Skip to contents

Fisher's z-Transformation

Source: R/fisherZ.R
fisherZ.Rd

Convert a Pearson correlation coefficient to Fisher's \(z\) scale and back. The transformation stabilizes the variance of the correlation coefficient and yields approximately normally distributed values.

Usage

fisherZ(rho)

fisherZInv(z)

Arguments

rho

numeric vector. Pearson correlation coefficient(s), typically in the interval \([-1, 1]\). Values of \(\pm 1\) are mapped to \(\pm \infty\).

z

numeric vector. Fisher \(z\)-transformed values.

Value

A numeric vector as follows:

fisherZ

Fisher \(z\)-transformed values.

fisherZInv

correlation coefficients.

Details

The forward transformation is defined as $$ z = \tanh^{-1}(r) = \frac{1}{2}\log\left(\frac{1 + r}{1 - r}\right), $$ and the inverse transformation as $$ r = \tanh(z). $$

Fisher's \(z\)-transformation is commonly used to construct confidence intervals and perform hypothesis tests for correlation coefficients.

See also

corCI(), cor.test()

Other test.correlation: corTest()

Examples

# Forward and inverse transformation
r <- seq(-0.9, 0.9, by = 0.1)
z <- fisherZ(r)
fisherZInv(z)
#>  [1] -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1  0.0  0.1  0.2  0.3  0.4  0.5
#> [16]  0.6  0.7  0.8  0.9

# Round-trip accuracy
all.equal(r, fisherZInv(fisherZ(r)))
#> [1] TRUE