Confidence Interval and t-Test from Pearson's Correlation Coefficient

This function performs a t-test for a given Pearson's product-moment correlation coefficient r derived from a sample of size n from a bivariate normal population, and calculates the population confidence interval.

Usage

cor_coef.test(
  r,
  n,
  alternative = c("two.sided", "less", "greater"),
  conf.level = 0.95
)

Arguments

r: numeric providing the value of Pearson's product-moment correlation coefficient r.
n: integer providing the number of pairs of observations from which r was derived; minimum 4.
alternative: a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less". You can specify just the initial letter.
conf.level: numeric between 0 and 1, the confidence level required; default 0.95.

Value

A list with class "htest" containing the following components: -

statistic: the value of the test statistic.
parameter: the degrees of freedom for the test statistic.
p.value: the p-value of the test.
conf.int: confidence interval of the correlation coefficient (95% or other specified level).
estimate: the correlation coefficient as provided in r.
null.value: the value of the association measure under the null hypothesis, always 0.
alternative: a character string describing the alternative hypothesis.
method: the character string "Pearson's product-moment correlation".
data.name: a character string giving the name of the data.

Details

The t-statistic is given by the correlation coefficient r divided by its standard error calculated using: -

$$SE = \displaystyle \sqrt{\frac{1 - r^2}{df}}$$

To calculate the confidence interval, firstly a value $Z_r$ is calculated from r using the Fisher transformation (inverse hyperbolic tangent): -

$$Z_r = \displaystyle \frac{1}{2}log_e\left(\frac{1 + r}{1 - r}\right)$$

Log upper and lower bounds (L and U) are then calculated using: -

$$L = \displaystyle Z_r - \frac{Z_{(1 - alpha/2)}}{\sqrt{n - 3}}$$

$$U = \displaystyle Z_r + \frac{Z_{(1 - alpha/2)}}{\sqrt{n - 3}}$$

The confidence interval is then calculated using the hyperbolic tangent: -

$$\displaystyle lower = \frac{e^{2L} - 1}{e^{2L} + 1}, upper = \frac{e^{2U} - 1}{e^{2U} + 1}$$

Note

This function is much like cor.test and potentially useful if the correlation coefficient is available but not the original data.

Examples

## Data from cor.test() example (Hollander & Wolfe, 1973): -
 x <- c(44.4, 45.9, 41.9, 53.3, 44.7, 44.1, 50.7, 45.2, 60.1)
 y <- c( 2.6,  3.1,  2.5,  5.0,  3.6,  4.0,  5.2,  2.8,  3.8)

 (corxy <- cor(x, y))
#> [1] 0.5711816

 cor_coef.test(corxy, 9)  ## alternative two-sided by default
#> 
#> 	Pearson's product-moment correlation
#> 
#> data:  corxy
#> t = 1.8411, df = 7, p-value = 0.1082
#> alternative hypothesis: true correlation is not equal to 0
#> 95 percent confidence interval:
#>  -0.1497426  0.8955795
#> sample estimates:
#>       cor 
#> 0.5711816 
#> 
 cor.test(x, y)  ## Result should be identical
#> 
#> 	Pearson's product-moment correlation
#> 
#> data:  x and y
#> t = 1.8411, df = 7, p-value = 0.1082
#> alternative hypothesis: true correlation is not equal to 0
#> 95 percent confidence interval:
#>  -0.1497426  0.8955795
#> sample estimates:
#>       cor 
#> 0.5711816 
#> 

 cor_coef.test(corxy, 9, alternative = "less")
#> 
#> 	Pearson's product-moment correlation
#> 
#> data:  corxy
#> t = 1.8411, df = 7, p-value = 0.9459
#> alternative hypothesis: true correlation is less than 0
#> 95 percent confidence interval:
#>  -1.0000000  0.8669786
#> sample estimates:
#>       cor 
#> 0.5711816 
#> 
 cor.test(x, y, alternative = "less")  ## Result should be identical
#> 
#> 	Pearson's product-moment correlation
#> 
#> data:  x and y
#> t = 1.8411, df = 7, p-value = 0.9459
#> alternative hypothesis: true correlation is less than 0
#> 95 percent confidence interval:
#>  -1.0000000  0.8669786
#> sample estimates:
#>       cor 
#> 0.5711816 
#> 

 cor_coef.test(corxy, 9, alternative = "greater")
#> 
#> 	Pearson's product-moment correlation
#> 
#> data:  corxy
#> t = 1.8411, df = 7, p-value = 0.05409
#> alternative hypothesis: true correlation is greater than 0
#> 95 percent confidence interval:
#>  -0.02223023  1.00000000
#> sample estimates:
#>       cor 
#> 0.5711816 
#> 
 cor.test(x, y, alternative = "greater")  ## Result should be identical
#> 
#> 	Pearson's product-moment correlation
#> 
#> data:  x and y
#> t = 1.8411, df = 7, p-value = 0.05409
#> alternative hypothesis: true correlation is greater than 0
#> 95 percent confidence interval:
#>  -0.02223023  1.00000000
#> sample estimates:
#>       cor 
#> 0.5711816 
#> 

 rm(corxy, x, y)