Linkage disequilibrium (LD) is the term used in genetics to indicate association between the qualitative random variables corresponding to alleles at different polymorphic sites. Measuring the levels of linkage disequilibrium is important for gene mapping and increasing our understanding of genome architecture. The current literature documents agreement only on the definition of measures of LD for biallelic markers. Consider two markers, with alleles A, a and B, b. Their haplotype distribution can be synthetically described as:

π = B b A x p − x p a q − x 1 − p − q + x 1 − p q 1 − q 1 .       ( 1 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFapaCcqGH9aqpfaqabeabeqfaubqaaaqaaiabdkeacbqaaiabdkgaIbqaaaqaaiabdgeabbqaaiabdIha4bqaaiabdchaWjabgkHiTiabdIha4bqaaiabdchaWbqaaiabdggaHbqaaiabdghaXjabgkHiTiabdIha4bqaaiabigdaXiabgkHiTiabdchaWjabgkHiTiabdghaXjabgUcaRiabdIha4bqaaiabigdaXiabgkHiTiabdchaWbqaaaqaaiabdghaXbqaaiabigdaXiabgkHiTiabdghaXbqaaiabigdaXaaacqGGUaGlcaWLjaGaaCzcamaabmaabaGaeGymaedacaGLOaGaayzkaaaaaa@54F3@

Fixing the marginals p and q, the distribution π is completely identified by the probability x of the haplotype (A, B). The discrepancy of a generic π from the distribution under linkage equilibrium, can be quantified simply by D = (x - pq). Measures of LD are defined as the standardized values of D. Two common such measures are R2=(x−pq)2pq(1−p)(1−q) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGsbGudaahaaWcbeqaaiabikdaYaaakiabg2da9maalaaabaGaeiikaGIaemiEaGNaeyOeI0IaemiCaaNaemyCaeNaeiykaKYaaWbaaSqabeaacqaIYaGmaaaakeaacqWGWbaCcqWGXbqCcqGGOaakcqaIXaqmcqGHsislcqWGWbaCcqGGPaqkcqGGOaakcqaIXaqmcqGHsislcqWGXbqCcqGGPaqkaaaaaa@44F3@ and D′=(x−pq)Dmax MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGebargaqbaiabg2da9maalaaabaGaeiikaGIaemiEaGNaeyOeI0IaemiCaaNaemyCaeNaeiykaKcabaGaemiraqecbiGae8xBa0Mae8xyaeMae8hEaGhaaaaa@3B02@, where Dmax is min(p(1 - q), q(1 - p)) when the numerator is positive, and min(pq, (1 - p) (1 - q)) otherwise. The definition of R² can be understood by considering the alleles as realizations of quantitative random variables (with values 0 and 1), among which we calculate a correlation coefficient. The measure R² ranges between 0 and 1, and it is equal to 1 only when two entries of the table in (1) are equal to 0. The measure D' ranges, by definition, between -1 and 1, and its absolute value is equal to 1 whenever one entry of the table in (1) is equal to 0. There is a large literature discussing the choice of these measures (see, for example, 1). Typically, R² is preferred when the focus is on the predictability of one polymorphism given the other (and hence it is often used in power studies for association designs). D', instead, is the measure of choice to assess recombination patterns (haplotypes blocks have often been defined on the basis of D'). Despite their effectiveness, these measures suffer from two limitations: (a) they are not easily generalizable to multiallelic markers; (b) they are defined on the population haplotype distribution, and their performance can be rather unsatisfactory when applied to the empirical distribution derived from a finite sample.

With regard to point (a), it is clear that the definitions of R² and D' are based on properties of the joint distribution of two biallelic markers and their generalization is not immediate. A partial solution is to evaluate R² or D' on all 2 × 2 sub-tables obtainable from the joint distribution of two multiallelic markers and then summarize these results in one value. However, this measure is not easily interpretable and does not have good small sample properties (see the discussion in the following paragraphs).

Finally, let us remark how the problem of defining measures of disequilibrium generalizable to I × J tables remains actual, even if the current high density genotyping efforts are focused on biallelic markers as SNPs. Often we are interested in studying the relation between SNPs haplotypes at different loci: these can be considered as qualitative variables with multiple levels, just as multiallelic markers.

