CrypticIBDcheck: an R package for checking cryptic relatedness in nominally unrelated individuals
 Annick NembotSimo^{1},
 Jinko Graham^{1} and
 Brad McNeney^{1}Email author
DOI: 10.1186/1751047385
© NembotSimo et al.; licensee BioMed Central Ltd. 2013
Received: 27 November 2012
Accepted: 21 January 2013
Published: 6 February 2013
Abstract
Background
In population association studies, standard methods of statistical inference assume that study subjects are independent samples. In genetic association studies, it is therefore of interest to diagnose undocumented close relationships in nominally unrelated study samples.
Results
We describe the R package CrypticIBDcheck to identify pairs of closelyrelated subjects based on genetic marker data from singlenucleotide polymorphisms (SNPs). The package is able to accommodate SNPs in linkage disequibrium (LD), without the need to thin the markers so that they are approximately independent in the population. Sample pairs are identified by superposing their estimated identitybydescent (IBD) coefficients on plots of IBD coefficients for pairs of simulated subjects from one of several common close relationships.
Conclusions
The methods implemented in CrypticIBDcheck are particularly relevant to candidategene association studies, in which dependent SNPs cluster in a relatively small number of genes spread throughout the genome. The accommodation of LD allows the use of all available genetic data, a desirable property when working with a modest number of dependent SNPs within candidate genes. CrypticIBDcheck is available from the Comprehensive R Archive Network (CRAN).
Keywords
Cryptic relatedness IBD estimation Linkage disequilibrium Gene drop simulationBackground
It is well known that the results of genetic association studies may be confounded by the presence of undocumented relationships – a phenomenon referred to as cryptic relatedness (e.g., [1, 2]). For example, [3] found that tests of association between genetic markers and quantitative phenotypes such as serum LDL tended to have inflated significance when relationships between individuals from a large Hutterite kindred were not accounted for. Before making any inference with the data, it is therefore important to understand cryptic relatedness in the study sample. To facilitate this understanding, we introduce CrypticIBDcheck, an R[4] package for exploring the presence of close relationships in a homogeneous sample of nominally unrelated individuals. Although several methods for exploring cryptic relatedness have been implemented (reviewed below), none are geared for data from candidategene association studies. CrypticIBDcheck fills this need. For ease of interpretation, the package implements exploratory displays based on popular measures of geneidentity by descent. However, a unique feature of these displays is that they accommodate population linkage disequilibrium (LD) amongst genetic markers. The accommodation of LD allows the use of data on all available markers rather than on a subset whose alleles are approximately independent in the population. This feature is attractive in candidategene association studies, where markers within genes are in LD but the number of genes is too small to select an independent subset of markers that is informative for relationship. Using the simulated data set analyzed in the Examples section we have found it possible to distinguish parentoffspring or full sibling pairs from unrelated individuals using as few as 60 candidate genes (average of five SNPs per gene). We return to the issue of how many SNPs are appropriate for analysis with CrypticIBDcheck in the Conclusions section.
The relatedness between two individuals may be defined in terms of the proportion of loci at which they share zero, one or two alleles that are identicalbydescent (IBD). We refer to these proportions as the actual IBDsharing coefficients, or IBD coefficients. The alleles from two individuals are IBD if they are descended from a common ancestor in a given reference population (e.g., [5]). Though alleles from each of two individuals may match or be identicalbystate (IBS), they are not necessarly IBD.
CrypticIBDcheck uses estimated IBD coefficients to summarize possible relationships among pairs of study subjects. The approach is exploratory and graphicallybased, similar to the GRR approach of [6] and the approach of [7] implemented in the ibdPlot() function of the GWASTools Bioconductor package.
GRR calculates and displays the mean and variance of IBS allele sharing over polymorphic loci for each pair of individuals. Pairs of known relationships form reference clusters on the plot, allowing the user to identify errors in reported relationships. In association studies of nominally unrelated individuals, however, there are no reference clusters available. In principle, reference clusters could be obtained theoretically from the joint distribution of the IBS mean and variance estimators, but it is unclear how to derive this distribution in the presence of LD.
