15  Converting to Cohen’s $d$

15.1 From Independent Samples $t$-statistic

To calculate a between subject standardized mean difference ($d_p$, i.e., pooled standard deviation standardizer), we can use the sample size in each group ($n_1$ and $n_2$) as well as the $t$-statistic from an independent sample t-test and plug it into the following formula:

$$\begin{array}{}\text{(15.1)}& d_p=t\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\end{array}$$

Using the t_to_d function in the effectsize package we can convert $t$ to $d_p$.

# Example:
# unpaired t-statistic = 3.25
# n1 = 50, n2 = 40

library(effectsize)

t <- 3.25
n1 <- 50
n2 <- 40

t_to_d(t, df_error = n1+n2-2, paired = FALSE)

d    |       95% CI
-------------------
0.69 | [0.26, 1.12]

15.2 From Paired Sample $t$-statistic

To calculate a within-subject standardized mean difference ($d_z$, i.e., difference score standardizer), we can use the sample size in each group ($n_1$ and $n_2$) as well as the $t$-statistic from an paired sample t-test and plug it into the following formula:

$$\begin{array}{}\text{(15.2)}& d_z=\frac{t}{\sqrt{n}}\end{array}$$

Using the t_to_d function in the effectsize package we can convert $t$ to $d_z$.

# Example:
# paired t-statistic = 3.25
# n = 50

t <- 3.25
n <- 50

t_to_d(t, df_error = n-1, paired = TRUE)

d    |       95% CI
-------------------
0.46 | [0.17, 0.76]

15.3 From Pearson Correlation

If a Pearson correlation is calculated between a continuous score and a dichotomous score, this is considered a point-biserial correlation. The point-biserial correlation can be converted into a $d_p$ value using the following formula:

$$\begin{array}{}\text{(15.3)}& d_p=\frac{r}{\sqrt{1-r^2}}\sqrt{\frac{n_1+n_2-2}{n_1}+\frac{n_1+n_2-2}{n_2}}\end{array}$$ Or if sample sizes within each group are unknown (or equal), the equation simplifies to be approximately,

$$\begin{array}{}\text{(15.4)}& d_p\approx \frac{r\sqrt{4}}{\sqrt{1-r^2}}\end{array}$$

Using the r_to_d function in the effectsize package we can convert $r$ to $d_p$.

# Example:
# r = 3.25
# n1 = 50, n2 = 40

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

r_to_d(r = r, n1 = n1, n2 = n2)

[1] 1.148913

15.4 From Odds-Ratio

An odds-ratio from a contingency table can also be converted to a $d_p$. Note that this formula is an approximation:

$$\begin{array}{}\text{(15.5)}& d_p=\frac{\log (OR)\sqrt{3}}{\pi }\end{array}$$

Using the oddsratio_to_d function in the effectsize package we can convert $OR$ to $d_p$.

# Example:
# OR = 1.46

OR <- 1.46

oddsratio_to_d(OR = OR)

[1] 0.2086429