Computing the minimum recombinant haplotype configuration from incomplete genotype data on a pedigree by integer linear programming.

We study the problem of reconstructing haplotype configurations from genotypes on pedigree data with missing alleles under the Mendelian law of inheritance and the minimum-recombination principle, which is important for the construction of haplotype maps and genetic linkage/association analyses. Our previous results show that the problem of finding a minimum-recombinant haplotype configuration (MRHC) is in general NP-hard. This paper presents an effective integer linear programming (ILP) formulation of the MRHC problem with missing data and a branch-and-bound strategy that utilizes a partial order relationship and some other special relationships among variables to decide the branching order. Nontrivial lower and upper bounds on the optimal number of recombinants are introduced at each branching node to effectively prune the search tree. When multiple solutions exist, a best haplotype configuration is selected based on a maximum likelihood approach. The paper also shows for the first time how to incorporate marker interval distance into a rule-based haplotyping algorithm. Our results on simulated data show that the algorithm could recover haplotypes with 50 loci from a pedigree of size 29 in seconds on a Pentium IV computer. Its accuracy is more than 99.8% for data with no missing alleles and 98.3% for data with 20% missing alleles in terms of correctly recovered phase information at each marker locus. A comparison with a statistical approach SimWalk2 on simulated data shows that the ILP algorithm runs much faster than SimWalk2 and reports better or comparable haplotypes on average than the first and second runs of SimWalk2. As an application of the algorithm to real data, we present some test results on reconstructing haplotypes from a genome-scale SNP dataset consisting of 12 pedigrees that have 0.8% to 14.5% missing alleles.     

HAPLORE: a program for haplotype reconstruction in general pedigrees without recombination

MOTIVATION: Haplotype reconstruction is an essential step in genetic linkage and association studies. Although many methods have been developed to estimate haplotype frequencies and reconstruct haplotypes for a sample of unrelated individuals, haplotype reconstruction in large pedigrees with a large number of genetic markers remains a challenging problem. METHODS: We have developed an efficient computer program, HAPLORE (HAPLOtype REconstruction), to identify all haplotype sets that are compatible with the observed genotypes in a pedigree for tightly linked genetic markers. HAPLORE consists of three steps that can serve different needs in applications. In the first step, a set of logic rules is used to reduce the number of compatible haplotypes of each individual in the pedigree as much as possible. After this step, the haplotypes of all individuals in the pedigree can be completely or partially determined. These logic rules are applicable to completely linked markers and they can be used to impute missing data and check genotyping errors. In the second step, a haplotype-elimination algorithm similar to the genotype-elimination algorithms used in linkage analysis is applied to delete incompatible haplotypes derived from the first step. All superfluous haplotypes of the pedigree members will be excluded after this step. In the third step, the expectation-maximization (EM) algorithm combined with the partition and ligation technique is used to estimate haplotype frequencies based on the inferred haplotype configurations through the first two steps. Only compatible haplotype configurations with haplotypes having frequencies greater than a threshold are retained. RESULTS: We test the effectiveness and the efficiency of HAPLORE using both simulated and real datasets. Our results show that, the rule-based algorithm is very efficient for completely genotyped pedigree. In this case, almost all of the families have one unique haplotype configuration. In the presence of missing data, the number of compatible haplotypes can be substantially reduced by HAPLORE, and the program will provide all possible haplotype configurations of a pedigree under different circumstances, if such multiple configurations exist. These inferred haplotype configurations, as well as the haplotype frequencies estimated by the EM algorithm, can be used in genetic linkage and association studies. AVAILABILITY: The program can be downloaded from http://bioinformatics.med.yale.edu.     

Association of genetic traits to estimated haplotypes from SNP genotypes using EM algorithm and Markov Chain Monte Carlo techniques

In order to find association of genetic traits to some haplotypes from closely spaced multilocus phase-unknown genotypes we use a three-stage approach. Haplotype frequencies and the most likely haplotype pair for each individual are estimated from random samples of individual or small (nuclear) family genotypes via an EM algorithm. If the most likely haplotype pair configuration of the whole sample outweighs the less likely ones, we may consider the estimated haplotypes as alleles of a multi-allelic marker and carry out the conventional statistics, TDT or ANOVA for quantitative traits. If the most likely haplotype pair configuration and the less likely ones do not differ much in their weight, we sample the TDT or ANOVA statistic over all haplotype pair configurations using the Metropolis-Hastings algorithm. Applications of our method to simulated data as well as real data are given.     