The ibdPlot() function in GWASTools may be applied to view estimated IBD coefficients along with reference clusters for the unobserved, true IBD coefficients based on theoretical moments of their distribution [8]. However, in candidategene studies with a modest number of singlenucleotide polymorphisms (SNPs), errors introduced by estimation of IBD coefficients cannot be ignored. Hence, reference distributions for the true IBD coefficients do not adequately represent those for estimated IBD coefficents.
The idea behind CrypticIBDcheck is to identify closelyrelated study pairs by displaying their estimated IBD coefficients together with those from simulated pairs of known relationships. The simulated reference pairs provide an empirical joint distribution of the IBD estimators under selected relationships which, in turn, suggest possible relationships amongst study pairs. Working with simulated pairs from known relationships avoids having to derive the joint distribution of the IBD estimators when the genetic markers are in LD. Simulated pairs are obtained by gene drop on a relationshipspecific pedigree, with pedigreefounder haplotypes drawn from a fitted haplotype model that accounts for LD [9]. We have implemented simulation of the following common relationships: monozygotic twins/sample duplicates, parentoffspring, full siblings, second degree (i.e., half siblings, avuncular or grandparentgrandchild) and first cousins. However, users may also specify their own custom relationships (see the Examples section).
The paper is structured as follows. In the Methods section we describe the IBD estimators and methods for gene drop simulation in the presence of LD to obtain reference clusters. The Results and Discussion section describes implementation details provides two examples of how to use the package. The paper ends with a Conclusions section that includes ideas for future work.
Methods
IBD estimation
There are two common approaches to estimating IBD coefficients: maximum likelihood [10–12] and the method of moments [13–15]. Typically, maximumlikelihood estimators (MLEs) are more biased than methodofmoments estimators (MMEs), especially when the number of loci is small; they are also more computationally expensive [14]. However, MMEs are less precise than MLEs and can fall outside the biologically meaningful parameter space [11].
In this section, we review a popular method of moments approach to estimating IBD coefficients introduced by [15] and implemented in PLINK. This approach assumes that the individuals are from the same homogeneous, randommating and noninbred population. Alleles from two individuals are considered to be IBD if they are descended from a common ancestor in some base population that we take to be relatively recent. All alleles in this base population are defined to be nonIBD. Given a SNP with alleles A and a, a pair of individuals that are, say, AA and aa, respectively, will be denoted (AA, aa).
The methodofmoments estimators of IBD coefficients for a given pair are obtained by substituting estimators of the conditional SNPspecific IBS probabilities, P_{ m }(I = iZ = z), pertaining to any pair and the pair’s marginal SNPspecific IBS probabilities, P_{ m }(I = i), into the identities and then solving for P(Z = i).
The marginal SNPspecific IBS probabilities, P_{ m }(I = i) for a pair of subjects may be estimated by the indicator function for whether the pair has I = i at the SNP. An unbiased estimator of $\sum _{m}{\widehat{P}}_{m}(I=i)$ is therefore the count of SNPs at which the pair shares i alleles IBS. Estimates of the SNPspecific conditional IBS probabilities, P_{ m }(I = iZ = z), are based on data from all subjects in the sample. Derivation of unbiased estimators of P_{ m }(I = iZ = z) is more involved. To simplify notation, we temporarily drop the SNP subscript m. If p and q = 1−p denote the frequencies of A and a in the base population, then P(I = iZ = z) is a function of p and q. For example, two individuals share 0 alleles IBS if they are either (AA, aa) or (AA, aa). Given that Z = 0, the probabilities of these genotypes are p^{2}q^{2} and q^{2}p^{2}, respectively, leading to P(I = 0Z = 0) = 2p^{2}q^{2}. The plugin estimators of conditional IBS probabilities, such as P(I = 0Z=0), obtained by inserting estimators $\widehat{p}$ and $\widehat{q}$ are biased [Additional file 1]. Unbiased estimators, expressed as the plugin estimator multiplied by a correction factor, may be derived as described next.