With regard to point (b), R² and D' are defined and studied assuming that the population haplotype distribution is known 2. In practice, this is rarely (if ever) the case: the sample haplotype frequencies provide an estimate of the population frequencies, and these estimates are used, following the plug-in principle, to obtain estimates of R² and D'. This approach encounters some difficulties in the case of D'. If a SNP has a low minor allele frequency, it is quite possible that the rare haplotype that carries it, is not observed in a small sample. This leads to a D' being equal to 1, irrespective of the level of linkage disequilibrium. Detailed analysis of this phenomena is available in 3 and 4. To avoid spurious results, researchers often calculate empirical confidence intervals for D' using resampling schemes (see, for example, 5). While this certainly takes care of the variability of D', it does not result in a "correct" measure of linkage disequilibrium and it clearly comes with a substantial computational cost. Moreover, the fact that the values of D' are inflated in the presence of rare alleles makes it difficult to obtain a multiallelic version of D' based on pooling statistics: as mentioned above, it is quite likely for a particular haplotype of multiallelic markers to have very low population frequency, resulting in zero observed counts. Volume measures 678 both take effectively into account the variability due to sample size and are immediately applicable to multiallelic markers. Let us first recall the main idea of volume measures. For distributions like (1), volume measures can be described as a different strategy for normalizing D with reference to the class C MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFce=qaaa@3825@ of distributions with the same marginals p and q. Rather than dividing the observed D by its maximum value over distributions in C MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFce=qaaa@3825@, one evaluates the proportion of distributions in C MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFce=qaaa@3825@ that have a smaller difference from the distribution under independence than D. In general, we can consider any quantification of the discrepancy between a generic distribution π and the distribution under independence. When π is known, a volume measure is defined as the probability that a distribution selected uniformly among all possible ones with the same margins as π results in a lower discrepancy from equilibrium. If, however, the population haplotype distribution π is unknown, and a sample of size n is available, volume measures are defined directly on the contingency table summarizing the data, avoiding spurious effects due to the sample size. To clarify this point, let us consider again the case of two biallelic markers. If the population distribution (1) is known, we define Dvol, the volume measure equivalent to D', as the ratio of two volumes V₁/V₂. V₁ is the volume of the space of all distributions with marginals equal to p and q, and Pr(0, 0) = z such that (x - pq) (z - pq) ≥ 0 and |x - pq| > |z - pq|. V₂ is the volume of the space of distributions that satisfy all but the last one of the constraints for V₁. A simple geometric argument shows that Dvol = D'. (See Additional file 1 for a graphical illustration). Suppose now, instead, that we do not know the population distribution (1), but we obtain a sample of size n from it, leading to the contingency table F

F = B b A f 11 f 12 r 1 a f 21 f 22 r 2 c 1 c 2 n , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGgbGrcqGH9aqpfaqabeabeqfaubqaaaqaaiabdkeacbqaaiabdkgaIbqaaaqaaiabdgeabbqaaiabdAgaMnaaBaaaleaacqaIXaqmcqaIXaqmaeqaaaGcbaGaemOzay2aaSbaaSqaaiabigdaXiabikdaYaqabaaakeaacqWGYbGCdaWgaaWcbaGaeGymaedabeaaaOqaaiabdggaHbqaaiabdAgaMnaaBaaaleaacqaIYaGmcqaIXaqmaeqaaaGcbaGaemOzay2aaSbaaSqaaiabikdaYiabikdaYaqabaaakeaacqWGYbGCdaWgaaWcbaGaeGOmaidabeaaaOqaaaqaaiabdogaJnaaBaaaleaacqaIXaqmaeqaaaGcbaGaem4yam2aaSbaaSqaaiabikdaYaqabaaakeaacqWGUbGBaaGaeiilaWcaaa@4E4C@

with row sums equal to r₁, r₂ and column sums c₁, c₂. Consider now the set Ω of tables T, with row and column sums r = (r₁, r₂) and c = (c₁, c₂). We define Dvol(F) as the fraction of contingency tables T in Ω that lead to a value |t₁₁ - c₁r₁/n| smaller than |f₁₁ - c₁r₁/n| among those for which (t₁₁ - r₁c₁/n) (f₁₁ - r₁c₁/n) > 0.

