Cubic exact solutions for the estimation of pairwise haplotype frequencies: implications for linkage disequilibrium analyses and a web tool 'CubeX'
 Tom R Gaunt^{1}Email author,
 Santiago Rodríguez^{1} and
 Ian NM Day^{1}
DOI: 10.1186/147121058428
© Gaunt et al; licensee BioMed Central Ltd. 2007
Received: 05 February 2007
Accepted: 02 November 2007
Published: 02 November 2007
Abstract
Background
The frequency of a haplotype comprising one allele at each of two loci can be expressed as a cubic equation (the 'Hill equation'), the solution of which gives that frequency. Most haplotype and linkage disequilibrium analysis programs use iterationbased algorithms which substitute an estimate of haplotype frequency into the equation, producing a new estimate which is repeatedly fed back into the equation until the values converge to a maximum likelihood estimate (expectationmaximisation).
Results
We present a program, "CubeX", which calculates the biologically possible exact solution(s) and provides estimated haplotype frequencies, D', r^{2} and χ^{2} values for each. CubeX provides a "complete" analysis of haplotype frequencies and linkage disequilibrium for a pair of biallelic markers under situations where sampling variation and genotyping errors distort sample HardyWeinberg equilibrium, potentially causing more than one biologically possible solution. We also present an analysis of simulations and real data using the algebraically exact solution, which indicates that under perfect sample HardyWeinberg equilibrium there is only one biologically possible solution, but that under other conditions there may be more.
Conclusion
Our analyses demonstrate that lower allele frequencies, lower sample numbers, population stratification and a possible D' value of 1 are particularly susceptible to distortion of sample HardyWeinberg equilibrium, which has significant implications for calculation of linkage disequilibrium in small sample sizes (eg HapMap) and rarer alleles (eg paucimorphisms, q < 0.05) that may have particular disease relevance and require improved approaches for meaningful evaluation.
Background
Linkage disequilibrium (LD) describes the condition that occurs when alleles at different loci are nonrandomly associated in a given population. Under LD the frequency (f_{11}) of a haplotype (h_{11}) representing the "1" allele at two loci is significantly more or less than the product of the respective allele frequencies. Characterisation of LD is important in medical genetics, influencing association mapping of trait loci and providing information on interactions between genes [1, 2]. LD is the result of a shared history of mutation and recombination, and other factors including: genetic drift, population growth, admixture, population structure, the ages of the polymorphisms, the physical distance separating them and the effects of selective pressure [3].
that is adapted from Hill's equation (4 [4] with the constants defined under Methods. With ${\widehat{f}}_{11}$ and the allele frequencies, all four haplotype frequencies can be calculated, thus estimating the unknown proportions of the middle cell.
Several approaches exist for solving equation (1), the solution of which enables estimation of haplotype frequencies and LD coefficients. The first approach uses iterationbased algorithms. An initial estimate of haplotype frequency (either random, or based on the known haplotype numbers) is substituted into the equation, providing a new estimate. This is then fed back into the equation and the expectationmaximisation (EM) process repeated until the values converge. This is the basis both of the algorithm described by Hill in 1974 for the estimation of pairwise haplotype frequencies [4], and of other EM algorithms that enable the estimation of multilocus haplotype frequencies. Many programs exist that utilise variations on this approach, including: GOLD [5], GOLDsurfer [6], MIDAS [7], Haploview [8] and many others reviewed in [9–12]. The potential problem for these approaches is that algorithms may converge on one of the alternative roots of the cubic equation (a local maximum rather than the global maximum).
Other approaches include parsimony, eg HAPAR [13] and Bayesian algorithms, eg PHASE [14–16]. Parsimony and Bayesian methods are both better suited to estimating individual haplotypes than EM approaches, while Bayesian and EM methods are useful for estimating population frequencies [11].