Let X and Y be the counts of the alleles A and a, respectively, so that the allele frequency estimators are $\widehat{p}=X/T$ and $\widehat{q}=Y/T$, where T is twice the number of observed genotypes in the population random sample. The estimators of the conditional IBS probabilities P(I = iZ = z) may be motivated by the following model. The genotype of each individual in the present population is obtained from two independent draws from an infinite base population of alleles. Consequently, the T alleles of a population random sample of study subjects can be viewed as a random sample from the base population. Moreover, conditional on IBD status, any pair of individuals in the present population can be viewed as independent allelic draws from the base population, with the number of draws determined by their IBD status.
The other conditional IBS probabilities are estimated in an analogous manner and their expressions are provided in Table one of [15].
Adjustments to bound these estimators to values consistent with their interpretation as IBD proportions were proposed in [15]. We have not made these adjustments in our graphical displays.
Gene drop simulation with LD
The package provides a graphical display that can be used to identify related sample pairs by plotting the estimated IBD coefficients $\widehat{P}(Z=1)$ versus $\widehat{P}(Z=0)$. To assess the variability of these estimators the points of the IBD plot are superposed on reference clusters obtained from one of the following relationships: unrelated, monozygotic twins/duplicates, parentoffspring, full siblings, half siblings and first cousins. These reference clusters are obtained by gene drop simulation that accounts for LD [16]. A strength of this approach is that we do not need to assume independence of marker loci. In candidategene association studies, this feature is important because of the dependence among a relatively small number of SNPs. Ignoring the dependence among SNPs within genes produces reference clusters that are too tight relative to the true variability, and can lead to falsepositive results. We return to this point in the examples.
A graphical model is an approach to modeling the joint distribution of a set of dependent random variables when many independences or conditional independences exist between subsets of the variables. In the case of LD, it is expected that the joint distribution of alleles allong haplotypes shows such a structure. A flexible graphical model of haplotype frequencies that captures LD between loci is described in [17]. The model is fit to data from subjects that can be regarded as a population random sample; e.g., the controls in a casecontrol study of a rare disease. Model parameters are estimated by use of a stochastic optimization algorithm [16].
Once the LD model is fit, it is used to sample haplotypes for the founders of a pedigree. Data on the remaining members of the pedigree are simulated by gene drop. Gene drop is a method for randomly generating the genotypes of related individuals in a pedigree. Alleles are “dropped” from the founders through the pedigree according to Mendel’s laws. Multilocus gene drop incorporates the process of recombination. To illustrate the simulation procedure, consider a parentoffspring relationship. A pedigree that encompasses this relationship is one comprised of two parents and the offspring. The founders are the parents. Parental haplotypes are simulated from the fitted LD model and are then dropped to the offspring. To mimic real data with missing genotypes, selected genotypes for a simulated individual are set to missing according to the missing genotype pattern of a randomlysampled study subject.
Programs for fitting LD models and performing gene drop simulations are available in the Java Programs for Statistical Genetics and Computational Statistics (JPSGSC) library developed by Alun Thomas (http://balance.med.utah.edu/wiki/index.php/JPSGCS). We use the R package rJPSGCS[18] to access these programs from R.
Results and Discussion
Implementation
The main function in CrypticIBDcheck is IBDcheck(), which estimates IBD coefficients for pairs of study subjects and optionally for simulated pairs of subjects and returns an object of class IBD. The plot method for the IBD class displays the IBD coefficients for pairs of study subjects, along with prediction ellipses for known relationship pairs.
The arguments of IBDcheck() are constructed by the functions new.IBD(), filter.control() and sim.control(). The function new.IBD produces an object of class IBD suitable for input to IBDcheck(). At a minimum, such an object includes the genetic data as a snp.matrix object from the chopsticks package [19], a data frame of SNP information that includes chromosome and physical map positions of each SNP, and a data frame of subject information that includes a logical vector indicating whether (TRUE) or not (FALSE) each subject is to be used to estimate the conditional IBS probabilities and fit the LD model. Optionally, the SNP information may include genetic map positions, in centiMorgans. If genetic map positions are missing, they are inferred assuming the physical positions are from build 36 of the human genome. Users with SNP data on diploid nonhuman organisms, such as mouse or drosophila, must supply their own genetic map positions. The documentation for new.IBD() and the examples below provide further details. The function filter.control() sets options for quality control filtering of data by SNPs and by subjects, while the function sim.control() sets options that control simulation of subjects by gene drop. The respective help files and the examples below provide further details. As the fitting of LD models in IBDcheck() can be computationally demanding, users have the option of splitting computations across a snow cluster [20], as described in Additional file 2. The output of IBDcheck() is an object of class IBD, which includes the estimated IBD coefficients for pairs of study subjects and for simulated pairs with known relationship.
