Fisher transformation
In
Definition
Given a set of N bivariate sample pairs (Xi, Yi), i = 1, ..., N, the
Here stands for the covariance between the variables and and stands for the standard deviation of the respective variable. Fisher's z-transformation of r is defined as
where "ln" is the
If (X, Y) has a
and standard deviation
where N is the sample size, and ρ is the true correlation coefficient.
This transformation, and its inverse
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
This article may require cleanup to meet Wikipedia's quality standards. The specific problem is: the steps of the derivation are not laid out completely. (July 2021) |
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 , say . Finding the first term in the large- expansion of the corresponding skewness results[5] in
Setting and solving the corresponding differential equation for yields the inverse hyperbolic tangent function.
Similarly expanding the mean m and variance v of , one gets
- m =
and
- v =
respectively.
The extra terms are not part of the usual Fisher transformation. For large values of and small values of 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)/v1/2, yields:
which has, to an excellent approximation, a
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 0.588 to 0.921. 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]
Other uses
While the Fisher transformation is mainly associated with the
See also
- Data transformation (statistics)
- Meta-analysis (this transformation is used in meta analysis for stabilizing the variance)
- Partial correlation
- Pearson correlation coefficient § Inference
References
- 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.
- ISSN 0035-9246.
- JSTOR 2683819.
- S2CID 120592303.
- ^ r-squared calculator
- JSTOR 2332329.
- JSTOR 2983768.
- JSTOR 2685369.
- S2CID 244594067.
- )
- ISBN 9781118445112.