This Dvol value will be different from D', calculated by treating f_ij/n as population frequencies. For example, the fact that one table entry is equal to zero will not be sufficient to guarantee Dvol = 1. To make this point clearer, however, it is appropriate to consider a more general and precise definition. In the remainder of the paper, we will only concern ourselves with volume measures defined on haplotype sample frequencies.

Consider the table of observed haplotypes counts F:

F = B 1 B 2 ⋯ B J A 1 f 11 f 12 ⋯ f 1 J r 1 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ A I f I 1 f I 2 ⋯ f I J r I c 1 c 2 ⋯ c J n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@71F1@

where B_i represent alleles at marker B and A_i alleles at marker A. Let Ω be the set of all tables T with row and column sums equal to r₁, ..., r_I and c₁, ..., c_J, respectively. Given a criterion to quantify the discrepancy between F and the table expected under independence (linkage equilibrium), a volume measure is defined as the proportion of tables T ∈ Ω that lead to a smaller discrepancy value. If the recorded discrepancy is the biggest possible, then the volume measure will have value close to 1 (the exact value 1 will be attained as the sample size increases to ∞). Conversely, if all other tables lead to larger discrepancies, the volume measure will be zero.

One may notice that this definition of volume measure is similar to one minus the p-value for a test of independence. Indeed, volume measures are related to the "volume test," an original notion introduced by Hotelling 8, and the effect of sample size on the measures is very much the same as its effect on a p-value. The key difference between volume measures and variants of the commonly used Fisher's exact test for independence is that in the case of volume measures, the relevant proportion of tables is evaluated assuming that all tables with the same margins are equally probable, while in the case of Fisher's exact test tables are generated under the hypothesis of independence. Because of this, volume measures and Fisher's exact tests answer two very different questions: the first compares the observed table to all tables with the same margins, while the second one assesses the likelihood of the observed table under independence. A thorough discussion of the different interpretations and uses of these two approaches can be found in 7. In order to concretely evaluate volume measures, one has to choose a criterion for discrepancy and be able to explore the space of tables with fixed margins to evaluate the required proportions. We start illustrating the first point by focusing on three specific measures: a) Dvol, which is defined only on 2 × 2 tables and coincides with D' when the population haplotype distribution is known; b) Mvol, which is a generalization of Dvol to multiallelic markers; c) Hvol, which is based on expected homozygosity and captures information that is close to the one described by R², although it can be defined on tables with any number of entries.

When I = J = 2, let Ω₁ = {T : t_i+ = r_i, t_+j = c_j, (t₁₁ - r₁c₁/n)(f₁₁ - r₁c₁/n) > 0}. We then define Dvol as

D v o l ( F ) = 1 | Ω 1 | ∑ T ∈ Ω 1 1 { M ( T ) < M ( F ) } , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGebarcqWG2bGDcqWGVbWBcqWGSbaBcqGGOaakcqWGgbGrcqGGPaqkcqGH9aqpdaWcaaqaaiabigdaXaqaaiabcYha8jabfM6axnaaBaaaleaacqaIXaqmaeqaaOGaeiiFaWhaamaaqafabaGaeGymaeZaaSbaaSqaaiabcUha7jabd2eanjabcIcaOiabdsfaujabcMcaPiabgYda8iabd2eanjabcIcaOiabdAeagjabcMcaPiabc2ha9bqabaaabaGaemivaqLaeyicI4SaeuyQdC1aaSbaaWqaaiabigdaXaqabaaaleqaniabggHiLdGccqGGSaalaaa@51F8@

where M(T)=∑i,j(tij−ricj/n)2ricj/n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGnbqtcqGGOaakcqWGubavcqGGPaqkcqGH9aqpdaaeqaqaamaalaaabaGaeiikaGIaemiDaq3aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGHsislcqWGYbGCdaWgaaWcbaGaemyAaKgabeaakiabdogaJnaaBaaaleaacqWGQbGAaeqaaOGaei4la8IaemOBa4MaeiykaKYaaWbaaSqabeaacqaIYaGmaaaakeaacqWGYbGCdaWgaaWcbaGaemyAaKgabeaakiabdogaJnaaBaaaleaacqWGQbGAaeqaaOGaei4la8IaemOBa4gaaaWcbaGaemyAaKMaeiilaWIaemOAaOgabeqdcqGHris5aaaa@4FC0@.

