On measures of association among genetic variables

Association based on Kullback-Leibler distance

The Kullback-Leibler (KL) logarithmic distance (Kullback 1968) is defined as the sum of two KL discrepancies between the distributions: one under association and the other under independence. The first discrepancy measures departure from association when the latter is true, and the other measures departure from independence when this attribute holds. The two discrepancies are positive by construction and are defined as

DAI=Discrepancy(association→independence)=Ep(x,y)(logαxy)=∫∫(logαxy)p(x,y)dxdy,

which measures discrepancy when it is true that X and Y are associated, and

DIA=Discrepancy(independence→association)=Ep(x)p(y)(logαxy−1)=∫∫(logαxy−1)p(x)p(y)dxdy,

measuring discrepancy when independence holds; the range of integration depends on the random processes involved. If the two distributions are identical, both discrepancies are null; on the other hand, large discrepancies reflect stochastic dependence. Above, E_p(x,y) and E_p(x)p(y) denote expectations taken under the assumptions that (X,Y) follow either a bivariate distribution (association) or that these variables are independently distributed.

A connection with the likelihood ratio statistic is fairly direct. Suppose that the parameters of the distributions under independence and association are estimated by maximum likelihood (ML), and that these estimates are regarded as if they were ‘true’ values. Then, log α_xy, with the densities evaluated at the corresponding ML estimates would be the log of a ratio between maximized likelihoods. In D_AI one would be taking expectations of the log-likelihood ratio over all possible realizations expected if the ‘full model’ (association) were true; on the other hand, in D_IA the expectations of the reciprocal of the log-likelihood would be taken under the assumption of independence. In this context, D_AI and D_IA are averages of log-likelihood ratios, as opposed to an evaluation at a single realization (in ML estimation parameters vary whereas observations are taken as fixed). Note that D_AI and D_IA do not involve any requirement of ‘nesting’ of models, contrary to likelihood ratio tests under regularity conditions, but this is a different issue.

The discrepancies are such that D_AI ≠ D_IA, and a unique, symmetric KL distance is defined (Kullback 1968) as

KL=DAI+DIA.(2)

Then, an index of association (taking values between 0 and 1) is obtained by normalizing D_IA as

θ=DIADAI+DIA,(3)

with

1−θ=DAIDAI+DIA,

provided that KL is not null. A large value of D_IA relative to D_AI (or the opposite) is indicative of situations in which there is more association than what would be expected under independence so that θ or 1 − θ would be close to 1. Thus, values of θ or of 1 − θ away from 12 provide evidence of discrepancy between the hypotheses of random ensembles of (x,y) pairs vs. associated ensembles.

There is no closed form for D_IA or D_AI for many distributions, in which case these metrics must be calculated numerically or estimated using Monte Carlo methods, given some parameter values. If association is weak, the two logarithmic discrepancies and KL are nearly null, and this can lead to very unstable, even absurd, Monte Carlo estimates of θ. Also, the KL distance and the two discrepancies are logarithmic, and this perhaps provides a less intuitive notion of ‘distance’ than one based directly on α_xy. An alternative ‘distance’ measure is presented immediately below.

Association based on relative distance

Define the relative discrepancies

DAI'=∫∫αxyp(x,y)dxdy=∫∫p2(x,y)p(x)p(y)dxdy,(4)

DIA'=∫∫αxy−1p(x)p(y)dxdy=∫∫p2(x)p2(y)p(x,y)dxdy(5)

and

θ'=DIA'DAI'+DIA'.(6)

The metric DAI' gives the expected value of α_xy under association and DIA' gives the expectation of αxy−1 under the assumption of independence. These expectations are both positive, by construction, and values of θ^′ away from 12 can be construed as reflecting departure from independence. For example, θ'=15 or 45 would suggest association. The connection with likelihood ratios surfaces again; for example, DIA' is the average likelihood-ratio over all possible realizations of the data under the independence assumption.

In what follows, the proposed indexes are examined for some distributions that are used often in quantitative and population genetics research.

