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.     

Haplotype Inference for Population Data with Genotyping Errors

Inference of haplotypes is important in genetic epidemiology studies. However, all large genotype data sets have errors due to the use of inexpensive genotyping machines that are fallible and shortcomings in genotyping scoring softwares, which can have an enormous impact on haplotype inference. In this article, we propose two novel strategies to reduce the impact induced by genotyping errors in haplotype inference. The first method makes use of double sampling. For each individual, the “GenoSpectrum” that consists of all possible genotypes and their corresponding likelihoods are computed. The second method is a genotype clustering algorithm based on multi‐genotyping data, which also assigns a “GenoSpectrum” for each individual. We then describe two hybrid EM algorithms (called DS‐EM and MG‐EM) that perform haplotype inference based on “GenoSpectrum” of each individual obtained by double sampling and multi‐genotyping data. Both simulated data sets and a quasi real‐data set demonstrate that our proposed methods perform well in different situations and outperform the conventional EM algorithm and the HMM algorithm proposed by Sun, Greenwood, and Neal (2007, Genetic Epidemiology 31 , 937–948) when the genotype data sets have errors.     

Bayesian haplotype inference via the Dirichlet process.

The problem of inferring haplotypes from genotypes of single nucleotide polymorphisms (SNPs) is essential for the understanding of genetic variation within and among populations, with important applications to the genetic analysis of disease propensities and other complex traits. The problem can be formulated as a mixture model, where the mixture components correspond to the pool of haplotypes in the population. The size of this pool is unknown; indeed, knowing the size of the pool would correspond to knowing something significant about the genome and its history. Thus methods for fitting the genotype mixture must crucially address the problem of estimating a mixture with an unknown number of mixture components. In this paper we present a Bayesian approach to this problem based on a nonparametric prior known as the Dirichlet process. The model also incorporates a likelihood that captures statistical errors in the haplotype/genotype relationship trading off these errors against the size of the pool of haplotypes. We describe an algorithm based on Markov chain Monte Carlo for posterior inference in our model. The overall result is a flexible Bayesian method, referred to as DP-Haplotyper, that is reminiscent of parsimony methods in its preference for small haplotype pools. We further generalize the model to treat pedigree relationships (e.g., trios) between the population's genotypes. We apply DP-Haplotyper to the analysis of both simulated and real genotype data, and compare to extant methods.     

Estimation of haplotype frequencies,linkage-disequilibrium measures,and combination of haplotype copies in each pool by use of pooled DNA data

Inference of haplotypes is important for many genetic approaches, including the process of assigning a phenotype to a genetic region. Usually, the population frequencies of haplotypes, as well as the diplotype configuration of each subject, are estimated from a set of genotypes of the subjects in a sample from the population. We have developed an algorithm to infer haplotype frequencies and the combination of haplotype copies in each pool by using pooled DNA data. The input data are the genotypes in pooled DNA samples, each of which contains the quantitative genotype data from one to six subjects. The algorithm infers by the maximum-likelihood method both frequencies of the haplotypes in the population and the combination of haplotype copies in each pool by an expectation-maximization algorithm. The algorithm was implemented in the computer program LDPooled. We also used the bootstrap method to calculate the standard errors of the estimated haplotype frequencies. Using this program, we analyzed the published genotype data for the SAA (n=156), MTHFR (n=80), and NAT2 (n=116) genes, as well as the smoothelin gene (n=102). Our study has shown that the frequencies of major (frequency >0.1 in a population) haplotypes can be inferred rather accurately from the pooled DNA data by the maximum-likelihood method, although with some limitations. The estimated D and D' values had large variations except when the /D/ values were >0.1. The estimated linkage-disequilibrium measure rho2 for 36 linked loci of the smoothelin gene when one- and two-subject pool protocols were used suggested that the gross pattern of the distribution of the measure can be reproduced using the two-subject pool data.     

A block-free hidden Markov model for genotypes and its application to disease association.

We present a new stochastic model for genotype generation. The model offers a compromise between rigid block structure and no structure altogether: It reflects a general blocky structure of haplotypes, but also allows for "exchange" of haplotypes at nonboundary SNP sites; it also accommodates rare haplotypes and mutations. We use a hidden Markov model and infer its parameters by an expectation-maximization algorithm. The algorithm was implemented in a software package called HINT (haplotype inference tool) and tested on 58 datasets of genotypes. To evaluate the utility of the model in association studies, we used biological human data to create a simple disease association search scenario. When comparing HINT to three other models, HINT predicted association most accurately.     