The incomplete perfect phylogeny haplotype problem

The problem of resolving genotypes into haplotypes, under the perfect phylogeny model, has been under intensive study recently. All studies so far handled missing data entries in a heuristic manner. We prove that the perfect phylogeny haplotype problem is NP-complete when some of the data entries are missing, even when the phylogeny is rooted. We define a biologically motivated probabilistic model for genotype generation and for the way missing data occur. Under this model, we provide an algorithm, which takes an expected polynomial time. In tests on simulated data, our algorithm quickly resolves the genotypes under high rates of missing entries.     

Bayesian Haplotype Inference for Multiple Linked Single-Nucleotide Polymorphisms

Haplotypes have gained increasing attention in the mapping of complex-disease genes, because of the abundance of single-nucleotide polymorphisms (SNPs) and the limited power of conventional single-locus analyses. It has been shown that haplotype-inference methods such as Clark's algorithm, the expectation-maximization algorithm, and a coalescence-based iterative-sampling algorithm are fairly effective and economical alternatives to molecular-haplotyping methods. To contend with some weaknesses of the existing algorithms, we propose a new Monte Carlo approach. In particular, we first partition the whole haplotype into smaller segments. Then, we use the Gibbs sampler both to construct the partial haplotypes of each segment and to assemble all the segments together. Our algorithm can accurately and rapidly infer haplotypes for a large number of linked SNPs. By using a wide variety of real and simulated data sets, we demonstrate the advantages of our Bayesian algorithm, and we show that it is robust to the violation of Hardy-Weinberg equilibrium, to the presence of missing data, and to occurrences of recombination hotspots.     

Haplotype inference in general pedigrees using the cluster variation method

We present CVMHAPLO, a probabilistic method for haplotyping in general pedigrees with many markers. CVMHAPLO reconstructs the haplotypes by assigning in every iteration a fixed number of the ordered genotypes with the highest marginal probability, conditioned on the marker data and ordered genotypes assigned in previous iterations. CVMHAPLO makes use of the cluster variation method (CVM) to efficiently estimate the marginal probabilities. We focused on single-nucleotide polymorphism (SNP) markers in the evaluation of our approach. In simulated data sets where exact computation was feasible, we found that the accuracy of CVMHAPLO was high and similar to that of maximum-likelihood methods. In simulated data sets where exact computation of the maximum-likelihood haplotype configuration was not feasible, the accuracy of CVMHAPLO was similar to that of state of the art Markov chain Monte Carlo (MCMC) maximum-likelihood approximations when all ordered genotypes were assigned and higher when only a subset of the ordered genotypes was assigned. CVMHAPLO was faster than the MCMC approach and provided more detailed information about the uncertainty in the inferred haplotypes. We conclude that CVMHAPLO is a practical tool for the inference of haplotypes in large complex pedigrees.     

Haplotype-based linkage disequilibrium mapping via direct data mining

MOTIVATION: With the availability of large-scale, high-density single-nucleotide polymorphism markers and information on haplotype structures and frequencies, a great challenge is how to take advantage of haplotype information in the association mapping of complex diseases in case-control studies. RESULTS: We present a novel approach for association mapping based on directly mining haplotypes (i.e. phased genotype pairs) produced from case-control data or case-parent data via a density-based clustering algorithm, which can be applied to whole-genome screens as well as candidate-gene studies in small genomic regions. The method directly explores the sharing of haplotype segments in affected individuals that are rarely present in normal individuals. The measure of sharing between two haplotypes is defined by a new similarity metric that combines the length of the shared segments and the number of common alleles around any marker position of the haplotypes, which is robust against recent mutations/genotype errors and recombination events. The effectiveness of the approach is demonstrated by using both simulated datasets and real datasets. The results show that the algorithm is accurate for different population models and for different disease models, even for genes with small effects, and it outperforms some recently developed methods.     

