normality - senecio7の日記

Nicholson G, Smith AV, Jónsson F, Gústafsson Ó, Stefánsson K & Donnelly P 2002 Assessing population differentiation and isolation from single-nucleotide polymorphism data. J R Stat Soc B 64:695-715.

in the Balding-Nichols (BN) model, allele frequencies at a particular locus in each population are assumed to vary independently about 'mean' values, according to a symmetric Dirichlet distribution (beta for biallelic systems like SNPs), with an additional parameter for each locus–population pair specifying the variance structure
we believe that the different perspective here (explicitly transient) is appropriate for the different genetic systems (SNPs) on which we focus
there is a literature (reviewed in Beaumont (2001)) on distinguishing between the BN model and the exact pure drift model for real data
the model that is introduced in this paper agrees with the BN model to ﬁrst and second moments
an important difference in principle is that because it has mass outside [0, 1] the use of a normal, but not of a beta, distribution allows for variation to be lost (population allele frequencies of 0 or 1) in some contemporaneous populations
a feature that is apparently important in the second data set below

we model the dependence structure among the population allele frequencies α as follows
first introduce another collection of unobserved quantities, one for each locus
π_i, i = 1, 2,..., L
in the population genetics discussion below these will play the role of the allele frequencies in a population ancestral to those sampled
in addition, introduce parameters c_j, j = 1, 2,..., P, one for each population
these will be the parameters that we aim to estimate
they specify how far (in the sense of variance) each population's allele frequencies tend to be from typical values
formally, conditionally on π, c,
α_ij ~ normal{π_i, c_jπ_i(1 − π_i)} ... (2)
here and throughout we shall sometimes specify continuous distributions for quantities which are restricted to [0, 1]
by this we mean the distribution whose density on (0, 1) is the relevant density, with atoms at 0 and 1 whose size is the total mass of the relevant distribution on (−∞, 0) and (1, ∞) respectively
to complete the hierarchy we put independent priors on π and c
π₁,..., π_L are independent and identically distributed with density f ... (3)
c₁,..., c_L are independent and identically distributed with density g ... (4)

the version of the Wright–Fisher diffusion in question has inﬁnitesimal mean 0 and inﬁnitesimal variance at z given by z(1 − z)
provided that τ is small, the increment in the diffusion, from starting value π, over time τ, is approximately normally distributed, in this case with mean 0 and variance τπ(1 − π)
α ~ normal{π, τπ(1 − π)} ... (7)
which is consistent with the marginal distribution implied by equation (2)
both the natural Markov model and the diffusion approximation have absorbing boundaries corresponding to either all or none of the chromosomes in the population carrying the variant in question
provided that the distributional statement (7) is interpreted as described after expression (2), then expression (7) remains a reasonable approximation
see Beaumont (2001) and references therein for an exact treatment
in this setting, the parameter c_j for population j has a very natural interpretation as the time, on the diffusion timescale, for which the population has undergone genetic drift
in the special case in which the major effect is the bottleneck at founding, then c_j can be interpreted as the inverse of the size of the bottleneck
expression (7) is effectively just the normal approximation to the binomial distribution for the number of copies of the allele of interest after the bottleneck

Wright (1951) described F_ST as 'the correlation between random gametes, drawn from the same subpopulation, relative to the total'
unfortunately this definition is not precise
some of the subsequent confusion in the literature stems from different interpretations
one conceptual dichotomy between approaches arises from differences (usually implicit rather than explicit) in what is being conditioned on
a common definition is
F_ST = (Q₂ − Q₃) / (1 − Q₃) ... (9)
where Q₂ and Q₃ are respectively the probability that two copies of the region on different chromosomes sampled from within and between populations are the same

in what Balding has called the descriptive approach, often associated with Nei and colleagues, these probabilities are thought of as relating only to the sampling process by which the chromosomes are chosen
they are deﬁned conditionally on the population allele frequencies
as a consequence, F_ST is a function of the population allele frequencies
in practice it is often 'evaluated' simply by replacing these by the obvious estimates from the sample data

other approaches (model based in Balding's terminology) interpret the probabilities in equation (9) as relating to repetitions of the entire evolutionary process, rather than simply over repetitions of sampling from the extant populations
F_ST would be thought of as a statistical parameter
the goal is to estimate it from data, and/or to relate it (by probability calculations) to parameters which directly specify the evolutionary model
the most common estimation procedure (see for example Weir (1996)) is often formulated by analogy with analysis of variance
it is equivalent (Rousset, 2001) to a method-of-moments approach
the probabilities Q₂ and Q₃ in equation (9) are estimated by the frequencies of identical pairs of chromosomes at the locus, within and between populations respectively, in the sample, and the estimates substituted into equation (9)

writing α for the allele frequency in a population, a rearrangement of equation (9) gives
F_ST = var(α) / (E(α) {1 − E(α)}) ... (10)
in the descriptive framework the randomness relates simply to which population may happen to be chosen
in a model-based framework the expectations are over the evolutionary model
note the standard multiplicative parameterization of var(α) relative to E(α) {1 − E(α)} to which we referred earlier
formula (10) bears a close similarity to the marginal variance structure implied by our model (2)
if we were to insist on a common value of c across populations, then equation (10) would obtain, with F_ST replaced by c, provided that it was interpreted as being conditional on π
in this sense, particularly as different approaches involve different conditioning anyway, our parameters c_j might be thought of as analogous to F_ST-values, but with one for each population