Efficient reconstruction of haplotype structure via perfect phylogeny

Each person's genome contains two copies of each chromosome, one inherited from the father and the other from the mother. A person's genotype specifies the pair of bases at each site, but does not specify which base occurs on which chromosome. The sequence of each chromosome separately is called a haplotype. The determination of the haplotypes within a population is essential for understanding genetic variation and the inheritance of complex diseases. The haplotype mapping project, a successor to the human genome project, seeks to determine the common haplotypes in the human population. Since experimental determination of a person's genotype is less expensive than determining its component haplotypes, algorithms are required for computing haplotypes from genotypes. Two observations aid in this process: first, the human genome contains short blocks within which only a few different haplotypes occur; second, as suggested by Gusfield, it is reasonable to assume that the haplotypes observed within a block have evolved according to a perfect phylogeny, in which at most one mutation event has occurred at any site, and no recombination occurred at the given region. We present a simple and efficient polynomial-time algorithm for inferring haplotypes from the genotypes of a set of individuals assuming a perfect phylogeny. Using a reduction to 2-SAT we extend this algorithm to handle constraints that apply when we have genotypes from both parents and child. We also present a hardness result for the problem of removing the minimum number of individuals from a population to ensure that the genotypes of the remaining individuals are consistent with a perfect phylogeny. Our algorithms have been tested on real data and give biologically meaningful results. Our webserver (http://www.cs.columbia.edu/compbio/hap/) is publicly available for predicting haplotypes from genotype data and partitioning genotype data into blocks.     

A new statistical method for haplotype reconstruction from population data

Current routine genotyping methods typically do not provide haplotype information, which is essential for many analyses of fine-scale molecular-genetics data. Haplotypes can be obtained, at considerable cost, experimentally or (partially) through genotyping of additional family members. Alternatively, a statistical method can be used to infer phase and to reconstruct haplotypes. We present a new statistical method, applicable to genotype data at linked loci from a population sample, that improves substantially on current algorithms; often, error rates are reduced by > 50%, relative to its nearest competitor. Furthermore, our algorithm performs well in absolute terms, suggesting that reconstructing haplotypes experimentally or by genotyping additional family members may be an inefficient use of resources.     

Mathematical properties and bounds on haplotyping populations by pure parsimony

Although the haplotype data can be used to analyze the function of DNA, due to the significant efforts required in collecting the haplotype data, usually the genotype data is collected and then the population haplotype inference (PHI) problem is solved to infer haplotype data from genotype data for a population. This paper investigates the PHI problem based on the pure parsimony criterion (HIPP), which seeks the minimum number of distinct haplotypes to infer a given genotype data. We analyze the mathematical structure and properties for the HIPP problem, propose techniques to reduce the given genotype data into an equivalent one of much smaller size, and analyze the relations of genotype data using a compatible graph. Based on the mathematical properties in the compatible graph, we propose a maximal clique heuristic to obtain an upper bound, and a new polynomial-sized integer linear programming formulation to obtain a lower bound for the HIPP problem.     

Haplotyping in pedigrees via a genetic algorithm.

Genome-wide screening for localization of disease genes necessitates the efficient reconstruction of haplotypes of members of a pedigree from genotype data at multiple loci. We propose a genetic algorithmic approach to haplotyping and show that it works fast, efficiently and reliably. This algorithm uses certain principles of biological evolution to find optimal solutions to complex problems. The optimality criterion used in the present problem is the minimum number of recombinations over possible haplotype configurations of members of a pedigree. The proposed algorithm is much less demanding in terms of data and assumption requirements compared to the currently used likelihood-based methods of haplotype reconstruction. It also provides multiple optimal haplotype configurations of a pedigree, if such multiple optima exist.     

