Frequency spectrum neutrality tests: one for all and all for one

Abstract

Neutrality tests based on the frequency spectrum (e.g., Tajima's D or Fu and Li's F) are commonly used by population geneticists as routine tests to assess the goodness-of-fit of the standard neutral model on their data sets. Here, I show that these neutrality tests are specific instances of a general model that encompasses them all. I illustrate how this general framework can be taken advantage of to devise new more powerful tests that better detect deviations from the standard model. Finally, I exemplify the usefulness of the framework on SNP data by showing how it supports the selection hypothesis in the lactase human gene by overcoming the ascertainment bias. The framework presented here paves the way for constructing novel tests optimized for specific violations of the standard model that ultimately will help to unravel scenarios of evolution.

Figure 1.—

Estimators of θ. A graphical view of the weight vectors of four typical estimators of θ (for n = 30). All values of the normalized vector sum to 1. In the top four panels, the -vectors that are defined for the unfolded frequency spectrum () are given, whereas the two bottom ones are the -vectors that are defined for the folded frequency spectrum (). For estimators that can be defined in terms of both and (here and ), the latter can be computed from the former with (when i ≠ n − i) or (when i = n − i).

Figure 2.—

Neutrality tests. A graphical view of the weight vectors of four typical neutrality tests (for n = 30). Because the -vectors used for neutrality tests are computed as a difference between two normalized vectors, all values of sum to 0. In the top four panels, the -vectors that are defined for the unfolded frequency spectrum () are given, whereas the two bottom ones are the -vectors that are defined for the folded frequency spectrum (). For estimators that can be defined in terms of both and (here D and F *), the latter can be computed from the former in the way that can be deduced from ω_i.

Figure 3.—

Example of a severe bottleneck. (a) The mean and the standard deviation of the spectrum that is observed in simulations (n = 30, 10⁴ replicates) of a standard model or of a recent severe bottleneck (reduction of f = 1/100 for a time T_l = 0.1). In both times after the bottleneck (T_b = 0.03 and T_b = 0.3), the observed trend is similar: an excess in low frequency of , though stronger for T_b = 0.03. (b) Left, the weight vector of a new neutrality test (here T_Ω) is reported. It focuses its sensitivity on low frequencies: . (b) Right, the power of four neutrality tests is compared in detecting a severe bottleneck as a function of the time elapsed after the bottleneck. The new test shows enhanced power to detect the bottleneck (more power for a longer time).

Figure 4.—

Isolation with migration. (a) The mean and the standard deviation of the spectrum that is observed in simulations (n = 30, 10⁴ replicates) of a standard model or of an isolation with migration model (two populations equally sampled, n_a = n_b = 15 that were a single ancestral panmictic population at time T_i = 3). In both sampling migration rates between the two populations (M = 0.1 and M = 1), the observed trend is similar: an excess of , though much stronger for M = 0.1. (b) Left, the weight vector of a new neutrality test (T_ω) that focuses its sensitivity on i = 15. The weight vector used here is , where is obtained using a binomial with p = 0.5 and n = 30. (b) Right, the power of four neutrality tests is compared when detecting the population structure as a function of the migration rate. The new test displays much more power to detect the population structure.