An alternative approach would be exact solution, such as Cardan's solution [17] of the generalized cubic equation (of which equation (1) is an example). This provides all roots to the cubic equation, from which we can select those that are both real (i.e. not a complex number) and biologically possible. If more than one solution exists then the likelihoods of the different solutions can be compared and an informed evaluation made of the result. Theoretically, the noniterative approach may be computationally less intensive and more accurate, but computational efficiency and accuracy will be software and platform dependent.
Implementation
Hill assumed random mating and Hardy Weinberg Equilibrium (HWE) [4]. Rearranging terms for consequent diplotype frequency expectations for two biallelic loci Luo and Suhai [18] obtained equation 1 given in the introduction (here redefining ${\widehat{f}}_{11}$ as x, a3 as a, a2 as b, a1 as c and a as d for convenience): ax^{3} + bx^{2} + cx + d = 0, where a = 4n; b = 2n (1  2p  2q)  2(2n_{11} + n_{12} + n_{21})  n_{22}; c = 2npq  (2n_{11} + n_{12} + n_{21})(1  2p  2q)  n_{22}(1  p  q); d = (2n_{11} + n_{12} + n_{21})pq; n = number of subjects; p = common allele freq of locus 1; q = common allele freq of locus 2; n_{11} is the number of subjects who are homozygous for the commoner allele at both loci; n_{12} are common homozygous at locus 1 and heterozygous at locus 2; n_{21} are heterozygous at locus 1 and common homozygous at locus 2; n_{22} are heterozygous at both loci [18]. Equation 1 can be solved exactly for x (with 1 to 3 real number solutions).
We have adopted the Nickalls treatment of the Cardan solution of the generalized cubic equation [17], and written a Python [19] program "CubeX" to solve equation 1 exactly. In CubeX, after calculation of constants ad from diplotypic data the following are calculated:
x_{ N }= b/(3a); δ^{2} = (b^{2} 3ac)/9a^{2}; h^{2} = 4a^{2}δ^{6}; y_{ N }= ax_{ N }^{3} + bx_{ N }^{2} + cx_{ N }+ d.
The discriminant Δ_{3} = y_{ N }^{2}  h^{2} is then used to determine the outcome in real roots (without having to go through complex number intermediates or ambiguities), with three possible outcomes:
Outcome 2: if y_{ N }^{2} = h^{2} there are three real roots (α, β and γ) and α and β are equal. For a value of $\mu =\sqrt[3]{\frac{{y}_{N}}{2a}}$:
α = x_{ N }+ μ
β = x_{ N }+ μ
γ = x_{ N } 2μ
Outcome 3: if y_{ N }^{2} <h^{2} there are three real roots (α, β and γ). Where $\theta =\frac{\mathrm{arccos}\phantom{\rule{0.25em}{0ex}}({y}_{N}/h)}{3}$:
α = x_{ N }+ 2δ cosθ
β = x_{ N }+ 2δ cos(2π/3 + θ)
γ = x_{ N }+ 2δ cos(4π/3 + θ)
D_{max} = min [p(1q),(1p)q] if D > 0 or D_{max} = min [pq, (1p)(1q)] if D < 0
D' = D/D_{max}
r^{2} = D^{2}/(p(1p)q(1q))
Diplotype frequencies based on the estimated haplotype frequencies are compared to the input diplotype frequencies by a χ^{2} test, which effectively tests sample deviation from the null hypothesis of HWE for the diplotypes formed of the four haplotypes. The number of degrees of freedom is equal to the number of observations (diplotype counts) minus four estimated parameters which are either three haplotypes (the fourth can be inferred) and D, or one haplotype, two allele frequencies and D. If nine different diplotypes are observed the number of degrees of freedom is therefore five. For each empty cell in the 3 × 3 the number of degrees of freedom is reduced by one. If the user knows there are only three haplotypes present (and therefore six diplotypes) then there are only three estimated parameters (D is inferred by the three haplotype frequencies) and 3 df. It is important to note that in the latter case neither cubic solution nor iteration is necessary as the haplotype frequencies can be directly counted from the diplotype data. If the user believes that there are only three alleles and hence six diplotypes, but there are nonzero values for any of the other three possible diplotypes, then reconsideration of the technical veracity of the data and of the homogeneity of the population sample would be wise.
