Estimating the correlation of exchangeable variables in assortative mating

Assortative mating studies often quantify similarity between parent variables using correlation. However, common correlation estimators are not robust to reordering within these parent pairs. These unordered pairs are exchangeable, meaning their joint distributions remain equal regardless of order. This study characterizes the impact of order bias on Pearson correlation estimates when variables are exchangeable, and introduces a new unbiased estimator called CorSym that is order-independent.

Exchangeable variables share identical marginal distributions for both variables, which CorSym accounts for. Standard correlation estimators, on the other hand, assume different distributions for the two variables, leading to biased orders that skew mean, variance, and covariance estimates. Through both theory and simulations, the research shows that order bias typically causes upward bias in Pearson correlation estimates. Simulations confirm CorSym's unbiased nature and validate its confidence intervals.

Applying the research to real data from 1000 Genomes admixed trios (parents and child), the study finds that global ancestry of parents aligns with exchangeability, as determined by Kolmogorov-Smirnov tests and a Binomial test for order bias. However, ANCESTOR, which estimates parental global ancestry from child's local ancestry, exhibits significant order biases in its output.

These biases result in substantial Pearson biases when calculating ancestry proportions. ANCESTOR also overestimates parent ancestry divergence and encounters another estimation artifact when using data directly from the parents. Overall, CorSym addresses an important estimation bias in assortative mating studies, offering unbiased and deterministic estimates that are independent of the arbitrary order of the data.