The third moments of the site frequency spectrum

A General Framework for Neutrality Tests Based on the Site Frequency Spectrum

One of the main necessities for population geneticists is the availability of sensitive statistical tools that enable to accept or reject the standard Wright-Fisher model of neutral evolution. A number of statistical tests have been developed to detect specific deviations from the null frequency spectrum in different directions (e.g., Tajima's D, Fu and Li's F and D tests, Fay and Wu's H). A general framework exists to generate all neutrality tests that are linear functions of the frequency spectrum. In this framework, it is possible to develop a family of optimal tests with almost maximum power against a specific alternative evolutionary scenario. In this paper we provide a thorough discussion of the structure and properties of linear and nonlinear neutrality tests. First, we present the general framework for linear tests and emphasise the importance of the property of scalability with the sample size (that is, the interpretation of the tests should not depend on the sample size), which, if missing, can lead to errors in interpreting the data. After summarising the motivation and structure of linear optimal tests, we present a more general framework for the optimisation of linear tests, leading to a new family of tunable neutrality tests. In a further generalisation, we extend the framework to nonlinear neutrality tests and we derive nonlinear optimal tests for polynomials of any degree in the frequency spectrum.

Multivariate phase-type theory for the site frequency spectrum

Linear functions of the site frequency spectrum (SFS) play a major role for understanding and investigating genetic diversity. Estimators of the mutation rate (e.g. based on the total number of segregating sites or average of the pairwise differences) and tests for neutrality (e.g. Tajima's D) are perhaps the most well-known examples. The distribution of linear functions of the SFS is important for constructing confidence intervals for the estimators, and to determine significance thresholds for neutrality tests. These distributions are often approximated using simulation procedures. In this paper we use multivariate phase-type theory to specify, characterize and calculate the distribution of linear functions of the site frequency spectrum. In particular, we show that many of the classical estimators of the mutation rate are distributed according to a discrete phase-type distribution. Neutrality tests, however, are generally not discrete phase-type distributed. For neutrality tests we derive the probability generating function using continuous multivariate phase-type theory, and numerically invert the function to obtain the distribution. A main result is an analytically tractable formula for the probability generating function of the SFS. Software implementation of the phase-type methodology is available in the R package PhaseTypeR, and R code for the reproduction of our results is available as an accompanying vignette.

Phase-type distributions in population genetics

Probability modelling for DNA sequence evolution is well established and provides a rich framework for understanding genetic variation between samples of individuals from one or more populations. We show that both classical and more recent models for coalescence (with or without recombination) can be described in terms of the so-called phase-type theory, where complicated and tedious calculations are circumvented by the use of matrix manipulations. The application of phase-type theory in population genetics consists of describing the biological system as a Markov model by appropriately setting up a state space and calculating the corresponding intensity and reward matrices. Formulae of interest are then expressed in terms of these aforementioned matrices. We illustrate this procedure by a number of examples: (a) Calculating the mean, (co)variance and even higher order moments of the site frequency spectrum in multiple merger coalescent models, (b) Analysing a sample of DNA sequences from the Atlantic Cod using the Beta-coalescent, and (c) Determining the correlation of the number of segregating sites for multiple samples in the two-locus ancestral recombination graph. We believe that phase-type theory has great potential as a tool for analysing probability models in population genetics. The compact matrix notation is useful for clarification of current models, and in particular their formal manipulation and calculations, but also for further development or extensions.

The neutral frequency spectrum of linked sites

We introduce the conditional Site Frequency Spectrum (SFS) for a genomic region linked to a focal mutation of known frequency. An exact expression for its expected value is provided for the neutral model without recombination. Its relation with the expected SFS for two sites, 2-SFS, is discussed. These spectra derive from the coalescent approach of Fu (1995) for finite samples, which is reviewed. Remarkably simple expressions are obtained for the linked SFS of a large population, which are also solutions of the multi-allelic Kolmogorov equations. These formulae are the immediate extensions of the well known single site θ∕f neutral SFS. Besides the general interest in these spectra, they relate to relevant biological cases, such as structural variants and introgressions. As an application, a recipe to adapt Tajima's D and other SFS-based neutrality tests to a non-recombining region containing a neutral marker is presented.