Results
Solutions are considered biologically possible when ${\widehat{f}}_{11}$ and the derived ${\widehat{f}}_{12}$, ${\widehat{f}}_{21}$ and ${\widehat{f}}_{22}$ all fall within the range 0 to 1 (i.e. ${\widehat{f}}_{11},{\widehat{f}}_{12},{\widehat{f}}_{21},{\widehat{f}}_{22}\in [0,1]$) and add up to 1. This constraint is tighter than those described elsewhere [22] as it relies on the inherent assumption of representative sampling and HWE, an extreme chance distortion of which could lead to three solutions at SNP allele frequencies of 0.5 in sample data drawn from a population (if all samples are heterozygous at both loci the following are possible: all could be diplotype 11/22, all could be diplotype 12/21, or there could be a combination of both).
Number of solutions to the cubic equation with simulated data
Number of solutions to the cubic equation with real data
Comparison of the cubic exact solution with other approaches
For the purposes of comparison we have analysed two datasets with PHASE [16], MIDAS [7] (Hill EM) and CubeX. The first is a dataset of directly haplotyped samples comprising 80 subjects from 3 ethnic groups (Asian, African and Caucasian) for APOE [25]. Although all but one SNP was in HardyWeinberg equilibrium, this dataset has the potential to invalidate some of the assumptions of the programs due to the mixture of ethnicity. However, this provides a useful substrate on which to test the influence of stratification on the outcome of the cubic exact solution. The second dataset is a set of multilocus phased data from HapMap CEU samples [23, 24] for the IGF2 gene region. Although these have not been directly haplotyped, the multilocus phased haplotypes are expected to be very accurate, and this dataset comprises Caucasians, so will not suffer from the same stratification issues. We tested the programs on pairwise subsets of these data.
Illustrative examples of comparison of CubeX with PHASE [16] and MIDAS [7] (Hill EM).
Haplotype frequencies (rounded to 5 decimal places)  Haplotype numbers (rounded to nearest haplotype)  

Example  SNP pair  Haplotype  REAL frequency  PHASE frequency  MIDAS frequency  CUBEX alpha  CUBEX beta  CUBEX gamma  REAL number  PHASE number  MIDAS number  CUBEX alpha  CUBEX beta  CUBEX gamma 
1  Pair1_2  AC  0.0875  0.08689  0.0875  na  0.0875  na  14  14  14  na  14  na 
Pair1_2  AT  0.725  0.72561  0.725  na  0.725  na  116  116  116  na  116  na  
Pair1_2  TC  0  0.00061  0  na  0  na  0  0  0  na  0  na  
Pair1_2  TT  0.1875  0.18689  0.1875  na  0.1875  na  30  30  30  na  30  na  
2  Pair1_5  AG  0.75  0.75318  0.75478  0.75478  na  0.75  120  121 *  121 *  121 *  na  120 
Pair1_5  AA  0.0625  0.05932  0.05772  0.05772  na  0.0625  10  9 *  9 *  9 *  na  10  
Pair1_5  TG  0.1875  0.18432  0.18272  0.18272  na  0.1875  30  29 *  29 *  29 *  na  30  
Pair1_5  TA  0  0.00318  0.00478  0.00478  na  0  0  1 *  1 *  1 *  na  0  
3  Pair1_9  AT  0.05625  0.06477  0.05633  na  0.0563  0.075  9  10 *  9  na  9  12 * 
Pair1_9  AC  0.75625  0.74773  0.75617  na  0.7562  0.7375  121  120 *  121  na  121  118 *  
Pair1_9  TT  0.01875  0.01023  0.01867  na  0.0187  0  3  2 *  3  na  3  0 *  
Pair1_9  TC  0.16875  0.17727  0.16883  na  0.1688  0.1875  27  28 *  27  na  27  30 *  
4  Pair2_3  CG  0.05625  0.04724  0.0465  0.0465  na  na  9  8 *  7 *  7 *  na  na 
