Bias in Rank Correlation Under Mixture Models

This study investigates the behavior of two prominent measures of rank correlation—Kendall’s tau and Spearman’s rho—under mixture models, particularly how they are biased when the sample is contaminated by observations from an unintended population. Using expressions for population versions of rank correlation, we derive that the bias under mixture is a polynomial in the mixing proportion p. The coefficients of these polynomials are sums of integrals of joint distributions of the mixture components. Interestingly, the bias is quadratic for tau but cubic for rho. For each degree polynomial, we derive a partition according to root behavior in the unit interval, yielding several possible scenarios for regions in which the bias will be positive or negative. We then demonstrate that within the Marshall–Olkin family of distributions, there exist choices of parameters that will give rise to every possible root behavior scenario through a computational experiment.