Bivariate normal distribution

Let X₁ and X₂ be two standardized normally distributed genetic values with correlation ρ. For this distribution DAI=−log1−ρ2 and DIA=ρ2(1−ρ2)−1+log1−ρ2 (e.g. Sorensen & Gianola 2002). When the correlation is null, the two discrepancies are 0; when the correlation is perfect, the discrepancies are both ∞. Thus, KL = ρ²(1 − ρ²)⁻¹, which is 0 when the two random variables are independent and ∞ under a perfect correlation. Then

θ=DIADAI+DIA=1+(1−ρ2)log1−ρ2ρ2,(7)

measures discrepancy away from the independence situation when the latter holds, relative to the KL distance between the two distributions. It can be shown that lim_ρ2→0(θ)=12 and lim_ρ2→1(θ)=1, providing the lower and upper bounds of θ respectively.

The values of θ and of 1 − θ are plotted against ρ in Fig. 1. The dotted line (‘holds water’) gives the relative discrepancy (θ) away from the independence models when this is true, and the dashed line (‘spills water’) depicts the trajectory of 1 − θ, i.e. the relative discrepancy under association; note that a given curve is a mirror image of the other. As the absolute value of the correlation increases θ → 1, as one would expect. Since an association index with values ranging between 0 and 1 may be easier to interpret, an alternative measure could be

γ=2DIADAI+DIA−1=2θ−1.

However, γ takes values in [0,1] provided that θ≥12, which holds for a bivariate Gaussian distribution, but not so in general. The relationship between γ and ρ for this Gaussian model is also shown in Fig. 1 (solid line), and the association is suggested as weaker when measured by θ than when assessed with the correlation ρ. For example, if |ρ|=12, θ is smaller than 0.15. In multivariate systems or in other distributions, θ can be much smaller than 12 so that γ takes negative values, as illustrated below, so it turns out that θ often provides an easier to interpret measure of association than does γ.

Multivariate normal distribution

The metrics under consideration can be applied to distributions with any number of dimensions, leading to measures of departure from independence in more general situations, such as when alleles in a multiple-locus system are studied. To illustrate, consider the K-dimensional multivariate Gaussian distribution (x∼N(0,R), where variates are in standard deviation units so that R is a correlation matrix; under independence (x∼N(0,I). It can be shown (Penny & Roberts 2002) that

KL=12tr(R−1+R)−K,(8)

Where tr(.) denotes the sum of the diagonal elements of a squared matrix such as R. Further,

DAI=12[tr(R)−log|R|−K],(9)

and

DIA=12[log|R|+tr(R−1)−K],(10)

so that

θ=12[log|R|+tr(R−1)−K]12tr(R−1+R)−K.(11)

Examples of several multivariate normal distributions are given next.

Equi-correlated trivariate normal distribution

Let three variables be equi-correlated with correlation ρ, so that

R=[1ρρρ1ρρρ1],

with |R|=2ρ3−3ρ2+1, tr(R)=3 and tr(R−1)=3(ρ+1)/(1+ρ−2ρ2). Using (8)-(11) yields

DAI=−log(ρ−1)2(2ρ+1),

DIA=3ρ21+ρ−2ρ2−DAI,

KL=3ρ21+ρ−2ρ2,

and

θ=1−(ρ−1)(1+2ρ)log(ρ−1)2(1+2ρ)3ρ2,(12)

With 0 < θ < 1. For example, for ρ = 0.25, 0.50 and 0.75, the index of association θ takes values 0.49, 0.54 and 0.66 respectively. Figure 2 depicts θ, 1 − θ and θ as a function of the coefficient of correlation. At ρ = 0, θ = 0.5, but then it decreases slightly as the correlation increases such that γ does not attain a positive value until γ  reaches a value of about 0.3. Note that θ is not symmetric with respect to ρ (the same is true of |R|); for instance, when ρ=−14, θ = 0.59 so that association would be viewed as stronger than when ρ=14, since θ = 0.49 then. This is a consequence of the values that the discrepancies D_IA and D_AI take at varying values of ρ.