The trimmed-haplotype test for linkage disequilibrium

Single-marker linkage-disequilibrium (LD) methods cannot fully describe disequilibrium in an entire chromosomal region surrounding a disease allele. With the advent of myriad tightly linked microsatellite markers, we have an opportunity to extend LD analysis from single markers to multiple-marker haplotypes. Haplotype analysis has increased statistical power to disclose the presence of a disease locus in situations where it correctly reflects the historical process involved. For maximum efficiency, evidence of LD ought to come not just from a single haplotype, which may well be rare, but in addition from many similar haplotypes that could have descended from the same ancestral founder but have been trimmed in succeeding generations. We present such an analysis, called the "trimmed-haplotype method." We focus on chromosomal regions that are small enough that disequilibrium in significant portions of them may have been preserved in some pedigrees and yet that contain enough markers to minimize coincidental occurrence of the haplotype in the absence of a disease allele: perhaps regions 1-2 cM in length. In general, we could have no idea what haplotype an ancestral founder carried generations ago, nor do we usually have a precise chromosomal location for the disease-susceptibility locus. Therefore, we must search through all possible haplotypes surrounding multiple locations. Since such repeated testing obliterates the sampling distribution of the test, we employ bootstrap methods to calculate significance levels. Trimmed-haplotype analysis is performed on family data in which genotypes have been assembled into haplotypes. It can be applied either to conventional parent-affected-offspring triads or to multiplex pedigrees. We present a method for summarizing the LD evidence, in any pedigree, that can be employed in trimmed-haplotype analysis as well as in other methods.     

Haplotypic QTL mapping in an outbred pedigree

An offspring genome can be viewed as a mosaic of chromosomal segments or haplotypes contributed by multiple founders in any quantitative trait locus (QTL) detection study but tracing these is especially complex to achieve for outbred pedigrees. QTL haplotypes can be traced from offspring back to individual founders in outbred pedigrees by combining founder-origin probabilities with fully informative flanking markers. This haplotypic method was illustrated for QTL detection using a three-generation pedigree for a woody perennial plant, Pinus taeda L. Growth rate was estimated using height measurements from ages 2 to 10 years. Using simulated and actual datasets, power of the experimental design was shown to be efficient for detecting QTLs of large effect. Using interval mapping and fully informative markers, a large QTL accounting for 11.3% of the phenotypic variance in the growth rate was detected. This same QTL was expressed at all ages for height, accounting for 7.9-12.2% of the phenotypic variance. A mixed-model inheritance was more appropriate for describing genetic architecture of growth curves in P. taeda than a strictly polygenic model. The positive QTL haplotype was traced from the offspring to its contributing founder, GP3, then the haplotypic phase for GP3 was determined by assaying haploid megagametophytes. The positive QTL haplotype was a recombinant haplotype contributed by GP3. This study illustrates the combined power of fully informative flanking markers and founder origin probabilities for (1) estimating QTL haplotype magnitude, (2) tracing founder origin and (3) determining haplotypic transmission frequency.     

Genetic association mapping via evolution-based clustering of haplotypes

Multilocus analysis of single nucleotide polymorphism haplotypes is a promising approach to dissecting the genetic basis of complex diseases. We propose a coalescent-based model for association mapping that potentially increases the power to detect disease-susceptibility variants in genetic association studies. The approach uses Bayesian partition modelling to cluster haplotypes with similar disease risks by exploiting evolutionary information. We focus on candidate gene regions with densely spaced markers and model chromosomal segments in high linkage disequilibrium therein assuming a perfect phylogeny. To make this assumption more realistic, we split the chromosomal region of interest into sub-regions or windows of high linkage disequilibrium. The haplotype space is then partitioned into disjoint clusters, within which the phenotype–haplotype association is assumed to be the same. For example, in case-control studies, we expect chromosomal segments bearing the causal variant on a common ancestral background to be more frequent among cases than controls, giving rise to two separate haplotype clusters. The novelty of our approach arises from the fact that the distance used for clustering haplotypes has an evolutionary interpretation, as haplotypes are clustered according to the time to their most recent common ancestor. Our approach is fully Bayesian and we develop a Markov Chain Monte Carlo algorithm to sample efficiently over the space of possible partitions. We compare the proposed approach to both single-marker analyses and recently proposed multi-marker methods and show that the Bayesian partition modelling performs similarly in localizing the causal allele while yielding lower false-positive rates. Also, the method is computationally quicker than other multi-marker approaches. We present an application to real genotype data from the CYP2D6 gene region, which has a confirmed role in drug metabolism, where we succeed in mapping the location of the susceptibility variant within a small error.     

