Fisher transformation

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.

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

${\displaystyle r={\frac {\operatorname {cov} (X,Y)}{\sigma _{X}\sigma _{Y}}}={\frac {\sum _{i=1}^{N}(X_{i}-{\bar {X}})(Y_{i}-{\bar {Y}})}{{\sqrt {\sum _{i=1}^{N}(X_{i}-{\bar {X}})^{2}}}{\sqrt {\sum _{i=1}^{N}(Y_{i}-{\bar {Y}})^{2}}}}}.}$

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

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

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

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

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

${\displaystyle {1 \over {\sqrt {N-3}}},}$

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

This transformation, and its inverse

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

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

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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 ${\displaystyle r}$, say ${\displaystyle G(r)}$. Finding the first term in the large-${\displaystyle N}$ expansion of the corresponding skewness ${\displaystyle \kappa _{3}}$ results^[5] in

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

Setting ${\displaystyle \kappa _{3}=0}$ and solving the corresponding differential equation for ${\displaystyle G}$ yields the inverse hyperbolic tangent ${\displaystyle G(\rho )=\operatorname {artanh} (\rho )}$ function.

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

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

and

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

respectively.

The extra terms are not part of the usual Fisher transformation. For large values of ${\displaystyle \rho }$ and small values of ${\displaystyle N}$ 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:

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

which has, to an excellent approximation, a standard normal distribution.^[6]

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 ^{[clarification needed]}, 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.

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^[8] 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^[9] and Hawkins in 1989 discovered the asymptotic distribution of z for data from a distribution with bounded fourth moments.^[10]

An alternative to the Fisher transformation is to use the exact confidence distribution density for ρ given by^[11][12] $${\displaystyle \pi (\rho |r)={\frac {\Gamma (\nu +1)}{{\sqrt {2\pi }}\Gamma (\nu +{\frac {1}{2}})}}(1-r^{2})^{\frac {\nu -1}{2}}\cdot (1-\rho ^{2})^{\frac {\nu -2}{2}}\cdot (1-r\rho )^{\frac {1-2\nu }{2}}F\!\left({\frac {3}{2}},-{\frac {1}{2}};\nu +{\frac {1}{2}};{\frac {1+r\rho }{2}}\right)}$$ where ${\displaystyle F}$ is the Gaussian hypergeometric function and ${\displaystyle \nu =N-1>1}$ .

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.^[13] 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
• Pearson correlation coefficient § Inference

• R implementation