Tetra-variate normal distributions

Alternative measures can be derived from eigenvalues of a correlation matrix; however, this is sensible only for distributions in which linear relationships between variables are meaningful. The index θ, derived from notions of statistical distance, does not have this limitation because KL discrepancies have a general meaning and can be calculated for any distribution, at least numerically, given values of the parameters. Naturally, any single measure of association in a multivariate distribution becomes less transparent as the system of variables increases in dimension.

Let K = 4 and suppose the variables are equi-correlated with coefficient ρ. Then, |R|=−(ρ−1)3(3ρ+1), tr(R−1)=−(8ρ+4)/[(ρ−1)(3ρ+1)] and

θ=1−(ρ−1)(1+3ρ)log−(ρ−1)3(1+3ρ)6ρ2.(13)

Figure 3 displays how θ and 1 − θ vary against ρ. At θ = 0 one has θ=1−θ=12; θ decreases somewhat subsequently and then increases, attaining 0.5 again at about ρ = 0.45. The values of the index of association suggest that the independence and association models are not ‘too distant’ unless the correlation is below −0.10, if negative, or above ρ = 0.50, if positive.

Let now K = 4 and take as correlation matrix

R=[10.9τ0.5τ0.1τ0.9τ10.2τ0.1τ0.5τ0.2τ10.1τ0.1τ0.1τ0.1τ1],

where 0 ≤ τ ≤ 1, so that when t = 0 the four variables are independent. For t = 1, log |R| = −2.5459 and tr(R⁻¹)  = 24.8980, yielding θ = 0.8782. Likewise, for τ=12, log |R| = −0.2960 and tr(R⁻¹) = 4.6540, so that θ  = 0.5473. This illustrates that the strength of the association is proportional to the strength of inter-correlation, and that θ provides a metric for measuring departure from independence.

We finish this section by posing the following question: what does θ say that is not informed by all possible pairwise correlations? Again, let K = 4 and take as correlation matrices

R1=[1014001014140114014141];R2=[100.600100.60.6010.600.60.61].

The salient point is that the correlation structure is such that while pairs of variables (1,3), (2,4) and (3,4) are correlated, (1,2) are independent.

What is the overall strength of the association in the system? Calculations yield θ₁ = 0.54 and θ₂ = 0.91 for the networks of variables characterized by ρ=14 and ρ=610, respectively. While the correlations measure association between pairs of nodes, θ gives an indication of the overall degree of association.

Bivariate t-distribution

As mentioned, there are situations in which random variables are jointly dependent, but yet, their correlation is 0, which would incorrectly suggest that X does not inform about Y. This highlights limitations of the correlation parameter as a metric for statistical dependence between sets of variables. One such case is the multivariate-t distribution, in which variables can be statistically dependent, yet uncorrelated. The univariate and multivariate t-distributions appear in robust linear regression models for quantitative genetic analysis based on pedigrees (e.g., Strandén & Gianola 1998; Sorensen & Gianola 2002; Rosa et al. 2003). The univariate-t distribution has also been used as a prior in Bayesian linear regression models for genome-enabled prediction of single traits (Meuwissen et al. 2001; Gianola et al. 2009).

Suppose two variables, e.g. SNP effects on a pair of traits, follow a bivariate t-distribution with a null mean vector, ν degrees of freedom and a scale matrix that is equal to a 2 × 2 identity matrix; here, the two effects are uncorrelated but not independent, as shown in the Appendix. For this situation, the density ratio (1) is

αx1x2=Γ(2+υ2)Γ(υ2)[1+x12+x22υ]−(2+υ2)Γ2(1+υ2)[(1+x12υ)(1+x22υ)]−(1+υ2),

so that the relative discrepancies in (4) and (5) become

DAI'=Γ(2+υ2)Γ(υ2)Γ2(1+υ2)Ep(x1,x2|υ){[1+x12+x22υ]−(2+υ2)[(1+x12υ)(1+x22υ)]−(1+υ2)},

and

DIA'=Γ2(1+υ2)Γ(2+υ2)Γ(υ2)Ep(x1|υ)p(x2|υ){[(1+x12υ)(1+x22υ)]−(1+υ2)(1+x12+x22υ)−(2+υ2)}.

