=1 | pages=3&ndash;32|url=http://digital.library.adelaide.edu.au/dspace/bitstream/2440/15169/1/14.pdf}}</ref> (aka '''Fisher z-transformation''') applied to the [[Pearson product-moment correlation coefficient|sample correlation coefficient]] ''r''.


==Definition==
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The transformation is defined by:
 
:<math>z = {1 \over 2}\ln\left({1+r \over 1-r}\right) = \operatorname{arctanh}(r),</math>
 
where "ln" is the [[natural logarithm]] function and "arctanh" is the [[inverse hyperbolic function]].
 
If (''X'',&nbsp;''Y'') has a [[bivariate normal distribution]], and if the (''X''<sub>''i''</sub>,&nbsp;''Y''<sub>''i''</sub>) pairs used to form ''r'' are independent for ''i''&nbsp;=&nbsp;1,&nbsp;...,&nbsp;''n'', then ''z'' is approximately [[normal distribution|normally distributed]] with mean
 
:<math>{1 \over 2}\ln\left({{1+\rho} \over {1-\rho}}\right),</math>
 
and [[Standard error (statistics)|standard error]]
 
:<math>{1 \over \sqrt{N-3}},</math>
 
where ''N'' is the sample size.  
 
This transformation, and its inverse,
 
:<math>{{\exp(2z)-1} \over {\exp(2z)+1}},</math>
 
can be used to construct a [[confidence interval]] for&nbsp;''ρ''.
 
== 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, 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&ndash;247 | journal=Biometrika | jstor=2332329 | issue=1/2 | publisher=Biometrika Trust}}</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&ndash;225 | jstor=2983768 |issue=2 |publisher=Blackwell Publishing}}</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-statistic|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 | publisher=American Statistical Association | jstor=2685369}}</ref>
 
==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. A similar result for the [[asymptotic distribution]] applies, but with a minor adjustment factor: see the latter article for details.
 
==References==
<references/>
 
{{statistics}}
 
{{DEFAULTSORT:Fisher Transformation}}
[[Category:Statistical theory]]
[[Category:Transforms]]

Latest revision as of 04:53, 12 October 2014


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