IBD objects are graphically displayed by the plot method of the class; the documentation for this method is available through help(‘‘plot.IBD''). Plots are of $\widehat{P}(Z=1)$ versus $\widehat{P}(Z=0)$ for pairs of study subjects, with prediction ellipses for known relationships superposed, if requested by the user. The prediction ellipses are produced from estimated IBD coefficients for a userspecified number (default 200) of simulated pairs of known relationships, assuming the distribution of estimated IBD coefficients is approximately bivariate Normal. The default setting for IBDcheck() is to omit simulated pairs from the object. When simulated pairs are omitted, plotting produces a single interactive display of estimated IBD coefficients for pairs of study subjects, on which points may be identified by clicking with the mouse. On the other hand, when the IBD object includes simulated pairs, the function returns a series of plots, which the user is prompted to view and interact with successively. The first plot to appear is nonclickable and shows the estimated IBD coefficients for all pairs of study subjects, along with the prediction ellipse for unrelated, simulated pairs. Subsequent plots are clickable and correspond to each relationship requested in the call to IBDcheck(). These relationshipspecific plots are for identifying pairs of study subjects which could have the relationship. The plotting regions are restricted to the neighborhood of the prediction ellipse for the simulated pairs of that relationship, which is also drawn. If, however, the plotting region overlaps with the prediction ellipse for simulated unrelated pairs, the ellipse for simulated unrelated pairs is drawn as well. Points falling within the prediction ellipse for the relationship and outside the prediction ellipse for unrelated pairs are automatically flagged. In addition, users may click on points of study pairs that appear to be related but are not automatically flagged. The plot method produces a data frame of information on pairs that have been flagged on the different plots, either automatically or interactively by the user through clicking the mouse.
Examples
We illustrate the features of the CrypticIBDcheck package using the genetic data Nhlsim that comes with the package. These data were simulated to mimic the characteristics of SNP genotypes in subjects of European ancestry from a candidategene, casecontrol study of nonHodgkin Lymphoma [21]. The data set is a list comprised of (i) a snp.matrix object called snp.data with genotypes for 108 controls and 100 cases; (ii) a vector chromosome of chromosome numbers for each SNP; (iii) a vector physmap of physical map positions of each SNP, from build 36 of the human genome; and (iv) a binary vector csct with value one for cases and zero for population controls. The binary vector csct is used to select controls for fitting LD models and estimating conditional IBS probabilities. All of the information in Nhlsim is required to run IBDcheck().
We present two examples. In the first (Default analysis with LD model fitting and gene drops), we illustrate basic use of IBDcheck() to fit LD models and do gene drop simulations. Once the user requests simulations, there are a number of parameters, such as the types of relationships to simulate, that control the simulations. Each simulation parameter has a default value, as described in the help file for sim.control(). In the first example we use these default settings. In the second example (Additional gene drops using previouslyfit LD models), we illustrate reuse of fitted LD models to perform additional gene drop simulations, this time for a userspecified relationship. For examples of how to use IBDcheck() to explore genomewide data, we refer readers to the package vignette IBDcheck‐hapmap that illustrates an analysis of genomewide data from HapMap, using thinning of markers to reduce the computational burden.