The expectations above do not have closed forms but can be evaluated using Monte Carlo methods: given the value of the degrees of freedom parameter, random numbers can be drawn from a bivariate t-distribution and from two independent univariate t-distributions, to approximate D′_AI and D′_IA.

The index of association θ′ in (6) was computed for t-distributions with μ = 1, 2, 4, 6, 8, 20 and 100 000. The last value produces an approximation of the joint distribution of two independent normal variates, whereas the setting with μ = 1 evaluates distance between a ‘bivariate Cauchy’ distribution and the distribution of two identically and independently distributed Cauchy random variables. The t-processes with μ = 1, 2, 4 do not have a finite variance, but the distributions are proper (i.e., the density integrates to 1). As shown in Table 1, with sets of 1 million random numbers used for evaluating D′_AI, D′_IA and θ′, the ‘distance’ D′_AI from association to independence decreases as ν increases, approaching 1, whereas the distance from independence to association D′_IA increases, approaching 1 as well. With ν = 100 000 the random variables are essentially independent since θ'=1−θ'=12, leading to γ′ = 0, the setting ν = 1 produces the strongest possible association, with θ′ = 0 and both 1 − θ′ and θ′ equal to 1.

Table 1. Departures of zero-mean bivariate t-distributions with a 2 × 2 identity matrix as scale parameter (so their correlation is null) from independence as measured by D′_AI, D′_IA and by the indexes of association θ′ = D′_IA/(D′_AI + D′_IA), 1 − θ′ and γ ′ = 2(1 − θ′) − 1 (when θ'=12 the random variables are independent)

Degrees of freedom	DAI'	DIA'	D'=DAI'+DIA'	θ′	1 − θ′	γ′
1	7.28 × 109	0.4376	7.28 × 109	0.0000	1.0000	1.0000
2	63438837	0.6424	6348838	0.0000	1.0000	1.0000
4	20.7443	0.7947	21.5390	0.0369	0.9631	0.9262
6	2.6299	0.8560	3.4859	0.2456	0.7544	0.5088
8	1.2715	0.8889	2.1604	0.4115	0.5885	0.1770
20	1.0584	0.9528	2.0112	0.4738	0.5262	0.0524
100 000	1.0000	1.0000	2.0000	0.5000	0.5000	0.0000

Bivariate beta distributions

A common finding in studies of population differentiation using Wright's F-statistics (Wright 1931; Cockerham 1969) with single nucleotide polymorphisms (SNPs) is that estimates of allelic frequencies within a bin of adjacent SNPs are correlated (e.g. Akey et al. 2002; Weir et al. 2005; Akey 2009). This is typically due to stochastic dependence between alleles at linked loci (producing LD), but evolutionary processes causing dependence among the true frequencies themselves have been suggested; see Ohta (1982) for a population genetics model based on this concept. For example, a correlation between alleles at randomly drawn pairs of loci arises if alleles are conditionally (given the allelic frequencies) independent but with allelic frequencies varying at random according to a beta distribution.

Suppose that pairs of loci are sampled at random from some conceptual population of equi-correlated loci and that all pairs of ‘true’ frequencies define a probability distribution. Wright (1937) found that a beta distribution arose from a diffusion equation used to study changes in allele frequencies in finite populations, so the beta process is well grounded in population genetics. Beta distributions have been used in studies of population differentiation as mixing processes to create randomness of allelic frequencies, leading to beta-binomial likelihoods or to unrecognized posterior distributions that must be resolved using sampling procedures, e.g., Holsinger (1999), Balding (2003), Beaumont & Balding (2004) and Gianola et al. (2010).

Similarly, it would seem sensible to assume that allelic frequencies of pairs of loci within the hypothetical population have an association that can be modelled with bivariate beta distributions to represent random variation of true, albeit unknown, allelic frequencies. For example, suppose that there are S half-sib families that can be viewed as drawn as random from some population and that we are interested in modeling association between pairs of allelic frequencies stemming from such sampling scheme. Here, distributions proposed by Sarmanov (1966) and Olkin & Liu (2003) are discussed and generalized as candidate processes that could be used to model this type of association.

