Supporting Information Supporting methods Estimating ancestry components in Admixture and Structure: Estimated individual ancestry from supervised and unsupervised clustering in Admixture are highly concordant for the autosomes for both migration events, and for the X chromosome for the Neolithic migration

Estimating ancestry components in Admixture and Structure: Estimated individual ancestry from supervised and unsupervised clustering in Admixture are highly concordant for the autosomes for both migration events, and for the X chromosome for the Neolithic migration scenario. For X-chromosomal ancestry estimated for the steppe migration, however, reference individuals do not emerge as clusters in unsupervised Admixture, therefore results cannot be used in this framework. For both the X chromosome and autosomes, unsupervised Structure estimates are similar to those from Admixture. Perhaps surprisingly, supervised Structure produces different estimates for both individual and population-level ancestry estimates; however, as we show below, this deviation is likely an effect of a small sample size, as increasing the sample size for supervised Structure leads to results similar to Admixture and unsupervised Structure for an example case with high coverage data. In all Structure analyses, we use 10,000 burn-in iterations, followed by 50,000 iterations. As our inference methods rely on the mean ancestry in the population (and are supported by the variance), we focus on comparisons of these summary statistics. The population mean and variance of ancestry for supervised Structure, supervised and unsupervised Admixture are highly concordant (no data for BA X chromosome in unsupervised Admixture). However, the mean ancestry estimated in supervised Structure is qualitatively different, and the variance is substantially lower. That is, downstream analyses using the first three methods would produce similar estimates of the levels of sex bias during the two migrations, while inference from supervised Structure would be qualitatively different, though notably, still in the same direction we observe for both migration events. Below, we investigated various factors, particularly sample size, that can cause the deviation in supervised Structure results by investigating HapMap data. We conclude that the differing finding using supervised Structure for the ancient individuals is due to sensitivity to low sample sizes for the supervised Structure algorithm and we use estimates from supervised Admixture for inference in the main text, and reiterate that inference using unsupervised Structure would be highly similar because of the similar mean and variance of ancestry estimated. Ancestry for each individual is presented as the average estimated individual ancestry using ten independent seeds, considering the X chromosome and autosomes separately.