Haplotype inference for present-absent genotype data using previously identified haplotypes and haplotype patterns

MOTIVATION: Killer immunoglobulin-like receptor (KIR) genes vary considerably in their presence or absence on a specific regional haplotype. Because presence or absence of these genes is largely detected using locus-specific genotyping technology, the distinction between homozygosity and hemizygosity is often ambiguous. The performance of methods for haplotype inference (e.g. PL-EM, PHASE) for KIR genes may be compromised due to the large portion of ambiguous data. At the same time, many haplotypes or partial haplotype patterns have been previously identified and can be incorporated to facilitate haplotype inference for unphased genotype data. To accommodate the increased ambiguity of present-absent genotyping of KIR genes, we developed a hybrid approach combining a greedy algorithm with the Expectation-Maximization (EM) method for haplotype inference based on previously identified haplotypes and haplotype patterns. RESULTS: We implemented this algorithm in a software package named HAPLO-IHP (Haplotype inference using identified haplotype patterns) and compared its performance with that of HAPLORE and PHASE on simulated KIR genotypes. We compared five measures in order to evaluate the reliability of haplotype assignments and the accuracy in estimating haplotype frequency. Our method outperformed the two existing techniques by all five measures when either 60% or 25% of previously identified haplotypes were incorporated into the analyses. AVAILABILITY: The HAPLO-IHP is available at http://www.soph.uab.edu/Statgenetics/People/KZhang/HAPLO-IHP/index.html. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.     

Estimation of genotype error rate using samples with pedigree information--an application on the GeneChip Mapping 10K array

Currently, most analytical methods assume all observed genotypes are correct; however, it is clear that errors may reduce statistical power or bias inference in genetic studies. We propose procedures for estimating error rate in genetic analysis and apply them to study the GeneChip Mapping 10K array, which is a technology that has recently become available and allows researchers to survey over 10,000 SNPs in a single assay. We employed a strategy to estimate the genotype error rate in pedigree data. First, the "dose-response" reference curve between error rate and the observable error number were derived by simulation, conditional on given pedigree structures and genotypes. Second, the error rate was estimated by calibrating the number of observed errors in real data to the reference curve. We evaluated the performance of this method by simulation study and applied it to a data set of 30 pedigrees genotyped using the GeneChip Mapping 10K array. This method performed favorably in all scenarios we surveyed. The dose-response reference curve was monotone and almost linear with a large slope. The method was able to estimate accurately the error rate under various pedigree structures and error models and under heterogeneous error rates. Using this method, we found that the average genotyping error rate of the GeneChip Mapping 10K array was about 0.1%. Our method provides a quick and unbiased solution to address the genotype error rate in pedigree data. It behaves well in a wide range of settings and can be easily applied in other genetic projects. The robust estimation of genotyping error rate allows us to estimate power and sample size and conduct unbiased genetic tests. The GeneChip Mapping 10K array has a low overall error rate, which is consistent with the results obtained from alternative genotyping assays.     

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.     

The impact of missing and erroneous genotypes on tagging SNP selection and power of subsequent association tests

OBJECTIVE: Single nucleotide polymorphisms (SNPs) serve as effective markers for localizing disease susceptibility genes, but current genotyping technologies are inadequate for genotyping all available SNP markers in a typical linkage/association study. Much attention has recently been paid to methods for selecting the minimal informative subset of SNPs in identifying haplotypes, but there has been little investigation of the effect of missing or erroneous genotypes on the performance of these SNP selection algorithms and subsequent association tests using the selected tagging SNPs. The purpose of this study is to explore the effect of missing genotype or genotyping error on tagging SNP selection and subsequent single marker and haplotype association tests using the selected tagging SNPs. METHODS: Through two sets of simulations, we evaluated the performance of three tagging SNP selection programs in the presence of missing or erroneous genotypes: Clayton's diversity based program htstep, Carlson's linkage disequilibrium (LD) based program ldSelect, and Stram's coefficient of determination based program tagsnp.exe. RESULTS: When randomly selected known loci were relabeled as 'missing', we found that the average number of tagging SNPs selected by all three algorithms changed very little and the power of subsequent single marker and haplotype association tests using the selected tagging SNPs remained close to the power of these tests in the absence of missing genotype. When random genotyping errors were introduced, we found that the average number of tagging SNPs selected by all three algorithms increased. In data sets simulated according to the haplotype frequecies in the CYP19 region, Stram's program had larger increase than Carlson's and Clayton's programs. In data sets simulated under the coalescent model, Carlson's program had the largest increase and Clayton's program had the smallest increase. In both sets of simulations, with the presence of genotyping errors, the power of the haplotype tests from all three programs decreased quickly, but there was not much reduction in power of the single marker tests. CONCLUSIONS: Missing genotypes do not seem to have much impact on tagging SNP selection and subsequent single marker and haplotype association tests. In contrast, genotyping errors could have severe impact on tagging SNP selection and haplotype tests, but not on single marker tests.     