Olkin-Liu bivariate beta distribution

First, we review how a specific bivariate beta distribution arises. As in Olkin & Liu (2003), let U, V and W be random variables following independent standard Gamma distributions with parameters a, b and c respectively. By construction, X=UU+W and Y=VV+W possess the beta distributions X: Beta(a,c) and Y: Beta(b,c) respectively. Clearly, X and Y are positively correlated (through W), with a strength that depends on the values of a, b, c. Olkin & Liu (2003) show that X and Y have a bivariate beta distribution with density function

p(x,y)=Γ(a+b+c)xa−1yb−1(1−x)b+c−1(1−y)a+c−1Γ(a)Γ(b)Γ(c)(1−xy)a+b+c,(14)

Where 0 < X < 1 and 0 < Y < 1. Moments E(X^kY^l) cannot be written in closed form but can be approximated numerically or using sampling methods. Large c and small a, b produce correlations close to 0, whereas large a, b or small c produce correlations close to 1 (Olkin & Liu 2003). For example, a correlation equal to 0.002 is obtained for a = b = 0.01 and c = 5, whereas the correlation is 0.91 for a = 2.5, b = 4 and c = 0.1. Figure 4 displays scatter plots of 5000 samples obtained from each of four bivariate beta distributions. Plot 1 represents a distribution in which the correlation is very low, and yet, there is considerable association between pairs of values near 0, illustrating inadequacy of correlation to reveal association. In plot 2 (resembling a ‘meteorite’) clustering takes place primarily at large values of X and Y. The two bottom plots depict bivariate beta distributions with similar correlations but with a completely different pattern of association. Clearly, correlation often fails as measure of statistical association.

In this bivariate beta model the density ratio takes the form

αxy=p(x,y)p(x)p(y)=Γ(a+b+c)Γ(c)Γ(a+c)Γ(b+c)(1−x)b(1−y)a(1−xy)a+b+c,(15)

Where p(x) and p(y) are the beta densities of X and Y: Beta(b,c). Then

DAI=log[Γ(a+b+c)Γ(c)Γ(a+c)Γ(b+c)]+Ep(x,y)[log(1−x)b(1−y)a(1−xy)a+b+c],

DIA=−log[Γ(a+b+c)Γ(c)Γ(a+c)Γ(b+c)]−Ep(x)p(y)[log(1−x)b(1−y)a(1−xy)a+b+c],

so that

−Ep(x)p(y){blog(1−x)+alog(1−y)−(a+b+c)log(1−xy)}.(16)

The corresponding D′_AI, D′_IA and KL′ can be calculated by taking expectations of density ratios as opposed to expectations of their logarithms.

Indexes of association based on D′_AI, D′_IA and KL′ were calculated for the following combinations of parameters, where ρ denotes the expected correlation: 1) a = 0.01, b = 0.01, c = 5 (ρ = 0.002); 2) a = 1, b = 0.5, c = 0.6 (ρ = 0.251), 3) a = 1, b = 1, c = 0.9 (ρ = 0.496) and 4) a = 2, b = 3, c = 0.4 (ρ = 0.750). Expectations under the independence model were approximated by drawing 1 million random numbers from independent X ∼ Beta(a,c) and Y ∼ Beta(b,c) distributions, and then averaging the evaluations of the appropriate expressions over the draws. In the association model, 1 million random triplets (U,V,W) were drawn from independent standard gamma variables with parameters a, b and c, to then form bivariate realizations as x=uu+w and y=vv+w. Table 3 gives the results, and all quantities (save for the true ρ_XY) are Monte Carlo estimates. The indexes of association θ′ and 1 − θ′ departed from 0.5 and approached 0 and 1, respectively, as the correlation in the bivariate beta distribution became stronger. Likewise, γ′ and γ′′ drifted away from 0, approaching |1|. In settings (3) and (4), θ′ and 1 − θ′ suggest a stronger association than what would be indicated by ρ.

