16 Converting to Pearson Correlation

16.1 From \(t\)-statistic

From a \(t\) statistic calculated from a correlational test, we can calculate the correlation coefficient using the following formula:

\[ r = \sqrt{\frac{t^2}{t^2 + n-2}} \tag{16.1}\]

Using the t_to_r function in the effectsize package we can convert \(t\) to \(r\).

# Example:
# t = 4.14, n = 50

library(effectsize)

t <- 4.14
n <- 50

t_to_r(t = t, df = n-2)

r    |       95% CI
-------------------
0.51 | [0.28, 0.67]

From a between groups Cohen’s \(d\) value (\(d_p\)), we can calculate the correlation coefficient from the following formula:

\[ r = \frac{d_p}{\sqrt{d_p^2+\frac{n_1+n_2-2}{n_1} + \frac{n_1+n_2-2}{n_2}}} \tag{16.2}\]

Using the d_to_r function in the effectsize package we can convert \(d_p\) to \(r\).

# Example:
# d = 0.60, n1 = 50, n2 = 70

d <- 0.60
n1 <- 50
n2 <- 70

d_to_r(d = d, n1 = n1, n2 = n2)

[1] 0.2858532

The correlation coefficient from an odds ratio can be calculated with the following formula:

\[ r = \frac{\log(OR)\times\sqrt{3}}{\pi\sqrt{\frac{3\log(OR)^2}{\pi^2}+\frac{n_1+n_2-2}{n_1} + \frac{n_1+n_2-2}{n_2}}} \tag{16.3}\]

Using the oddsratio_to_r function in the effectsize package we can convert \(OR\) to \(r\).

# Example:
# OR = 2.21, n1 = 50, n2 = 70

OR <- 2.21
n1 <- 50
n2 <- 70

oddsratio_to_r(OR=OR, n1 = n1, n2 = n2)

[1] 0.2124017