pointbiserial

The point-biserial correlation coefficient, typically denoted r_pb, is a measure of the strength and direction of the association between a dichotomous (binary) variable and a continuous variable. It is a special case of the Pearson correlation when one variable is binary coded as 0 and 1. It is commonly used in psychology, education, medicine, and social sciences to assess how a binary grouping relates to a quantitative outcome.

Calculation is based on the means of the continuous variable in the two groups defined by the binary variable. Let X be the binary variable with values 0 and 1, having proportions p and q = 1 − p, respectively. Let Y be the continuous variable with overall standard deviation s, M0 be the mean of Y when X = 0, and M1 be the mean of Y when X = 1. The point-biserial correlation is r_pb = (M1 − M0) * sqrt(p q) / s. Equivalently, r_pb is the Pearson correlation between X and Y when X is coded as 0 and 1.

Interpretation focuses on direction and strength. The value of r_pb ranges from -1 to 1. A positive r_pb indicates higher Y values are associated with the group coded as 1; a negative r_pb indicates higher Y values with the group coded as 0. The magnitude reflects the strength of the association, with guidelines similar to Pearson’s r, though context matters.

Assumptions and cautions include that Y should be measured on at least an interval scale, observations should be independent, and unequal group sizes can affect stability. Like all correlations, r_pb does not imply causation and captures linear association; non-linear relationships may not be well represented by r_pb.