**Point-biserial correlation** is used to measure the relationship between a binary variable, x, and a continuous variable, y.

Similar to the Pearson correlation coefficient, the point-biserial correlation coefficient takes on a value between -1 and 1 where:

- -1 indicates a perfectly negative correlation between two variables
- 0 indicates no correlation between two variables
- 1 indicates a perfectly positive correlation between two variables

This tutorial explains how to calculate the point-biserial correlation between two variables in R.

**Example: Point-Biserial Correlation in R**

Suppose we have a binary variable, x, and a continuous variable, y:

x <- c(0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0) y <- c(12, 14, 17, 17, 11, 22, 23, 11, 19, 8, 12)

We can use the built-in R function **cor.test() **to calculate the point-biserial correlation between the two variables:

#calculate point-biserial correlation cor.test(x, y) Pearson's product-moment correlation data: x and y t = 0.67064, df = 9, p-value = 0.5193 alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval: -0.4391885 0.7233704 sample estimates: cor 0.2181635

From the output we can observe the following:

- The point-biserial correlation coefficient is
**0.218** - The corresponding p-value is
**0.5193**

Since the correlation coefficient is positive, this indicates that when the variable x takes on the value “1” that the variable y tends to take on higher values compared to when the variable x takes on the value “0.”

However, since the p-value of this correlation is not less than .05, this correlation is not statistically significant.

Note that the output also provides a 95% confidence interval for the true correlation coefficient, which turns out to be:

**95% C.I. = (-0.439, 0.723)**

Since this confidence interval contains zero, this is further evidence that the correlation coefficient is not statistically significant.

**Note**: You can find the complete documentation for the** cor.test()** function here.

**Additional Resources**

The following tutorials explain how to calculate other correlation coefficients in R:

How to Calculate Partial Correlation in R

How to Calculate Rolling Correlation in R

How to Calculate Spearman Rank Correlation in R

How to Calculate Polychoric Correlation in R

Hello.Thank you very much for the synthax.I supposedcor.test doeasnt work for point.biserial correlation.Is there a seperate command?

Hi Armita…Yes, you’re correct. The `cor.test()` function in most statistical software packages typically calculates correlation tests for Pearson’s correlation coefficient (for continuous variables) or Spearman’s rank correlation coefficient (for ordinal variables). However, for point-biserial correlation, which measures the relationship between a continuous variable and a binary variable, you would typically use a different approach.

You can calculate the point-biserial correlation coefficient directly using the formula:

\[ r_{pb} = \frac{{M_1 – M_0}}{{\sqrt{{\frac{{N_1N_0}}{{N(N-1)}}}}}} \]

Where:

– \( M_1 \) is the mean of the continuous variable for the group with a value of 1 (e.g., the mean of the continuous variable for the group with the characteristic you’re interested in),

– \( M_0 \) is the mean of the continuous variable for the group with a value of 0,

– \( N_1 \) is the number of observations in the group with a value of 1,

– \( N_0 \) is the number of observations in the group with a value of 0,

– \( N \) is the total number of observations.

You can then perform hypothesis tests or calculate confidence intervals for the point-biserial correlation coefficient using standard formulas or statistical software that supports this analysis. Some statistical software packages may have specific functions or procedures for point-biserial correlation analysis. If you’re using R, for example, you might need to use a package like `psych` or `ltm` to calculate point-biserial correlation coefficients and conduct related tests.