Default analysis with LD model fitting and gene drops
Figure 1 shows the nonclickable plot of the estimated IBD coefficients, P(Z=1) versus P(Z=0), for all pairs of study subjects, with the prediction ellipse for unrelated pairs superposed. The level of the prediction ellipse is left at its default value of ellipse.coverage=0.95 and, for unrelated pairs, is adjusted to account for the majority of study pairs being unrelated. Specifically, a Bonferronitype adjustment, 1−(1−ellipse.coverage)/n_{ p }, is applied, where n_{ p } is the number of pairs of study subjects. One purpose of the prediction ellipse is to avoid confusing the display by adding points for simulated pairs. Another purpose is to avoid having to manually click points for study pairs that appear within a cloud of points from simulated pairs. We adopted a bivariate normal approximation to the prediction ellipse because it correctly identified the majority of points in experiments with simulated data (e.g., Figure 1). However, in Figure 1, several unrelated pairs appear outside the prediction ellipse, indicating that the distribution of estimated IBD coefficients is slightly heaviertailed than the bivariate normal approximation.
In this data set, there are no duplicate/MZ twins pairs and no pairs flagged as such (Figure 2, left panel). The two parentoffspring pairs in the Nhlsim data fall in the prediction ellipse for parentoffspring pairs (Figure 2, right panel). Similarly, all three fullsibling pairs in the Nhlsim data fall in the prediction ellipse for full siblings (Figure 3, left panel). The substantial overlap of the prediction ellipses for half siblings and unrelated pairs (Figure 3, right panel) indicates insufficient data to distinguish these two relationships. Though there are no halfsibling pairs in Nhlsim, one pair of unrelated subjects, sub31 and sub44, has atypical estimated IBD coefficients that fall within the prediction ellipse for half siblings but outside the prediction ellipse for unrelated pairs.
The unrelated pair flagged as a potential halfsibling pair is a falsepositive result. We observed (results not shown) that the number of falsepositive related pairs is greatly increased if we fail to take the LD between SNPs into account. Specifically, if we repeat the simulation of unrelated and halfsibling pairs of subjects assuming independent SNPs (fitLD=FALSE), we obtain 16 falsepositive half sibling pairs. These observations highlight that naïvely ignoring the dependence among SNPs produces reference clusters that are too tight relative to the true variability.
Additional gene drops using previouslyfit LD models
By far the most computationallydemanding step of IBDcheck() is the fitting of LD models. The fitted LD models are stored in plaintext files in the working directory and can be reused for future gene drops using the argument LDfiles of sim.control(), as we now illustrate. We also demonstrate how users can create their own relationships to use as reference clusters on the IBD plot.
where the argument filter=FALSE to filter. control() reflects the fact that there is no need to refilter the data. On the plot of cibd.user (not shown) the prediction ellipse for simulated motherdaughter pairs where the daughter is inbred is very similar to that from simulated pairs where the daughter is not inbred (Figure 2, right panel). However, relative to the noninbred case, the prediction ellipse in the inbred case is shifted slightly downward on the plot, reflecting the fact that the probability of 2 genes IBD is now nonzero and the probability of 1 gene IBD is therefore smaller.
Conclusions
CrypticIBDcheck is an R package for exploring cryptic relatedness in a homogeneous sample of nominally unrelated individuals. The main function of the package, IBDcheck(), computes estimates of IBD coefficients for pairs of study subjects and, optionally, for pairs of subjects simulated to have one of several known relationships. Simulated data for a given relationship are obtained by gene drop simulation on a pedigree that captures the relationship, with founder haplotypes simulated according to an LD model fit to the data. Objects of class IBD returned by IBDcheck() are displayed by the plot method of the class. Pairs of study subjects whose estimated IBD coefficients are consistent with one of the relationships requested in the call to IBDcheck() are flagged, either automatically or interactively by user mouseclicks, and returned in a data frame.
The methods implemented in CrypticIBDcheck are geared specifically towards exploring cryptic relatedness with data from candidategene association studies. These studies involve a relatively modest number of SNPs which are correlated because they are clustered within candidate genes. With a modest number of SNPs, the variability in the estimator of IBD coefficients cannot be ignored. Hence, reference distributions for true IBD coefficients do not adequately represent those for estimated IBD coefficients. In addition, thinning to an approximately independent and yet informative set of SNPs is not an option. Nor is ignoring LD and assuming SNPs are approximately independent. As illustrated in the Examples, ignoring LD leads to reference clusters that are too tight.
