Confidence Intervals for Pearson Correlation

Find the confidence intervals for a specified correlation based on Fisher's z-transformation.

CorCI(
  rho,
  n,
  conf.level = 0.95,
  alternative = c("two.sided", "less", "greater")
)

Arguments

rho: the Pearson's correlation coefficient
n: sample size used for calculating the confidence intervals
conf.level: confidence level for the returned confidence interval, restricted to lie between zero and one.
alternative: 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.

Value

rho, lower and upper confidence intervals (CorCI)

Details

The sampling distribution of Pearson's r is not normal. Fisher developed a transformation now called "Fisher's z-transformation" used for the calculation of normal distributed confidence intervals.

Author

William Revelle revelle@northwestern.edu,
slight modifications Andri Signorell andri@signorell.net based on R-Core code

Examples


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