Table 3. Departures of Olkin-Liu bivariate beta distributions from independence as measured by D′_AI, D′_IA and by the indexes of association θ′ and 1 − θ′ (when θ'=12 independence holds), for four combinations of parameters a, b and c. ρ^XY is the Monte Carlo estimate (1 million samples) of the true correlation ρ_XY. Means of variables X and Y are estimates from samples from association (A) or independence (I)models. The D′_AI and D′_IA parameters are estimates under the appropriate asumptions; KL′ = D′_AI + D′_IA; θ′ = D′_IA/(D′_AI + D′_IA); γ′ = 2θ′ − 1; γ′′ = 1 − 2q′

Item	a=0.01b=0.011.2pcc=5	a=0.1b=0.5c=0.6	a=1b=1c=0.9	a=2b=3c=0.4
ρXY	0.002	0.2510	0.4960	0.7500
ρ^XY	0.002	0.2499	0.4951	0.7304
ave. (X)	−.2pc12.01×10−3(A).5pc2.00×10−3(I)	−.15pc0.1429(A)−.25pc0.1431(I)	−.15pc0.5262(A)−.18pc0.5366(I)	−.15pc0.8335(A)−.3pc0.8333(I)
ave. (Y)	−.1pc1.98×10−3(A)−.2pc2.00×10−3(I)	−.15pc0.4548(A)−.2pc0.4542(I)	−.15pc0.5264(A)−.18pc0.5259(I)	−.15pc0.8825(A)−.3pc0.8823(I)
DAI'	0.9999	0.8709	0.4292	0.0745
DIA'	0.9999	0.8009	0.0779	1.944 × 10−5
KL′	1.9999	1.5781	0.5071	0.0745
θ′	0.5000	0.4481	0.1537	0.0003
1 − θ′	0.5000	0.5519	0.8463	0.9997
γ′	0.0000	−0.1037	−0.6926	−0.9995
γ′′	0.0000	0.1037	0.6926	0.9995

Olkin-Liu bivariate beta-binomial distribution

In studies of LD, focus is typically on correlation between alleles rather than between frequencies. We discuss how a correlation between alleles can arise when the association stems from frequencies, as could happen when the population consists of clusters of individuals resulting from some family structure. Consider bi-allelic locus l with allelic frequencies Pr(A_l)  = p_l and Pr(a_l) = 1 − p_l; A_l and a_l denote the two alleles at locus l. Suppose that a sample of N individuals is scored, and that the observed number of copies of A_l and a_l are nAl and nal, respectively, with nAl+nal=2N. Given p_l, the distribution of nAl is Binomial(2N,p_l). Now, let the allelic frequency p_l vary at random among clusters (e.g. sub-populations) according to a beta distribution with parameters (a,c). Then, the marginal distribution of the observed number of alleles is beta-binomial (e.g., Casella & George 1992; Sorensen & Gianola 2002), that is, the probability of observing nAl copies among the 2N alleles scored is

⪻(nAl|2N,a,c)=(2NnAl)Γ(a+c)Γ(a)Γ(c)Γ(nAl+a)Γ(nal+c)Γ(2N+a+c).(17)

Consider now a pair of loci, and let the observed number of copies of the four alleles be represented as DATA=(nA1,na1,nA2,na2); haplotype frequencies are not relevant in the model that follows. If, given the allelic frequencies p = {p_l}, the number of copies of A₁ and A₂ are independently distributed, i.e., the two loci are in linkage equilibrium, the distribution of the observed number of alleles is

l(DATA|p)=Πl=12(2NnAl)plnAl(1−pl)nal.(18)

Suppose now that this pair is a realization from a stochastic process where loci are sampled over a conceptual population of pairs, e.g. the two loci are drawn at random from a physical ‘bin’ of a certain length in kilobases. This process generates a covariance structure between allelic frequencies of pairs of loci within the ‘bin’. The alleles would be in LD, marginally, even though being in conditional (given the allelic frequencies) equilibrium. If the unobserved, ‘true’, allelic frequencies follow some bivariate distribution with density g(p₁,p₂|θ), where θ denotes its parameters, the density of the marginal distribution of the observed number of allelic counts under this association model would be