To provide guidance on the numbers of SNPs for which CrypticIBDcheck will be useful, we offer the following observations. A lower bound on the number of SNPs required is difficult to provide because some marker sets are more relationshipinformative than others of the same size, depending on characteristics such as marker allele frequencies and the patterns of marker linkage disequilibrium (LD). In our experiments with the example data set Nhlsim, containing 1249 SNPs from 209 genes, between 325350 SNPs from about 60 of the candidate genes appears to be adequate for identifying the parentoffspring and full sibling pairs that are present. There is no upper limit on the number of SNPs that may be used. However, the computational time for fitting LD models scales approximately linearly with the number of SNPs [16]. For the Nhlsim data, it took 40 minutes to fit the LD model. For the close relationships that we consider, genomewide data can be reduced to a set of approximately independent SNPs with no loss of resolution. An analysis of genotypes from 16,245 approximately independent SNPs in the HapMap Luhya sample took about 2 minutes to complete. This analysis is described the package vignette IBDcheck‐HapMap.
We offer the user complete flexibility with respect to the type of relationships and number of pairs of each relationship to be simulated. Users can choose from a number of close relationships builtin to IBDcheck(), or specify their own relationships, as illustrated in the section Additional gene drops using previouslyfit LD models.
A reviewer has pointed out that it would be useful to allow the reference distributions of IBD coefficients, represented as ellipses on the graphical displays, to be conditional on the patterns of missing genotypes in a particular pair of subjects. The intent is to allow for a twostage analysis. In the first stage, potentially related subjects are identified using the current implementation of reference distributions, which are mixtures over the patterns of missing genotypes in all pairs of subjects. In the second stage, the reference distributions can be customized to be conditional on the missing data patterns of a pair of subjects identified in the first stage. For an assessment of whether the pair of interest has a particular relationship, distributions conditional on that pair’s missing data patterns are the most appropriate. We plan to implement an option to use a specific missing data pattern to generate the reference distributions in a future release of the package.
Abbreviations
 IBD:

Identity by descent
 IBD:

Identity by state
 LD:

Linkage disequilibrium
 MLE:

Maximum likelihood estimator
 MME:

Method of moments estimator.
Declarations
Acknowledgements
We thank John Spinelli for access to the nonHodgkin lymphoma study data and the reviewers for their insightful comments. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and by the Mathematics of Information Technology and Complex Systems (Mitacs), Canadian Networks of Centres of Excellence.
Authors’ Affiliations
References
 Devlin B, Roeder K: Genomic control for association studies. Biometrics. 1999, 55: 9971004. 10.1111/j.0006341X.1999.00997.x.View ArticlePubMedGoogle Scholar
 Voight B, Pritchard J: Confounding from cryptic relatedness in casecontrol association studies. Plos Genet. 2005, 1: e3210.1371/journal.pgen.0010032.PubMed CentralView ArticlePubMedGoogle Scholar
 Newman DL, Abney M, McPeek MS, Ober C, Cox NJ: The importance of genealogy in determining genetic associations with complex traits. Am J Hum Genet. 2001, 69 (5): 11461148. 10.1086/323659.PubMed CentralView ArticlePubMedGoogle Scholar
 R Development Core Team: R: A Language and Environment for Statistical Computing. 2012, Vienna: R Foundation for Statistical Computing, [http://www.Rproject.org/]. [ISBN 3900051070]Google Scholar
