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.

Definition

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

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

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

{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)

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

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

This transformation, and its inverse

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.

Derivation

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

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

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

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

m =

\operatorname {artanh} (\rho )+{\frac {\rho }{2N}}+O(N^{-2})

and

v =

{\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 $\rho$ and small values of $N$ 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:

{\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]

Calculator for the confidence belt of r-squared values (or coefficient of determination/explanation or goodness of fit).^[7]

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, 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 0.656 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. 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] $\pi (\rho |r)={\frac {\Gamma (\nu +1)}{{\sqrt {2\pi }}\Gamma (\nu +{\frac {1}{2}})}}(1-r^{2})^{\frac {\nu -1}{2}}\cdot (1-\rho ^{2})^{\frac {\nu -2}{2}}\cdot (1-r\rho )^{\frac {1-2\nu }{2}}F\!\left({\frac {3}{2}},-{\frac {1}{2}};\nu +{\frac {1}{2}};{\frac {1+r\rho }{2}}\right)$ where $F$ is the Gaussian hypergeometric function and $\nu =N-1>1$ .

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.^[13] A similar result for the asymptotic distribution applies, but with a minor adjustment factor: see the cited article for details.

References

^ Fisher, R. A. (1915). "Frequency distribution of the values of the correlation coefficient in samples of an indefinitely large population". Biometrika. 10 (4): 507–521. doi:10.2307/2331838. hdl:2440/15166. JSTOR 2331838.
^ Fisher, R. A. (1921). "On the 'probable error' of a coefficient of correlation deduced from a small sample" (PDF). Metron. 1: 3–32.
^ Rick Wicklin. Fisher's transformation of the correlation coefficient. September 20, 2017. https://blogs.sas.com/content/iml/2017/09/20/fishers-transformation-correlation.html. Accessed Feb 15,2022.
^ Hotelling, Harold (1953). "New Light on the Correlation Coefficient and its Transforms". Journal of the Royal Statistical Society, Series B (Methodological). 15 (2): 193–225. doi:10.1111/j.2517-6161.1953.tb00135.x. ISSN 0035-9246.
^ Winterbottom, Alan (1979). "A Note on the Derivation of Fisher's Transformation of the Correlation Coefficient". The American Statistician. 33 (3): 142–143. doi:10.2307/2683819. ISSN 0003-1305. JSTOR 2683819.
^ Vrbik, Jan (December 2005). "Population moments of sampling distributions". Computational Statistics. 20 (4): 611–621. doi:10.1007/BF02741318. S2CID 120592303.
^ r-squared calculator
^ Gayen, A. K. (1951). "The Frequency Distribution of the Product-Moment Correlation Coefficient in Random Samples of Any Size Drawn from Non-Normal Universes". Biometrika. 38 (1/2): 219–247. doi:10.1093/biomet/38.1-2.219. JSTOR 2332329.
^ Hotelling, H (1953). "New light on the correlation coefficient and its transforms". Journal of the Royal Statistical Society, Series B. 15 (2): 193–225. JSTOR 2983768.
^ Hawkins, D. L. (1989). "Using U statistics to derive the asymptotic distribution of Fisher's Z statistic". The American Statistician. 43 (4): 235–237. doi:10.2307/2685369. JSTOR 2685369.
^ Taraldsen, Gunnar (2021). "The Confidence Density for Correlation". Sankhya A. doi:10.1007/s13171-021-00267-y. ISSN 0976-8378. S2CID 244594067.
^ Taraldsen, Gunnar (2020). "Confidence in Correlation". doi:10.13140/RG.2.2.23673.49769. {{cite journal}}: Cite journal requires |journal= (help)
^ Zar, Jerrold H. (2005). "Spearman Rank Correlation: Overview". Encyclopedia of Biostatistics. doi:10.1002/9781118445112.stat05964. ISBN 9781118445112.

External links

R implementation

[1] Fisher, R. A. (1915). "Frequency distribution of the values of the correlation coefficient in samples of an indefinitely large population". Biometrika. 10 (4): 507–521. doi:10.2307/2331838. hdl:2440/15166. JSTOR 2331838.

[2] Fisher, R. A. (1921). "On the 'probable error' of a coefficient of correlation deduced from a small sample" (PDF). Metron. 1: 3–32.

[3] Rick Wicklin. Fisher's transformation of the correlation coefficient. September 20, 2017. https://blogs.sas.com/content/iml/2017/09/20/fishers-transformation-correlation.html. Accessed Feb 15,2022.

[4] Hotelling, Harold (1953). "New Light on the Correlation Coefficient and its Transforms". Journal of the Royal Statistical Society, Series B (Methodological). 15 (2): 193–225. doi:10.1111/j.2517-6161.1953.tb00135.x. ISSN 0035-9246.

[5] Winterbottom, Alan (1979). "A Note on the Derivation of Fisher's Transformation of the Correlation Coefficient". The American Statistician. 33 (3): 142–143. doi:10.2307/2683819. ISSN 0003-1305. JSTOR 2683819.

[6] Vrbik, Jan (December 2005). "Population moments of sampling distributions". Computational Statistics. 20 (4): 611–621. doi:10.1007/BF02741318. S2CID 120592303.

[7] r-squared calculator

[8] Gayen, A. K. (1951). "The Frequency Distribution of the Product-Moment Correlation Coefficient in Random Samples of Any Size Drawn from Non-Normal Universes". Biometrika. 38 (1/2): 219–247. doi:10.1093/biomet/38.1-2.219. JSTOR 2332329.

[9] Hotelling, H (1953). "New light on the correlation coefficient and its transforms". Journal of the Royal Statistical Society, Series B. 15 (2): 193–225. JSTOR 2983768.

[10] Hawkins, D. L. (1989). "Using U statistics to derive the asymptotic distribution of Fisher's Z statistic". The American Statistician. 43 (4): 235–237. doi:10.2307/2685369. JSTOR 2685369.

[11] Taraldsen, Gunnar (2021). "The Confidence Density for Correlation". Sankhya A. doi:10.1007/s13171-021-00267-y. ISSN 0976-8378. S2CID 244594067.

[12] Taraldsen, Gunnar (2020). "Confidence in Correlation". doi:10.13140/RG.2.2.23673.49769. {{cite journal}}: Cite journal requires |journal= (help)

[13] Zar, Jerrold H. (2005). "Spearman Rank Correlation: Overview". Encyclopedia of Biostatistics. doi:10.1002/9781118445112.stat05964. ISBN 9781118445112.

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