(19)

If a bivariate Olkin-Liu beta distribution with parameters θ = (a,b,c) is assumed, the joint density of allelic frequencies p₁ and p₂ as in (14). Letting

C(a,b,c)=Γ(a+b+c)Γ(a)Γ(b)Γ(c),

the marginal distribution (19) takes the form

pA(DATA)=[Πl=12(2NnAl)]C(a,b,c)

(20)

This integral cannot be written in closed form, but it can be evaluated using Monte Carlo methods such as importance sampling (e.g. Sorensen & Gianola 2002). One possible scheme is outlined in the Appendix.

Under this distribution

αnA1nA2=pA(DATA)pI(DATA),

Where p_I(DATA) is the distribution under independence between observed allele numbers, and given by the product of two beta-binomial distributions, that is

pI(DATA)=(2NnA1)Γ(a+c)Γ(a)Γ(c)Γ(nA1+a)Γ(na1+c)Γ(2N+a+c)

×(2NnA2)Γ(b+c)Γ(b)Γ(c)Γ(nA2+b)Γ(na2+c)Γ(2N+b+c).(21)

Then

αnA1nA2=Γ(a+b+c)Γ(c)Γ(a+c)Γ(b+c)Γ(2N+a+c)Γ(2N+b+c)Γ(2N+a+b+c)×

Γ(nA1+a)Γ(na1+b+c)Γ(nA1+a)Γ(na1+c)Γ(nA2+b)Γ(na2+c)EB1,B2[h(p1,p2)](22)

where EB1,B2[h(p1,p2)] is defined in the Appendix. Preceding the expectation, the first term involves the parameters of the bivariate beta distribution, the second involves sample size (N) as well, and the third one depends on observed allelic counts.

The logarithmic and relative distances between distributions are calculated as

DAI=∑nA1=02N∑nA2=02Nlog(αnA1nA2)pA(DATA),

DIA=−∑nA1=02N∑nA2=02Nlog(αnA1nA2)pI(DATA),

DAI'=∑nA1=02N∑nA2=02NαnA1nA2pA(DATA),

and

DIA'=∑nA1=02N∑nA2=02NαnA1nA2−1pI(DATA),

With θ and θ′ calculated as before. Calculations proceed on an entirely numerical basis.

A generalization of the beta process to a situation with more than two loci is in Olkin & Liu (2003). For K loci the joint density of the allelic frequencies takes the form

g(p1,p2,…,pK|θ)=Γ(a1+a2+…+aK)Πi=1KΓ(ai)Πi=1Kpiai−1(1−pi)ai+1   [1+∑i=1Kpi1−pi]−(a1+a2+…+aK).

The marginal distribution of allelic counts is obtained by mixing a multinomial distribution over the multivariate beta density above, but results are not available in closed form, so a numerical solution is needed again.

Lee-Sarmanov bivariate beta distribution

Lee (1996) proposed a different bivariate beta distribution following Sarmanov (1966); see Danaher & Hardie (2005). A Lee-Sarmanov bivariate beta distribution for allelic frequencies p₁ and p₂ has density

g(p1,p2|θ*)=B1(a1,b1)B2(a2,b2)[1+w(p1−μ1)(p2−μ2)],(23)

Where B₁(.) and B₂(.) are beta densities as before and w is a parameter (w = 0 produces independence); θ* = (a₁,b₁,a₂,b₂,w). Above, μ1=a1a1+b1 and μ2=a2a2+b2 are the expected values of p₁ and p₂ respectively. The marginal densities of p₁ and p₂ under this model are B₁(.) and B₂(.). Further,

E(p1p2)=μ1μ2+wVar(p1)Var(p2),

Cov(p1,p2)=wVar(p1)Var(p2),

so that

Corr(p1,p2)=wVar(p1)Var(p2),