CrypticIBDcheck: an R package for checking cryptic relatedness in nominally unrelated individuals
© Nembot-Simo et al.; licensee BioMed Central Ltd. 2013
Received: 27 November 2012
Accepted: 21 January 2013
Published: 6 February 2013
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.
We describe the R package CrypticIBDcheck to identify pairs of closely-related subjects based on genetic marker data from single-nucleotide 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 identity-by-descent (IBD) coefficients on plots of IBD coefficients for pairs of simulated subjects from one of several common close relationships.
The methods implemented in CrypticIBDcheck are particularly relevant to candidate-gene 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).
KeywordsCryptic relatedness IBD estimation Linkage disequilibrium Gene drop simulation
Identity by descent
Identity by state
Maximum likelihood estimator
Method of moments estimator.
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,  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 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 candidate-gene association studies. CrypticIBDcheck fills this need. For ease of interpretation, the package implements exploratory displays based on popular measures of gene-identity 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 candidate-gene 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 parent-offspring 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 identical-by-descent (IBD). We refer to these proportions as the actual IBD-sharing 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., ). Though alleles from each of two individuals may match or be identical-by-state (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 graphically-based, similar to the GRR approach of  and the approach of  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 . However, in candidate-gene studies with a modest number of single-nucleotide 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 closely-related 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 relationship-specific pedigree, with pedigree-founder haplotypes drawn from a fitted haplotype model that accounts for LD . We have implemented simulation of the following common relationships: monozygotic twins/sample duplicates, parent-offspring, full siblings, second degree (i.e., half siblings, avuncular or grandparent-grandchild) 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.
There are two common approaches to estimating IBD coefficients: maximum likelihood [10–12] and the method of moments [13–15]. Typically, maximum-likelihood estimators (MLEs) are more biased than method-of-moments estimators (MMEs), especially when the number of loci is small; they are also more computationally expensive . However, MMEs are less precise than MLEs and can fall outside the biologically meaningful parameter space .
In this section, we review a popular method of moments approach to estimating IBD coefficients introduced by  and implemented in PLINK. This approach assumes that the individuals are from the same homogeneous, random-mating and non-inbred 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 non-IBD. 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 method-of-moments estimators of IBD coefficients for a given pair are obtained by substituting estimators of the conditional SNP-specific IBS probabilities, P m (I = i|Z = z), pertaining to any pair and the pair’s marginal SNP-specific IBS probabilities, P m (I = i), into the identities and then solving for P(Z = i).
The marginal SNP-specific 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 is therefore the count of SNPs at which the pair shares i alleles IBS. Estimates of the SNP-specific conditional IBS probabilities, P m (I = i|Z = z), are based on data from all subjects in the sample. Derivation of unbiased estimators of P m (I = i|Z = 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 = i|Z = 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 = 0|Z = 0) = 2p 2 q 2. The plug-in estimators of conditional IBS probabilities, such as P(I = 0|Z=0), obtained by inserting estimators and are biased [Additional file 1]. Unbiased estimators, expressed as the plug-in 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 and , where T is twice the number of observed genotypes in the population random sample. The estimators of the conditional IBS probabilities P(I = i|Z = 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 .
Adjustments to bound these estimators to values consistent with their interpretation as IBD proportions were proposed in . 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 versus . 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, parent-offspring, full siblings, half siblings and first cousins. These reference clusters are obtained by gene drop simulation that accounts for LD . A strength of this approach is that we do not need to assume independence of marker loci. In candidate-gene 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 false-positive 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 . The model is fit to data from subjects that can be regarded as a population random sample; e.g., the controls in a case-control study of a rare disease. Model parameters are estimated by use of a stochastic optimization algorithm .
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. Multi-locus gene drop incorporates the process of recombination. To illustrate the simulation procedure, consider a parent-offspring 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 randomly-sampled 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 to access these programs from R.
Results and Discussion
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 , 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 non-human 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 , 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 versus 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 user-specified 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 non-clickable 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 relationship-specific 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.
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 candidate-gene, case-control study of non-Hodgkin Lymphoma . 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 previously-fit LD models), we illustrate re-use of fitted LD models to perform additional gene drop simulations, this time for a user-specified relationship. For examples of how to use IBDcheck() to explore genome-wide data, we refer readers to the package vignette IBDcheck‐hapmap that illustrates an analysis of genome-wide 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 non-clickable 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 Bonferroni-type 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 heavier-tailed 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 parent-offspring pairs in the Nhlsim data fall in the prediction ellipse for parent-offspring pairs (Figure 2, right panel). Similarly, all three full-sibling 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 half-sibling 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 half-sibling pair is a false-positive result. We observed (results not shown) that the number of false-positive 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 half-sibling pairs of subjects assuming independent SNPs (fitLD=FALSE), we obtain 16 false-positive 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 previously-fit LD models
By far the most computationally-demanding step of IBDcheck() is the fitting of LD models. The fitted LD models are stored in plain-text files in the working directory and can be re-used 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 re-filter the data. On the plot of cibd.user (not shown) the prediction ellipse for simulated mother-daughter 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 non-inbred 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 non-zero and the probability of 1 gene IBD is therefore smaller.