For general I × J tables, recall that Ω denotes the set of all contingency tables with the same row and column sums as F: Ω = {T : t_i+ = r_i, t_+j = c_j}. Then, we define

M v o l ( F ) = 1 | Ω | ∑ T ∈ Ω 1 { M ( T ) < M ( F ) } . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGnbqtcqWG2bGDcqWGVbWBcqWGSbaBcqGGOaakcqWGgbGrcqGGPaqkcqGH9aqpdaWcaaqaaiabigdaXaqaaiabcYha8jabfM6axjabcYha8baadaaeqbqaaiabigdaXmaaBaaaleaacqGG7bWEcqWGnbqtcqGGOaakcqWGubavcqGGPaqkcqGH8aapcqWGnbqtcqGGOaakcqWGgbGrcqGGPaqkcqGG9bqFaeqaaaqaaiabdsfaujabgIGiolabfM6axbqab0GaeyyeIuoakiabc6caUaaa@4FC0@

The definition above should clarify how Mvol is closely related to Dvol, and the difference between the two is that Mvol does not consider the "sign" of the association, a notion that is undefined in generic I × J tables.

Letting H(T)=∑i,jtij2−∑iri2∑jcj2/n2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGibascqGGOaakcqWGubavcqGGPaqkcqGH9aqpdaaeqaqaaiabdsha0naaDaaaleaacqWGPbqAcqWGQbGAaeaacqaIYaGmaaaabaGaemyAaKMaeiilaWIaemOAaOgabeqdcqGHris5aOGaeyOeI0YaaabeaeaacqWGYbGCdaqhaaWcbaGaemyAaKgabaGaeGOmaidaaaqaaiabdMgaPbqab0GaeyyeIuoakmaaqababaGaem4yam2aa0baaSqaaiabdQgaQbqaaiabikdaYaaaaeaacqWGQbGAaeqaniabggHiLdGccqGGVaWlcqWGUbGBdaahaaWcbeqaaiabikdaYaaaaaa@4EF5@, we can define the measure Hvol:

H v o l ( F ) = sign ( H ( F ) ) ∑ T ∈ Ω 1 { | H ( T ) | < | H ( F ) | } 1 { H ( T ) H ( F ) ≥ 0 } ∑ T ∈ Ω 1 { H ( T ) H ( F ) ≥ 0 } . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@7E7B@

We have mentioned how Hvol captures information closely related to that of R². A careful discussion of the interpretation of LD measures based on homozygosity can be found in 9. Here it suffices to recall that joint homozygosity relates to a measure of agreement between the two markers and excess in homozygosity indicates that knowledge of the allele value at one marker increases predictive accuracy of the allele values at the other marker. The results of a recent empirical study conducted using homozygosity-based measures are documented in 10.

Note that all the above definitions use the strict inequality sign. The choice of this over ≤ is irrelevant for large n, but it makes a difference in the case of small n, where strict inequality allows us to better discriminate against apparent association due to small samples.

To evaluate these measures, we need to explore the space of all tables with the same margins. In the case of I = J = 2, this can be done by simple enumeration. For multiallelic tables enumeration is impractical. An obvious alternative is to restrict one's attention to a sample of possible tables. However, obtaining a sample of tables according to the uniform distribution among all tables with fixed margin (as opposed to according to the Fisher-Yates distribution) is not easy. It is indeed the computational difficulty associated with volume tests 7 and measures 6 that has substantially hindered their wide-spread application. Previous solutions have been proposed with Markov chain Monte Carlo algorithms in 11, as well as rejection sampling (see 12 for a review). The main contribution of this paper is that we have successfully implemented a sequential importance sampling (SIS) algorithm, originally introduced in 12, to evaluate volume measures accurately and in a timely manner. This implementation makes volume measures applicable to high throughput analysis.