Haplotype‐based genotyping‐by‐sequencing in oat genome research

In a de novo genotyping‐by‐sequencing (GBS) analysis of short, 64‐base tag‐level haplotypes in 4657 accessions of cultivated oat, we discovered 164741 tag‐level (TL) genetic variants containing 241224 SNPs. From this, the marker density of an oat consensus map was increased by the addition of more than 70000 loci. The mapped TL genotypes of a 635‐line diversity panel were used to infer chromosome‐level (CL) haplotype maps. These maps revealed differences in the number and size of haplotype blocks, as well as differences in haplotype diversity between chromosomes and subsets of the diversity panel. We then explored potential benefits of SNP vs. TL vs. CL GBS variants for mapping, high‐resolution genome analysis and genomic selection in oats. A combined genome‐wide association study (GWAS) of heading date from multiple locations using both TL haplotypes and individual SNP markers identified 184 significant associations. A comparative GWAS using TL haplotypes, CL haplotype blocks and their combinations demonstrated the superiority of using TL haplotype markers. Using a principal component‐based genome‐wide scan, genomic regions containing signatures of selection were identified. These regions may contain genes that are responsible for the local adaptation of oats to Northern American conditions. Genomic selection for heading date using TL haplotypes or SNP markers gave comparable and promising prediction accuracies of up to r = 0.74. Genomic selection carried out in an independent calibration and test population for heading date gave promising prediction accuracies that ranged between r = 0.42 and 0.67. In conclusion, TL haplotype GBS‐derived markers facilitate genome analysis and genomic selection in oat.     

An optimal algorithm for automatic genotype elimination

In an effort to accelerate likelihood computations on pedigrees, Lange and Goradia defined a genotype-elimination algorithm that aims to identify those genotypes that need not be considered during the likelihood computation. For pedigrees without loops, they showed that their algorithm was optimal, in the sense that it identified all genotypes that lead to a Mendelian inconsistency. Their algorithm, however, is not optimal for pedigrees with loops, which continue to pose daunting computational challenges. We present here a simple extension of the Lange-Goradia algorithm that we prove is optimal on pedigrees with loops, and we give examples of how our new algorithm can be used to detect genotyping errors. We also introduce a more efficient and faster algorithm for carrying out the fundamental step in the Lange-Goradia algorithm-namely, genotype elimination within a nuclear family. Finally, we improve a common algorithm for computing the likelihood of a pedigree with multiple loops. This algorithm breaks each loop by duplicating a person in that loop and then carrying out a separate likelihood calculation for each vector of possible genotypes of the loop breakers. This algorithm, however, does unnecessary computations when the loop-breaker vector is inconsistent. In this paper we present a new recursive loop breaker-elimination algorithm that solves this problem and illustrate its effectiveness on a pedigree with six loops.     

Distinct error rates for reference and nonreference genotypes estimated by pedigree analysis

Errors in genotype calling can have perverse effects on genetic analyses, confounding association studies, and obscuring rare variants. Analyses now routinely incorporate error rates to control for spurious findings. However, reliable estimates of the error rate can be difficult to obtain because of their variance between studies. Most studies also report only a single estimate of the error rate even though genotypes can be miscalled in more than one way. Here, we report a method for estimating the rates at which different types of genotyping errors occur at biallelic loci using pedigree information. Our method identifies potential genotyping errors by exploiting instances where the haplotypic phase has not been faithfully transmitted. The expected frequency of inconsistent phase depends on the combination of genotypes in a pedigree and the probability of miscalling each genotype. We develop a model that uses the differences in these frequencies to estimate rates for different types of genotype error. Simulations show that our method accurately estimates these error rates in a variety of scenarios. We apply this method to a dataset from the whole-genome sequencing of owl monkeys (Aotus nancymaae) in three-generation pedigrees. We find significant differences between estimates for different types of genotyping error, with the most common being homozygous reference sites miscalled as heterozygous and vice versa. The approach we describe is applicable to any set of genotypes where haplotypic phase can reliably be called and should prove useful in helping to control for false discoveries.     