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 mouse-clicks, and returned in a data frame.
The methods implemented in CrypticIBDcheck are geared specifically towards exploring cryptic relatedness with data from candidate-gene 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 relationship-informative 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 325-350 SNPs from about 60 of the candidate genes appears to be adequate for identifying the parent-offspring 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 . For the Nhlsim data, it took 40 minutes to fit the LD model. For the close relationships that we consider, genome-wide 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 built-in to IBDcheck(), or specify their own relationships, as illustrated in the section Additional gene drops using previously-fit 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 two-stage 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.
We thank John Spinelli for access to the non-Hodgkin 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.
- Devlin B, Roeder K: Genomic control for association studies. Biometrics 1999, 55:997–1004.PubMedView Article
- Voight B, Pritchard J: Confounding from cryptic relatedness in case-control association studies. Plos Genet 2005, 1:e32.PubMedView Article
- 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):1146–1148.PubMedView Article
- R Development Core Team: R: A Language and Environment for Statistical Computing. Vienna: R Foundation for Statistical Computing; 2012. [http://www.R-project.org/]. [ISBN 3–900051–07–0]
- Weir B, Anderson A, Hepler A: Genetic relatedness analysis: modern data and new challenges. Nat Rev Genet 2006, 7:771–780.PubMedView Article
- Abecasis G, Cherny S, Cookson W, Cardon L: GRR: Graphical Representation of Relationship Errors. Bioinformatics 2001,17(8):742–743.PubMedView Article
- 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]
- Hill WG, Weir BS: Variation in actual relationship as a consequence of Mendelian sampling and linkage. Genet Res (Camb) 2011, 93:47–64.View Article
- Thomas A: Assessment of SNP streak statistics using gene drop simulation with linkage disequilibrium. Genet Epidemiol 2010, 34:119–124.PubMedView Article
- Thompson E: The estimation of pairwise relationships. Ann Human Genet 1975, 39:173–188.View Article
- Milligan B: Maximum-likelihood estimation of relatedness. Genetics 2003, 163:1153–1167.PubMed
- Choi Y, Wijsman E, Weir B: Case-control association testing in the presence of unknown relationships. Genet Epidemiol 2009, 33:668–678.PubMedView Article
- Ritland K: Estimators for pairwise relatedness and individual inbreeding coefficients. Genet Res 1996, 67:175–185.View Article
- Lynch M, Ritland K: Estimation of pairwise relatedness with molecular markers. Genetics 1999, 152:1753–1766.PubMed
- Purcell S, Neale B, Todd-Brown K, Thomas L, Ferreira M, Bender D, Maller J, Sklar P, de Bakker P, Daly M, Sham P: PLINK: a tool set for whole-genome association and population-based linkage analyses. Am J Human Genet 2007, 81:559–575.View Article
- 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, 25:1287–1292.View Article
- 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, 53:1818–1828.PubMedView Article
- Blay S, Graham J, McNeney B, Nembot-Simo A: rJPSGCS: R-Interface to Gene Drop Java Programs for Statistical Genetics and Computational Statistics (JPSGCS). 2011. [http://CRAN.R-project.org/package=rJPSGCS], [R package version 0.2–5]
- Leung HT: chopsticks: The snp.matrix and X.snp.matrix Classes. 2011. [http://outmodedbonsai.sourceforge.net/]. [R package version 1.18.3]
- Tierney L, Rossini AJ, Sevcikova H, Li N: snow: Simple Network of Workstations. 2011. [http://CRAN.R-project.org/package=snow]. [R package version 0.3–8]
- Schuetz JM, Daley D, Graham J, Berry BR, Gallagher RP, Connors JM, Gascoyne RD, Spinelli JJ, Brooks-Wilson AR: Genetic variation in cell death genes and risk of Non-Hodgkin Lymphoma. PLoS ONE 2012,7(2):e31560. [http://www.plosone.org/article/info\%3Adoi\%2F10.1371\%2Fjournal.pone.0031560]PubMedView Article
- Loredo-Osti 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]
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.