To enumerate all tables in Ω₁ (I = J = 2), it is useful to notice that t₁₁ must satisfy

max(0, r₁ + c₁ - n) ≤ t₁₁ ≤ min(r₁, c₁),     (2)

and after t₁₁ is chosen, we can fill in other entries of the 2 × 2 table by the marginal sum constraints. Therefore we can enumerate tables in Ω₁ by assigning all possible integers satisfying (2) to t₁₁, and keeping those tables such that (t₁₁ - r₁c₁/n) has the same sign as in F.

We now consider the SIS procedure for I × J tables. Let u(T) be the uniform distribution over all tables in Ω. Then Mvol(F) can be treated as the expectation of the indicator function 1_{m(T)<m(F)} with respect to u(T). It is hard to sample directly from u(T). The idea of importance sampling is to sample tables from another proposal distribution g(T), and then estimate Mvol(F) by

∑ ℓ = 1 L 1 { M ( T ℓ ) < M ( F ) } u ( T ℓ ) g ( T ℓ ) ∑ ℓ = 1 L u ( T ℓ ) g ( T ℓ ) = ∑ ℓ = 1 L 1 { M ( T ℓ ) < M ( F ) } 1 g ( T ℓ ) ∑ ℓ = 1 L 1 g ( T ℓ ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@8926@

where T₁, ..., T_L are L independent and identically distributed (i.i.d.) samples from g(T). SIS generates a table cell by cell by decomposing the proposal distribution g(T) as

g(T)= g(t₁₁)g(t₂₁|t₁₁) ... g(t_IJ|t_I-1, _J, ..., t₁₁).

Notice that the support for the first entry t₁₁ is max(0, r₁ + c₁ - n) ≤ t₁₁ ≤ min(r₁, c₁). We sample an integer uniformly from the above range for in, i.e., g(t₁₁) is the uniform distribution on the support of t₁₁.

Recursively, suppose we have chosen t_i1 = ti1∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG0baDdaqhaaWcbaGaemyAaKMaeGymaedabaGaey4fIOcaaaaa@3184@ for i = 1, ..., k - 1. Then the support for t_k1 is max⁡(0,(t.1−∑i=1k−1ti1∗)−∑i=k+1Iri)≤tk1≤min⁡(rk,c1−∑i=1k−1ti1∗) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@75ED@. We sample an integer uniformly from the above range for t_k1. The procedure is continued until all the entries in the first column have been considered. Then we update the row sums by subtracting the realization of the first column from the original row sum, and sample the second column of the table in the same way.

The computing time and precision of the algorithm are different for 2 × 2 or larger size tables. For 2 × 2 tables, our algorithm simply lists all possible tables with fixed margins. CPU time is then proportional to the total number of tables: usually their enumeration takes a fraction of a second. The algorithm is exact and we do not have approximation errors in the output. For the general case of I × J tables, the CPU time depends on the number of generated uniform random variables: I × J × L for L Monte Carlo samples. It is important to keep in mind that the output of the algorithm is not exact, but an estimate of the true volumes ratio (so different runs will give slightly different results). The precision of the final estimate depends on the number L of Monte Carlo samples and how well the proposal distribution in the SIS algorithm approximates the target distribution for a given table. Indeed, the value of the parameter L has to be specified by the user. It is advisable to conduct multiple trial runs to estimate the precision of the estimate and select a value of L that assures an acceptable precision.

We now illustrate the performance of our algorithm and the relevance of volume measures with three examples.

We describe a novel implementation of a sequential importance sampling algorithm to evaluate volume measures of linkage disequilibrium. We focus on three measures. Dvol corresponds conceptually to D', but we show that Dvol is not inflated for small sample size. Mvol represents a generalization of Dvol that can be evaluated on generic I × J tables. Hvol is based on expected homozygosity and measures agreement between markers, so that it captures information similar to that of R². However, unlike R², Hvol can be evaluated on generic I × J tables.

The source code for evaluating the volume measures described in this paper is available at the following url: http://www.stat.uiuc.edu/~yuguo/software/volume/.

It is a C program that can be compiled with gcc and requires the libraries math.h, stdlib.h, malloc.h, time.h, and stdio.h.

YC has been responsible mainly for the algorithm and code development. CL has contributed mainly to data analysis. CS has supervised the project.