Distortion of genealogical properties when the sample is very large

Abstract

Study sample sizes in human genetics are growing rapidly, and in due course it will become routine to analyze samples with hundreds of thousands, if not millions, of individuals. In addition to posing computational challenges, such large sample sizes call for carefully reexamining the theoretical foundation underlying commonly used analytical tools. Here, we study the accuracy of the coalescent, a central model for studying the ancestry of a sample of individuals. The coalescent arises as a limit of a large class of random mating models, and it is an accurate approximation to the original model provided that the population size is sufficiently larger than the sample size. We develop a method for performing exact computation in the discrete-time Wright-Fisher (DTWF) model and compare several key genealogical quantities of interest with the coalescent predictions. For recently inferred demographic scenarios, we find that there are a significant number of multiple- and simultaneous-merger events under the DTWF model, which are absent in the coalescent by construction. Furthermore, for large sample sizes, there are noticeable differences in the expected number of rare variants between the coalescent and the DTWF model. To balance the trade-off between accuracy and computational efficiency, we propose a hybrid algorithm that uses the DTWF model for the recent past and the coalescent for the more distant past. Our results demonstrate that the hybrid method with only a handful of generations of the DTWF model leads to a frequency spectrum that is quite close to the prediction of the full DTWF model.

Fig. 1.

The percentage relative error in the number of singletons and doubletons between the coalescent and DTWF models, as a function of the sample size n. When the sample size is comparable to the current population size, the number of singletons predicted by the DTWF model is larger than the coalescent prediction by as much as 11%, whereas the number of doubletons predicted by the DTWF model is smaller than the coalescent prediction by about 4.8%. In model 4, we could not consider a sample size comparable to the population size (10⁶) because of computational burden, but we expect a similar extent of deviation as in models 1–3 as n increases. Note that the y-axis scale for model 4 is different from that for models 1–3. (A) Model 1. (B) Model 2. (C) Model 3. (D) Model 4.

Fig. 2.

The percentage relative error, with respect to the full DTWF model, in the number of singletons and doubletons in a hybrid algorithm with switching time t_s. The hybrid method uses the DTWF model for generations ≤t_s and the coalescent model in generations >t_s. The results are for model 3 in the case in which the sample size n is equal to the current effective population size N₀ = 67,627. The case of t_s = 0 corresponds to using the coalescent model only. This plot shows that the difference in the frequency spectrum between the full DTWF model and the hybrid algorithm decreases very rapidly as the switching time t_s increases.