Pair2_3  CT  0.03125  0.04026  0.041  0.041  na  na  5  6 *  7 *  7 *  na  na  
Pair2_3  TG  0.48125  0.49026  0.491  0.491  na  na  77  78 *  79 *  79 *  na  na  
Pair2_3  TT  0.43125  0.42224  0.4215  0.4215  na  na  69  68 *  67 *  67 *  na  na  
5  pair5_9  GT  0.075  0.07313  0.06664  na  0.0666  0.075  12  12  11 *  na  11 *  12 
pair5_9  GC  0.8625  0.86437  0.87086  na  0.8709  0.8625  138  138  139 *  na  139 *  138  
pair5_9  AT  0  0.00187  0.00836  na  0.0084  0  0  0  1 *  na  1 *  0  
pair5_9  AC  0.0625  0.06063  0.05414  na  0.0541  0.0625  10  10  9 *  na  9 *  10 
For the HapMap [23, 24] IGF2 region data (comprising SNPs rs3802971, rs734351, rs3213221, rs4244808, rs1003483, rs3741208, rs1004446, rs4320932 and rs7924316) CubeX gives only one solution in all cases, and there is little difference between the outcome of the three approaches (Additional File 2). This confirms that in situations of higher allele frequencies there is less of an issue with multiple biologically possible solutions to the cubic equation, and iterative approaches are completely acceptable.
Discussion
We have written an online program, "CubeX", to enable simple analysis of the biologically possible estimated haplotypes for pairs of biallelic markers. This program takes data from a pair of markers as a standard 3 × 3 table of nine diplotypes, generates cubic exact solutions to equation 1 and generates output in the format shown in Figure 3. The number of possible solutions is shown, followed by haplotype frequencies and LD statistics for those solutions. Below that a duplicate of the 3 × 3 input table is displayed with the addition of expected absolute diplotype frequencies calculated from the haplotype frequencies. The difference between these and the input data are subjected to a χ^{2} test, which effectively tests sample deviation from the null hypothesis of HWE for the diplotypes formed of the four haplotypes. However, the interpretation of solutions depends on the prior hypothesis. In the example in Figure 3, although solution γ exhibits a slightly worse χ^{2} fit than solution β, the former is consistent with a prior hypothesis of only three of the four haplotypes existing (see Figure 5 in reference [7]), which is biologically likely in the absence of recombination between any two loci. In fact, in all tested cases in Figure 2 generating more than one solution, the diplotype data included zero values in at least one corner cell and the two adjacent edge cells of the 3 × 3 (i.e. where one possible solution has a D' = 1, although it should be noted that more than one solution can occur without zero values if double heterozygotes are greatly overrepresented). This suggests that the principal issue is whether three or four haplotypes exist, and in these cases the prior hypothesis (based on distance and recombination rates) is of utmost importance. If input data for individual SNPs are significantly out of HWE a warning message is given at the top of the page. For completeness, the biologically impossible real number solutions are displayed at the bottom, along with minimum and maximum biologically possible values for ${\widehat{f}}_{11}$ and allele frequencies. This program provides a convenient utility for researchers to both analyse data for haplotype frequencies and LD statistics and to check previous analyses for potential problems caused by multiple solutions.
