Convert a correlation to a z score or z to r using the Fisher transformation.
FisherZ(rho)
FisherZInv(z)z value corresponding to r (in FisherZ)
r corresponding to z (in
FisherZInv)
The sampling distribution of Pearson's r is not normally distributed. Fisher developed a transformation now called "Fisher's z-transformation" that converts Pearson's r to the normally distributed variable z. The formula for the transformation is:
$$z_r = tanh^{-1}(r) = \frac{1}{2}log\left ( \frac{1+r}{1-r}\right )$$
cors <- seq(-.9, .9, .1)
zs <- FisherZ(cors)
rs <- FisherZInv(zs)
round(zs, 2)
#> [1] -1.47 -1.10 -0.87 -0.69 -0.55 -0.42 -0.31 -0.20 -0.10 0.00 0.10 0.20
#> [13] 0.31 0.42 0.55 0.69 0.87 1.10 1.47
n <- 30
r <- seq(0, .9, .1)
rc <- t(sapply(r, CorCI, n=n))
t <- r * sqrt(n-2) / sqrt(1-r^2)
p <- (1 - pt(t, n-2)) / 2
r.rc <- data.frame(r=r, z=FisherZ(r), lower=rc[,2], upper=rc[,3], t=t, p=p)
round(r.rc,2)
#> r z lower upper t p
#> 1 0.0 0.00 -0.36 0.36 0.00 0.25
#> 2 0.1 0.10 -0.27 0.44 0.53 0.15
#> 3 0.2 0.20 -0.17 0.52 1.08 0.07
#> 4 0.3 0.31 -0.07 0.60 1.66 0.03
#> 5 0.4 0.42 0.05 0.66 2.31 0.01
#> 6 0.5 0.55 0.17 0.73 3.06 0.00
#> 7 0.6 0.69 0.31 0.79 3.97 0.00
#> 8 0.7 0.87 0.45 0.85 5.19 0.00
#> 9 0.8 1.10 0.62 0.90 7.06 0.00
#> 10 0.9 1.47 0.80 0.95 10.93 0.00