Point-Biserial Correlation

What is a Point-biserial correlation?

Point-biserial correlation is a special case of Pearson correlation and examines the linear relationship between a dichotomous variable and a metric variable.

What is a dichotomous variable and what is a metric variable? A dichotomous variable has two categories, for example gender (male/female) or smoking status (smoker/non-smoker). A metric variable could be a person's weight or salary.

If we have a dichotomous variable and a metric variable and want to know if there is a correlation, we can use a point-biserial correlation. We need to check the assumptions first, but more about that later.

Calculate the point-biserial correlation

As noted above, the point-biserial correlation is a special case of Pearson correlation. But how can we calculate Pearson correlation when one variable is nominal? Let's look at an example.

Let's say we want to study the correlation between the number of hours spent studying for an exam and the exam result (pass/fail).

We collected data from 20 students, 12 of whom passed the test and 8 of whom failed. We recorded the number of hours each student studied for the exam.

To calculate the point-biserial correlation, we first convert the test result to numbers. We can assign a value of 1 to students who passed and 0 to students who failed. Reversing this coding changes the sign of the correlation, but not its magnitude.

Now we can either calculate the Pearson correlation between study time and test result, or use the equation for the point-biserial correlation.

For the 0/1 coding used here, the numerator must be the mean for the group coded 1 minus the mean for the group coded 0. The labels for the two group means in the displayed equation are reversed.

Point-biserial correlation and Pearson correlation

Whether we calculate the Pearson correlation or use the equation for the point-biserial correlation, we get the same result.

Calculation with numiqo

Let's look at this in numiqo. We have learning hours, the pass/fail test result, and the test result coded as 0 and 1. We treat the test result with zero and one as metric.

If we go to correlation and calculate the Pearson correlation for these two metric variables, we get a correlation coefficient of 0.31. If we calculate the point-biserial correlation for learning hours and exam result with "passed" and "failed," we also get a correlation of 0.31.

Point-biserial correlation coefficient

Like the Pearson correlation coefficient r, the point-biserial correlation coefficient r_pb also varies between -1 and 1.

With the coding used above, a negative coefficient means that the group coded 1 tends to have lower values of the metric variable than the group coded 0.

A positive coefficient means that the group coded 1 tends to have higher values. A coefficient of 0 means that the group means are equal, although the two groups may still differ in other ways. As with any correlation, this does not by itself show that one variable causes the other.

Hypotheses

Often, however, starting from a sample, we want to test a hypothesis about the population. In the case of correlation analysis, we can test whether the correlation coefficient is significantly different from 0.

The hypotheses for the point-biserial correlation are:

• Null hypothesis: The population correlation coefficient ρ_pb = 0.
• Alternative hypothesis: The population correlation coefficient ρ_pb ≠ 0.

Point-biserial correlation and the t-test for independent samples

The usual significance test for a point-biserial correlation gives the same p-value as the equal-variance independent t-test applied to the same data. Both tests ask whether the two population means are equal. Welch's t-test, which does not assume equal variances, can give a different p-value.

Whether we test a correlation hypothesis with the point-biserial correlation or a difference hypothesis with the t-test, we get the same p-value.

If we calculate a t-test in numiqo with the data under the tab "Hypothesis Tests", and we have the null hypothesis: "There is no difference between the fail and pass groups with respect to the variable Hours Studied", then we get a p-value of 0.179.

Likewise, if we calculate a point-biserial correlation under the tab "Correlation" and we have the null hypothesis: "There is no correlation between Hours Studied and Test Result", we also get a p-value of 0.179!

In our example, the p-value is greater than 0.05, which is commonly used as a significance level, and thus the null hypothesis is not rejected.

Assumptions for a point-biserial correlation

For point-biserial correlation, we need to distinguish between calculating the correlation coefficient and testing a hypothesis. To calculate the coefficient, we need a metric variable and a dichotomous variable. For the usual analysis, observations should be independent. Strong outliers in the metric variable can have a large effect on the coefficient.

For the usual small-sample significance test, the metric variable should be approximately normally distributed within each group. The test corresponding to the equal-variance independent t-test also assumes equal population variances. With large samples, moderate departures from normality are often less problematic, but outliers and strongly unequal group sizes still require care.

A naturally dichotomous variable, such as pass/fail, is appropriate here. Turning a continuous variable into two artificial groups discards information and can make the result harder to interpret.