sfsr

An R package for analysing population-genetic data represented as the site frequency spectrum (SFS)

Background

The site frequency spectrum (SFS; also known as the allele frequency spectrum) is the histogram over allele counts in a population. More formally, for a population consisting of $N$ chromosomes, the SFS is the vector $\mathbf{\zeta} = \zeta_0, \zeta_1, \dots, \zeta_N$ whose entries correspond to the number of sites at which $0, 1, \dots, N$ copies of the derived allele are segregating in the population. When the polarity (ancestral vs derived) of alleles is unknown, the SFS is said to be "folded;" the folded SFS is often denoted instead $\mathbf{\nu} = \nu_0, \dots, \nu_m$ (with $m = \ceil{\frac{N}{2}}$) and its entries are the count of the minor allele.

The SFS can be generalized to higher dimensions when more than one population is considered; in this case we may call it the joint SFS. In this case each entry $\mathbf{\zeta}_{i,j,\dots}$ gives the count of alleles with frequency $i$ in population 1, $j$ in population 2, etc. Given a joint SFS, the single-population SFS can always be obtained by "marginalizing" -- just summing over all other dimensions.

The SFS provides a rich summary of polymorphism data. Departures in the shape of the SFS are informative for demography and/or selection. Most common summary statistics for polymorphism data both within and between populations can be computed directly from the SFS.

Rationale

Although the SFS can be easily tabulated from discrete genotypes, it may be advantageous to treat it as a parameter to be estimated from data under some model of uncertainty and technical error. Such methods are implemented in software like ANGSD, but I found few resources for manipulating and analyzing the resulting SFS. Hence the sfsr package.

At the moment the package supports only unfolded SFS -- those for which alleles can be polarized without ambiguity as ancestral vs derived. This happens to be the case for problems I work on, and gets around the data-dependency of the "minor allele" designation.

Features

• File I/O
  • read ANGSD-style SFS from disk (read_sfs())
• Manipulation and transformation of SFS
  • marginalization (margnialize_sfs())
  • changes of ancestral-vs-derived polarity (repolarize_sfs())
• Common within-population summary statistics
  • Estimators of $\theta$
    • Tajima's pairwise, aka $\pi$ (theta_pi())
    • Watterson's (theta_w())
    • from singleton count (theta_zeta())
  • Neutrality statistics
    • Tajima's $D$ (tajimaD())
    • Fu and LI's $D$ (fuliD())
    • Fu and LI's $F$ (fuliF())
    • Achaz's $Y$, robust to sequencing errors (achazY())
• Between-population summary statistics
  • fixed vs polymorhic sites (fixpoly())
  • average divergence (d_xy())
  • Weir and Cockerham's $F_{st}$ (f_st())
• Hudson-Kreitman-Aguade test (hka_test())

Because bootstrap resampling from the SFS is a straightforward way to estimate uncertainty, SFS as reprsented by sfsr may carry bootstrap replicates as an attribute. In this case most summary statistics are automatically calculated across the bootstraps for convenient estimation of standard errors.