 Weir B, Anderson A, Hepler A: Genetic relatedness analysis: modern data and new challenges. Nat Rev Genet. 2006, 7: 771780. 10.1038/nrg1960.View ArticlePubMedGoogle Scholar
 Abecasis G, Cherny S, Cookson W, Cardon L: GRR: Graphical Representation of Relationship Errors. Bioinformatics. 2001, 17 (8): 742743. 10.1093/bioinformatics/17.8.742.View ArticlePubMedGoogle Scholar
 Gogarten SM, Laurie C, Bhangale T, Conomos MP, Laurie C, McHugh C, Painter I, Zheng X, Shen J, Swarnkar R: GWASTools: Tools for Genome Wide Association Studies. 2012, [R package version 1.2.0]Google Scholar
 Hill WG, Weir BS: Variation in actual relationship as a consequence of Mendelian sampling and linkage. Genet Res (Camb). 2011, 93: 4764. 10.1017/S0016672310000480.View ArticleGoogle Scholar
 Thomas A: Assessment of SNP streak statistics using gene drop simulation with linkage disequilibrium. Genet Epidemiol. 2010, 34: 119124. 10.1002/gepi.20440.PubMed CentralView ArticlePubMedGoogle Scholar
 Thompson E: The estimation of pairwise relationships. Ann Human Genet. 1975, 39: 173188. 10.1111/j.14691809.1975.tb00120.x.View ArticleGoogle Scholar
 Milligan B: Maximumlikelihood estimation of relatedness. Genetics. 2003, 163: 11531167.PubMed CentralPubMedGoogle Scholar
 Choi Y, Wijsman E, Weir B: Casecontrol association testing in the presence of unknown relationships. Genet Epidemiol. 2009, 33: 668678. 10.1002/gepi.20418.PubMed CentralView ArticlePubMedGoogle Scholar
 Ritland K: Estimators for pairwise relatedness and individual inbreeding coefficients. Genet Res. 1996, 67: 175185. 10.1017/S0016672300033620.View ArticleGoogle Scholar
 Lynch M, Ritland K: Estimation of pairwise relatedness with molecular markers. Genetics. 1999, 152: 17531766.PubMed CentralPubMedGoogle Scholar
 Purcell S, Neale B, ToddBrown K, Thomas L, Ferreira M, Bender D, Maller J, Sklar P, de Bakker, Daly M, Sham P: PLINK: a tool set for wholegenome association and populationbased linkage analyses. Am J Human Genet. 2007, 81: 559575. 10.1086/519795.View ArticleGoogle Scholar
 Thomas A: A method and program for estimating graphical models for linkage disequilibrium that scale linearly with the number of loci, and their application to gene drop simulation. Bioinformatics. 2009b, 25: 12871292. 10.1093/bioinformatics/btp146.View ArticleGoogle Scholar
 Thomas A: Estimation of graphical models whose conditional independence graphs are interval graphs and its application to modeling linkage disequilibrium. Comput Stat Data Anal. 2009, 53: 18181828. 10.1016/j.csda.2008.02.003.PubMed CentralView ArticlePubMedGoogle Scholar
 Blay S, Graham J, McNeney B, NembotSimo A: rJPSGCS: RInterface to Gene Drop Java Programs for Statistical Genetics and Computational Statistics (JPSGCS). 2011, [http://CRAN.Rproject.org/package=rJPSGCS], [R package version 0.25].Google Scholar
 Leung HT: chopsticks: The snp.matrix and X.snp.matrix Classes. 2011, [http://outmodedbonsai.sourceforge.net/]. [R package version 1.18.3].Google Scholar
 Tierney L, Rossini AJ, Sevcikova H, Li N: snow: Simple Network of Workstations. 2011, [http://CRAN.Rproject.org/package=snow]. [R package version 0.38].Google Scholar
 Schuetz JM, Daley D, Graham J, Berry BR, Gallagher RP, Connors JM, Gascoyne RD, Spinelli JJ, BrooksWilson AR: Genetic variation in cell death genes and risk of NonHodgkin Lymphoma. PLoS ONE. 2012, 7 (2): e3156010.1371/journal.pone.0031560. [http://www.plosone.org/article/info∖%3Adoi∖%2F10.1371∖%2Fjournal.pone.0031560]PubMed CentralView ArticlePubMedGoogle Scholar
 LoredoOsti J, Morgan K: Pedfiddler: A Set of Programs to Manipulate Pedigree Graphs. 2010, [http://www.stat.washington.edu/thompson/Genepi/Pedfiddler.shtml]. [Version 0.5].Google Scholar
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