FisherZ.RdConvert a correlation to a z score or z to r using the Fisher transformation or find the confidence intervals for a specified correlation.
FisherZ(rho)
FisherZInv(z)
CorCI(rho, n, conf.level = 0.95, alternative = c("two.sided", "less", "greater"))the Pearson's correlation coefficient
a Fisher z transformed value
sample size used for calculating the confidence intervals
is a character string, one of "greater",
"less", or "two.sided", or the initial letter of each,
indicating the specification of the alternative hypothesis.
"greater" corresponds to positive association, "less" to negative association.
confidence level for the returned confidence interval, restricted to lie between zero and one.
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:
$$z_r = tanh^{-1}(r) = \frac{1}{2}log\left ( \frac{1+r}{1-r}\right )$$
z value corresponding to r (in FisherZ)
r corresponding to z (in FisherZInv)
rho, lower and upper confidence intervals (CorCI)
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