Under perfect sample HWE the frequencies of all haplotypes can be directly inferred from the corresponding corner diplotypes of the 3 × 3. For example: ${n}_{11}=\text{n}{\widehat{f}}_{11}^{2}$, so ${\widehat{f}}_{11}=\sqrt{\frac{{n}_{11}}{n}}$. That being the case there are only two possible values for ${\widehat{f}}_{11}$, one positive and one negative, the latter being biologically impossible. Perfect sample HWE therefore results in only a single biologically possible solution to the cubic equation. In the case of extreme sample HWD where all samples fall within the middle cell of the 3 × 3, ${\widehat{f}}_{11}$ can contribute either a half, a quarter or none of the haplotypes to the middle cell. There are therefore three biologically possible solutions under conditions of extreme sample HWD. The results from real data confirm that in some cases more than one biologically possible solution to the cubic equation for haplotype frequency can exist. The simulations suggest that this occurs where small sample size, sampling errors or nonrandom mating result in a distortion of sample HWE, and demonstrates the importance of testing HWE before haplotype analyses. The greater the distortion of sample HWE the higher the allele frequency at which more than one solution can occur (hence, as described above, three solutions can occur at allele frequencies of 0.5 if all samples are heterozygous at both loci). In these cases the cubic exact algorithm gives all possible solutions and a test of HWE, while an iterationbased method would only give one. This supports the hypothesis that the cubic exact approach is superior to iterationbased methods in realworld datasets where sample data rarely fit exactly to HWE (note that sample may differ from population in HWE statistics – here we refer to sample HWE). This is particularly important in the analysis of low frequency SNPs and paucimorphisms [26–28], for which different solutions can significantly distort D' results, despite the relatively similar solutions giving similar r^{2} results. In all the observed data with two solutions there were no occasions in which r^{2} exceeded 0.3 for any biologically possible solution, and in most cases there is only a small difference in r^{2} between biologically possible solutions. The largest effect is on D'. On the basis of empirical data and using different approaches to inference Wong et al showed that coding SNPs with minor allele frequencies <0.06 are likely to be of functional importance [29], and rarer alleles, haplotypes and diplotypes of causal importance have emerged in numerous disease contexts (eg. inflammatory bowel disease, hemochromatosis). In addition to being applicable and giving exact evaluation for D' analysis of common SNPs, the cubic exact solution may prove of particular value for evaluating "postHapMap" and "postdbSNP" rarer haplotypes, for fully evaluating D' estimates from datasets with greater deviations from the random mating and HWE assumptions and for fully evaluating LD in small datasets.
Finally, we have demonstrated by comparison with PHASE [16] and MIDAS [7] (Hill EM) that in certain situations (low minor allele frequency, population stratification) the cubic exact approach can perform better for pairwise analyses than alternative approaches by indicating the existence of multiple solutions. However, our findings confirm that in most other situations iterative approaches are robust and accurate.
Conclusion
We present a comprehensive analysis of the consequences of different variables on the number of solutions to the cubic equation for haplotype frequency. Our analyses demonstrate that lower allele frequencies, lower sample numbers and a possible D' value of 1 can result in more than one solution. This has significant implications for the calculation of LD in small sample sizes and with rarer alleles that may have particular disease relevance. This evaluation provides essential information for an understanding of the limitations of LD estimation, which is particularly relevant for genomewide analyses (where sample sizes and allele frequencies can be low). Finally, we present a program "CubeX", freely available as an online program, which provides each of the biologically possible cubic exact solution(s) to equation 1 for haplotype frequency, enabling the user to identify the solution that best fits their prior hypothesis for number of haplotypes.
Availability and Requirements
Project name: CubeX
Project home page: http://www.oege.org/software/cubex
Operating system(s): Platform independent (webbased)
Programming language: Python http://www.python.org
Licence: CubeX licence available from http://www.oege.org/software/cubex
Any restrictions to use by nonacademics: royaltyfree use allowed within terms of licence
Abbreviations
 EM:

ExpectationMaximisation
 HWE:

HardyWeinberg Equilibrium
 LD:

Linkage Disequilibrium
Declarations
Acknowledgements
TRG is funded by a BHF (British Heart Foundation) Intermediate Fellowship (FS/05/065/19497), SR by a HOPE (Wessex Medical Trust) fellowship and work in our laboratory by the Medical Research Council (UK) (Programme Grant G9800748). We thank an anonymous reviewer for their suggestion of a comparison with PHASE on the APOE dataset [25].
Authors’ Affiliations
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