biserial

Biserial refers to a family of statistical measures used to describe the association between a binary (dichotomous) variable and a continuous variable, typically under the assumption that the binary outcome arises from dichotomizing an underlying normal latent variable.

Two common forms are the point-biserial correlation and the biserial correlation. For a binary variable Y (0/1) with p = P(Y = 1), and a continuous variable X with overall mean μ and standard deviation σ, and with μ1 and μ0 denoting the means of X for Y = 1 and Y = 0 respectively, the point-biserial correlation is r_pb = (μ1 − μ0) · sqrt(p q) / σ, where q = 1 − p. This is the direct, observed correlation between X and Y.

The biserial correlation, r_b, is an adjusted form that reflects an underlying latent continuous variable Z from which Y is derived by thresholding. If z_p is the standard normal quantile corresponding to p (z_p = Φ^−1(p)) and φ is the standard normal density, then r_b = (μ1 − μ0) · φ(z_p) / (σ · p q). This form tends to be larger than r_pb when p is near 0.5 and incorporates the assumed normal shape of the latent distribution.

Interpretation and use: Both coefficients range from −1 to 1 and indicate the strength and direction of association between the continuous variable and the binary variable. r_b is particularly used when researchers believe an underlying continuous propensity exists and Y reflects a threshold on that latent variable. Assumptions include approximate normality of X within Y groups and, for r_b, an appropriate latent-variable model. When those assumptions may not hold, alternative measures such as the phi coefficient or rank-based methods can be considered.