Comparison of haplotype inference methods using genotypic data from unrelated individuals

OBJECTIVE: Haplotypes are gaining popularity in studies of human genetics because they contain more information than does a single gene locus. However, current high-throughput genotyping techniques cannot produce haplotype information. Several statistical methods have recently been proposed to infer haplotypes based on unphased genotypes at several loci. The accuracy, efficiency, and computational time of these methods have been under intense scrutiny. In this report, our aim was to evaluate haplotype inference methods for genotypic data from unrelated individuals. METHODS: We compared the performance of three haplotype inference methods that are currently in use--HAPLOTYPER, hap, and PHASE--by applying them to a large data set from unrelated individuals with known haplotypes. We also applied these methods to coalescent-based simulation studies using both constant size and exponential growth models. The performance of these methods, along with that of the expectation-maximization algorithm, was further compared in the context of an association study. RESULTS: While the algorithm implemented in the software PHASE was found to be the most accurate in both real and simulated data comparisons, all four methods produced good results in the association study.     

Efficient inference of haplotypes from genotypes on a pedigree

We study haplotype reconstruction under the Mendelian law of inheritance and the minimum recombination principle on pedigree data. We prove that the problem of finding a minimum-recombinant haplotype configuration (MRHC) is in general NP-hard. This is the first complexity result concerning the problem to our knowledge. An iterative algorithm based on blocks of consecutive resolved marker loci (called block-extension) is proposed. It is very efficient and can be used for large pedigrees with a large number of markers, especially for those data sets requiring few recombinants (or recombination events). A polynomial-time exact algorithm for haplotype reconstruction without recombinants is also presented. This algorithm first identifies all the necessary constraints based on the Mendelian law and the zero recombinant assumption, and represents them using a system of linear equations over the cyclic group Z2. By using a simple method based on Gaussian elimination, we could obtain all possible feasible haplotype configurations. A C++ implementation of the block-extension algorithm, called PedPhase, has been tested on both simulated data and real data. The results show that the program performs very well on both types of data and will be useful for large scale haplotype inference projects.     

Estimating haplotype frequencies and standard errors for multiple single nucleotide polymorphisms

Estimating haplotype frequencies becomes increasingly important in the mapping of complex disease genes, as millions of single nucleotide polymorphisms (SNPs) are being identified and genotyped. When genotypes at multiple SNP loci are gathered from unrelated individuals, haplotype frequencies can be accurately estimated using expectation-maximization (EM) algorithms (Excoffier and Slatkin, 1995; Hawley and Kidd, 1995; Long et al., 1995), with standard errors estimated using bootstraps. However, because the number of possible haplotypes increases exponentially with the number of SNPs, handling data with a large number of SNPs poses a computational challenge for the EM methods and for other haplotype inference methods. To solve this problem, Niu and colleagues, in their Bayesian haplotype inference paper (Niu et al., 2002), introduced a computational algorithm called progressive ligation (PL). But their Bayesian method has a limitation on the number of subjects (no more than 100 subjects in the current implementation of the method). In this paper, we propose a new method in which we use the same likelihood formulation as in Excoffier and Slatkin's EM algorithm and apply the estimating equation idea and the PL computational algorithm with some modifications. Our proposed method can handle data sets with large number of SNPs as well as large numbers of subjects. Simultaneously, our method estimates standard errors efficiently, using the sandwich-estimate from the estimating equation, rather than the bootstrap method. Additionally, our method admits missing data and produces valid estimates of parameters and their standard errors under the assumption that the missing genotypes are missing at random in the sense defined by Rubin (1976).     

Linkage disequilibrium and haplotype homozygosity in population samples genotyped at a high marker density

OBJECTIVE: Analyze the information contained in homozygous haplotypes detected with high density genotyping. METHODS: We analyze the genotypes of approximately 2,500 markers on chr 22 in 12 population samples, each including 200 individuals. We develop a measure of disequilibrium based on haplotype homozygosity and an algorithm to identify genomic segments characterized by non-random homozygosity (NRH), taking into account allele frequencies, missing data, genotyping error, and linkage disequilibrium. RESULTS: We show how our measure of linkage disequilibrium based on homozygosity leads to results comparable to those of R(2), as well as the importance of correcting for small sample variation when evaluating D'. We observe that the regions that harbor NRH segments tend to be consistent across populations, are gene rich, and are characterized by lower recombination. CONCLUSIONS: It is crucial to take into account LD patterns when interpreting long stretches of homozygous markers.     

