Confidence Interval for a Pearson Correlation

Computes confidence intervals for a Pearson correlation coefficient using Fisher's $z$-transformation. Since the sampling distribution of the correlation coefficient is not normally distributed, Fisher's transformation is applied to obtain approximately normally distributed values, from which confidence intervals can be derived.

Usage

corCI(rho, n, conf.level = 0.95, sides = c("two.sided", "left", "right"))

Arguments

rho: numeric; Pearson correlation coefficient. Must be a single value in the interval $[-1, 1]$.
n: integer; sample size used to estimate the correlation. Must be at least 3.
conf.level: confidence level of the interval, a single number in $(0, 1)$. Defaults to 0.95; unlike the estimators in this package it cannot be NA, since the interval is all this function computes.
sides: character string specifying the sidedness of the confidence interval (one of "two.sided" (default), "left" or "right"). See DescToolsX::ConfidenceIntervals.

Value

A named numeric vector with elements:

est: point estimate (correlation coefficient rho).
lci: lower confidence interval bound.
uci: upper confidence interval bound.

Details

Fisher's $z$-transformation is defined as $$z = \frac{1}{2} \log\left(\frac{1 + r}{1 - r}\right),$$ which stabilizes the variance of the correlation coefficient. The transformed values are approximately normally distributed with standard error $1 / \sqrt{n - 3}$. Confidence intervals are computed on the transformed scale and then back-transformed.

A correlation lies in $[-1, 1]$, so the open side of a one-sided interval is reported at that boundary rather than at $\pm\infty$.

The transformation has nothing to say in two situations, and both are answered with NA bounds and a warning rather than with an interval that only looks informative: at $|\rho| = 1$, where $z$ is infinite and the interval would collapse onto the estimate and thereby rule out every other value, and at $n = 3$, where the standard error $1/\sqrt{n-3}$ is infinite and the interval would be the whole range.

Note

Based on code by William Revelle, adapted to conform to package standards.

Argument name

This function used to take alternative with the values "less" and "greater". It now takes sides, like every other interval function in the package, and sides names the side carrying the finite bound - so "left" is the former "greater" and "right" the former "less". The values produced are unchanged.

Examples

# Confidence interval for a single correlation
corCI(0.5, n = 30)
#>       est       lci       uci 
#> 0.5000000 0.1704314 0.7289586 

# one-sided: "left" carries the finite lower bound
corCI(0.5, n = 30, sides = "left")
#>       est       lci       uci 
#> 0.5000000 0.2286399 1.0000000 

# Compare multiple correlations
r <- seq(0, 0.9, by = 0.1)
t(sapply(r, corCI, n = 30))
#>       est         lci       uci
#>  [1,] 0.0 -0.36026922 0.3602692
#>  [2,] 0.1 -0.26999636 0.4442638
#>  [3,] 0.2 -0.17271392 0.5226130
#>  [4,] 0.3 -0.06757251 0.5958674
#>  [5,] 0.4  0.04642030 0.6645084
#>  [6,] 0.5  0.17043137 0.7289586
#>  [7,] 0.6  0.30584207 0.7895902
#>  [8,] 0.7  0.45429999 0.8467329
#>  [9,] 0.8  0.61778629 0.9006795
#> [10,] 0.9  0.79870459 0.9516908