Are molecular haplotypes worth the time and expense? A cost-effective method for applying molecular haplotypes

Because current molecular haplotyping methods are expensive and not amenable to automation, many researchers rely on statistical methods to infer haplotype pairs from multilocus genotypes, and subsequently treat these inferred haplotype pairs as observations. These procedures are prone to haplotype misclassification. We examine the effect of these misclassification errors on the false-positive rate and power for two association tests. These tests include the standard likelihood ratio test (LRTstd) and a likelihood ratio test that employs a double-sampling approach to allow for the misclassification inherent in the haplotype inference procedure (LRTae). We aim to determine the cost-benefit relationship of increasing the proportion of individuals with molecular haplotype measurements in addition to genotypes to raise the power gain of the LRTae over the LRTstd. This analysis should provide a guideline for determining the minimum number of molecular haplotypes required for desired power. Our simulations under the null hypothesis of equal haplotype frequencies in cases and controls indicate that (1) for each statistic, permutation methods maintain the correct type I error; (2) specific multilocus genotypes that are misclassified as the incorrect haplotype pair are consistently misclassified throughout each entire dataset; and (3) our simulations under the alternative hypothesis showed a significant power gain for the LRTae over the LRTstd for a subset of the parameter settings. Permutation methods should be used exclusively to determine significance for each statistic. For fixed cost, the power gain of the LRTae over the LRTstd varied depending on the relative costs of genotyping, molecular haplotyping, and phenotyping. The LRTae showed the greatest benefit over the LRTstd when the cost of phenotyping was very high relative to the cost of genotyping. This situation is likely to occur in a replication study as opposed to a whole-genome association study.     

Stacks 2: Analytical methods for paired‐end sequencing improve RADseq‐based population genomics

For half a century population genetics studies have put type II restriction endonucleases to work. Now, coupled with massively‐parallel, short‐read sequencing, the family of RAD protocols that wields these enzymes has generated vast genetic knowledge from the natural world. Here, we describe the first software natively capable of using paired‐end sequencing to derive short contigs from de novo RAD data. Stacks version 2 employs a de Bruijn graph assembler to build and connect contigs from forward and reverse reads for each de novo RAD locus, which it then uses as a reference for read alignments. The new architecture allows all the individuals in a metapopulation to be considered at the same time as each RAD locus is processed. This enables a Bayesian genotype caller to provide precise SNPs, and a robust algorithm to phase those SNPs into long haplotypes, generating RAD loci that are 400–800 bp in length. To prove its recall and precision, we tested the software with simulated data and compared reference‐aligned and de novo analyses of three empirical data sets. Our study shows that the latest version of Stacks is highly accurate and outperforms other software in assembling and genotyping paired‐end de novo data sets.     

A new method for haplotype inference including full-sib information

Recent literature has suggested that haplotype inference through close relatives, especially from nuclear families, can be an alternative strategy in determining linkage phase and estimating haplotype frequencies. In the case of no possibility to obtain genotypes for parents, and only full-sib information being used, a new approach is suggested to infer phase and to reconstruct haplotypes. We present a maximum-likelihood method via an expectation-maximization algorithm, called FSHAP, using only full-sib information when parent information is not available. FSHAP can deal with families with an arbitrary number of children, and missing parents or missing genotypes can be handled as well. In a simulation study we compare FSHAP with another existing expectation-maximization (EM)-based approach (FAMHAP), the conditioning approach implemented in FBAT and GENEHUNTER, which is only pedigree based and assumes linkage equilibrium. In most situations, FSHAP has the smallest discrepancy of haplotype frequency estimation and the lowest error rate in haplotype reconstruction, only in some cases FAMHAP yields comparable results. GENEHUNTER produces the largest discrepancy, and FBAT produces the highest error rate in offspring in most situations. Among the methods compared, FSHAP has the highest accuracy in reconstructing the diplotypes of the unavailable parents. Potential limitations of the method, e.g., in analyzing very large haplotypes, are indicated and possible solutions are discussed.