A comparison of phasing algorithms for trios and unrelated individuals

Knowledge of haplotype phase is valuable for many analysis methods in the study of disease, population, and evolutionary genetics. Considerable research effort has been devoted to the development of statistical and computational methods that infer haplotype phase from genotype data. Although a substantial number of such methods have been developed, they have focused principally on inference from unrelated individuals, and comparisons between methods have been rather limited. Here, we describe the extension of five leading algorithms for phase inference for handling father-mother-child trios. We performed a comprehensive assessment of the methods applied to both trios and to unrelated individuals, with a focus on genomic-scale problems, using both simulated data and data from the HapMap project. The most accurate algorithm was PHASE (v2.1). For this method, the percentages of genotypes whose phase was incorrectly inferred were 0.12%, 0.05%, and 0.16% for trios from simulated data, HapMap Centre d'Etude du Polymorphisme Humain (CEPH) trios, and HapMap Yoruban trios, respectively, and 5.2% and 5.9% for unrelated individuals in simulated data and the HapMap CEPH data, respectively. The other methods considered in this work had comparable but slightly worse error rates. The error rates for trios are similar to the levels of genotyping error and missing data expected. We thus conclude that all the methods considered will provide highly accurate estimates of haplotypes when applied to trio data sets. Running times differ substantially between methods. Although it is one of the slowest methods, PHASE (v2.1) was used to infer haplotypes for the 1 million-SNP HapMap data set. Finally, we evaluated methods of estimating the value of r(2) between a pair of SNPs and concluded that all methods estimated r(2) well when the estimated value was >or=0.8.     

Imputation of missing genotypes from sparse to high density using long-range phasing

Related individuals share potentially long chromosome segments that trace to a common ancestor. We describe a phasing algorithm (ChromoPhase) that utilizes this characteristic of finite populations to phase large sections of a chromosome. In addition to phasing, our method imputes missing genotypes in individuals genotyped at lower marker density when more densely genotyped relatives are available. ChromoPhase uses a pedigree to collect an individual's (the proband) surrogate parents and offspring and uses genotypic similarity to identify its genomic surrogates. The algorithm then cycles through the relatives and genomic surrogates one at a time to find shared chromosome segments. Once a segment has been identified, any missing information in the proband is filled in with information from the relative. We tested ChromoPhase in a simulated population consisting of 400 individuals at a marker density of 1500/M, which is approximately equivalent to a 50K bovine single nucleotide polymorphism chip. In simulated data, 99.9% loci were correctly phased and, when imputing from 100 to 1500 markers, more than 87% of missing genotypes were correctly imputed. Performance increased when the number of generations available in the pedigree increased, but was reduced when the sparse genotype contained fewer loci. However, in simulated data, ChromoPhase correctly imputed at least 12% more genotypes than fastPHASE, depending on sparse marker density. We also tested the algorithm in a real Holstein cattle data set to impute 50K genotypes in animals with a sparse 3K genotype. In these data 92% of genotypes were correctly imputed in animals with a genotyped sire. We evaluated the accuracy of genomic predictions with the dense, sparse, and imputed simulated data sets and show that the reduction in genomic evaluation accuracy is modest even with imperfectly imputed genotype data. Our results demonstrate that imputation of missing genotypes, and potentially full genome sequence, using long-range phasing is feasible.     

Inferring haplotypes from genotypes on a pedigree with mutations, genotyping errors and missing alleles

Inferring the haplotypes of the members of a pedigree from their genotypes has been extensively studied. However, most studies do not consider genotyping errors and de novo mutations. In this paper, we study how to infer haplotypes from genotype data that may contain genotyping errors, de novo mutations, and missing alleles. We assume that there are no recombinants in the genotype data, which is usually true for tightly linked markers. We introduce a combinatorial optimization problem, called haplotype configuration with mutations and errors (HCME), which calls for haplotype configurations consistent with the given genotypes that incur no recombinants and require the minimum number of mutations and errors. HCME is NP-hard. To solve the problem, we propose a heuristic algorithm, the core of which is an integer linear program (ILP) using the system of linear equations over Galois field GF(2). Our algorithm can detect and locate genotyping errors that cannot be detected by simply checking the Mendelian law of inheritance. The algorithm also offers error correction in genotypes/haplotypes rather than just detecting inconsistencies and deleting the involved loci. Our experimental results show that the algorithm can infer haplotypes with a very high accuracy and recover 65%-94% of genotyping errors depending on the pedigree topology.