Estimating the correlation of exchangeable variables in assortative mating
In studies of assortative mating, similarity between variables measured in parents is often quantified using correlation. The order of the parents within any given pair can be arbitrary in these applications, but common correlation estimators are not robust to reordering within pairs. These unordered variable pairs are exchangeable, since the joint distributions of both orders are equal, and a given order is biased if the one variable has a lower expectation than the other. In this work, we characterize the effect of order bias on Pearson correlation estimates assuming exchangeable variables, and develop a new unbiased estimator, CorSym, that does not depend on order within each pair. Exchangeable variables have equal marginal distributions for both variables, a property accounted for by CorSym. In contrast, standard correlation estimators assume the two variables have different distributions, so biased orders skew the underlying mean, variance and covariance estimates. We show, through theory and simulations, how order bias often results in upwardly biased Pearson correlation estimates. Simulations confirm CorSym is unbiased, and validate its estimated confidence intervals. Using real admixed trios (parents and a child) from 1000 Genomes, we first demonstrate that the global ancestry of fathers and mothers are consistent with exchangeability, using both Kolmogorov-Smirnov tests and a Binomial test for order bias. However, ANCESTOR, which estimates parental global ancestry from a child's local ancestry, produces significant order biases in its output that result in substantial Pearson biases, which CorSym overcomes. Compared to ancestry proportions calculated directly on the parents, ANCESTOR also overestimates parent ancestry divergence and experiences another estimation artifact. Overall, CorSym solves an important estimation bias likely to be encountered in the study of assortative mating, providing unbiased and deterministic estimates that do not depend on the arbitrary order of the data.