Converting-to-Odds-Ratio.qmd

# Converting to Odds Ratio

## From Cohen's $d$

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

$$
OR = \exp\left(\frac{d_p \pi}{\sqrt{3}}\right)
$$

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 $OR$.

```{r,echo=TRUE}
# 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)
```

## From a Pearson Correlation

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

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

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

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

Using the `r_to_oddsratio` function in the `effectsize` package we can convert $d$ to $OR$.

```{r,echo=TRUE}
# Example:
# r = .50, n1 = 50, n2 = 70

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

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