17  Converting to Odds Ratio

17.1 From Cohen’s \(d\)

We can calculate an odds-ratio from a between groups cohen’s \(d\) (\(d_p\)):

\[ OR = \exp\left(\frac{d_p \pi}{\sqrt{3}}\right) \tag{17.1}\]

Where \(\exp(\cdot)\) is an exponential transformation (this inverses the logarithm). Using the d_to_oddsratio function in the effectsize package we can convert \(d\) to \(OR\).

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

library(effectsize)

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

d_to_oddsratio(d = d, n1 = n1, n2 = n2)

[1] 2.969162

17.2 From a Pearson Correlation

We can calculate an odds ratio from a Pearson correlation using the following formula:

\[ OR = \exp\left(\frac{r\pi \sqrt{\frac{n_1+n_2-2}{n_1} + \frac{n_1+n_2-2}{n_2}}}{\sqrt{3(1-r^2)}}\right) \tag{17.2}\]

When sample sizes are equal, this equation can be simplified to be approximately,

\[ OR = \exp\left(\frac{r\pi \sqrt{4}}{\sqrt{3(1-r^2)}}\right) \tag{17.3}\]

Using the r_to_oddsratio function in the effectsize package we can convert \(d\) to \(OR\).

# Example:
# r = .50, n1 = 50, n2 = 70

r <- .40
n1 <- 50
n2 <- 70

r_to_oddsratio(r = r, n1 = n1, n2 = n2)

[1] 4.870584