Fisher transformation

Derivation

Hotelling gives a concise derivation of the Fisher transformation.

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 in

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

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

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

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

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:

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

which has, to an excellent approximation, a standard normal distribution.

Application

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 and N equals 30, 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.

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| < 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.

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 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 and Hawkins in 1989 discovered the asymptotic distribution of z for data from a distribution with bounded fourth moments.

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

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 cited article for details.