Sibship reconstruction from genetic data with typing errors

Likelihood methods have been developed to partition individuals in a sample into full-sib and half-sib families using genetic marker data without parental information. They invariably make the critical assumption that marker data are free of genotyping errors and mutations and are thus completely reliable in inferring sibships. Unfortunately, however, this assumption is rarely tenable for virtually all kinds of genetic markers in practical use and, if violated, can severely bias sibship estimates as shown by simulations in this article. I propose a new likelihood method with simple and robust models of typing error incorporated into it. Simulations show that the new method can be used to infer full- and half-sibships accurately from marker data with a high error rate and to identify typing errors at each locus in each reconstructed sib family. The new method also improves previous ones by adopting a fresh iterative procedure for updating allele frequencies with reconstructed sibships taken into account, by allowing for the use of parental information, and by using efficient algorithms for calculating the likelihood function and searching for the maximum-likelihood configuration. It is tested extensively on simulated data with a varying number of marker loci, different rates of typing errors, and various sample sizes and family structures and applied to two empirical data sets to demonstrate its usefulness.     

Estimating quantitative genetic parameters using sibships reconstructed from marker data

Previous techniques for estimating quantitative genetic parameters, such as heritability in populations where exact relationships are unknown but are instead inferred from marker genotypes, have used data from individuals on a pairwise level only. At this level, families are weighted according to the number of pairs within which each family appears, hence by size rather than information content, and information from multiple relationships is lost. Estimates of parameters are therefore not the most efficient achievable. Here, Markov chain Monte Carlo techniques have been used to partition the population into complete sibships, including, if known, prior knowledge of the distribution of family sizes. These pedigrees have then been used with restricted maximum likelihood under an animal model to estimate quantitative genetic parameters. Simulations to compare the properties of parameter estimates with those of existing techniques indicate that the use of sibship reconstruction is superior to earlier methods, having lower mean square errors and showing nonsignificant downward bias. In addition, sibship reconstruction allows the estimation of population allele frequencies that account for the relationships within the sample, so prior knowledge of allele frequencies need not be assumed. Extensions to these techniques allow reconstruction of half sibships when some or all of the maternal genotypes are known.     

Parentage and Sibship Inference From Multilocus Genotype Data Under Polygamy

Likelihood methods have been developed to partition individuals in a sample into sibling clusters using genetic marker data without parental information. Most of these methods assume either both sexes are monogamous to infer full sibships only or only one sex is polygamous to infer full sibships and paternal or maternal (but not both) half sibships. We extend our previous method to the more general case of both sexes being polygamous to infer full sibships, paternal half sibships, and maternal half sibships and to the case of a two-generation sample of individuals to infer parentage jointly with sibships. The extension not only expands enormously the scope of application of the method, but also increases its statistical power. The method is implemented for both diploid and haplodiploid species and for codominant and dominant markers, with mutations and genotyping errors accommodated. The performance and robustness of the method are evaluated by analyzing both simulated and empirical data sets. Our method is shown to be much more powerful than pairwise methods in both parentage and sibship assignments because of the more efficient use of marker information. It is little affected by inbreeding in parents and is moderately robust to nonrandom mating and linkage of markers. We also show that individually much less informative markers, such as SNPs or AFLPs, can reach the same power for parentage and sibship inferences as the highly informative marker simple sequence repeats (SSRs), as long as a sufficient number of loci are employed in the analysis.     

Genetic association analysis using sibship data: a multilevel model approach

Family based association study (FBAS) has the advantages of controlling for population stratification and testing for linkage and association simultaneously. We propose a retrospective multilevel model (rMLM) approach to analyze sibship data by using genotypic information as the dependent variable. Simulated data sets were generated using the simulation of linkage and association (SIMLA) program. We compared rMLM to sib transmission/disequilibrium test (S-TDT), sibling disequilibrium test (SDT), conditional logistic regression (CLR) and generalized estimation equations (GEE) on the measures of power, type I error, estimation bias and standard error. The results indicated that rMLM was a valid test of association in the presence of linkage using sibship data. The advantages of rMLM became more evident when the data contained concordant sibships. Compared to GEE, rMLM had less underestimated odds ratio (OR). Our results support the application of rMLM to detect gene-disease associations using sibship data. However, the risk of increasing type I error rate should be cautioned when there is association without linkage between the disease locus and the genotyped marker.     

Proximal spinal muscular atrophy (SMA) types II and III in the same sibship are not caused by different alleles at the SMA locus on 5q.

Proximal spinal muscular atrophy (SMA) is a group of progressive muscular diseases recently mapped to chromosome 5q. SMA is usually classified into types I-III, and there are cases of two types of SMA in the same sibship. Becker and others later proposed that these sibships might be due to the existence of several alleles at the same locus predisposing to the different forms of the disease. In a sample of four sibships in which both SMA type II and SMA type III occur, this hypothesis was clearly rejected for the SMA locus on 5q, by using information on the segregation of linked markers (P less than .001). Thus the difference between SMA type II and SMA type III is not due to different alleles at the SMA locus on 5q. This finding is suggestive of an involvement of other factors, genetic or environmental, in the determination of disease severity in SMA.     

Optimal sibship selection for genotyping in quantitative trait locus linkage analysis

In this paper we present a novel method for selecting optimally informative sibships of any size for quantitative trait locus (QTL) linkage analysis. The method allocates a quantitative index of potential informativeness to each sibship on the basis of observed trait scores and an assumed true QTL model. Any sample of phenotypically screened sibships can therefore be easily rank-ordered for selective genotyping. The quantitative index is the sibship's expected contribution to the non-centrality parameter. This expectation represents the weighted sum of chi(2) test statistics that would be obtained given the observed trait values over all possible sibship genotypic configurations; each configuration is weighted by the likelihood of it occurring given the assumed true genetic model. The properties of this procedure are explored in relation to the accuracy of the assumed true genetic model and sibship size. In comparison to previous methods of selecting phenotypically extreme sibships for genotyping, the proposed method is considerably more efficient and is robust with regard to the specification of the genetic model.     

Adding further power to the Haseman and Elston method for detecting linkage in larger sibships: weighting sums and differences

Haseman and Elston (H-E) proposed a robust test to detect linkage between a quantitative trait and a genetic marker. In their method the squared sib-pair trait difference is regressed on the estimated proportion of alleles at a locus shared identical by descent by sib pairs. This method has recently been improved by changing the dependent variable from the squared difference to the mean-corrected product of the sib-pair trait values, a significantly positive regression indicating linkage. Because situations arise in which the original test is more powerful, a further improvement of the H-E method occurs when the dependent variable is changed to a weighted average of the squared sib-pair trait difference and the squared sib-pair mean-corrected trait sum. Here we propose an optimal method of performing this weighting for larger sibships, allowing for the correlation between pairs within a sibship. The optimal weights are inversely proportional to the residual variances obtained from the two different regressions based on the squared sib-pair trait differences and the squared sib-pair mean-corrected trait sums, respectively, allowing for correlations among sib pairs. The proposed method is compared with the existing extension of the H-E approach for larger sibships. Control of the type I error probabilities for sibships of any size can be improved by using a generalized estimating equation approach and the robust sandwich estimate of the variance, or a Monte-Carlo permutation test.     

Likelihood-based genetic mark-recapture estimates when genotype samples are incomplete and contain typing errors

Genotypes produced from samples collected non-invasively in harsh field conditions often lack the full complement of data from the selected microsatellite loci. The application to genetic mark-recapture methodology in wildlife species can therefore be prone to misidentifications leading to both ‘true non-recaptures’ being falsely accepted as recaptures (Type I errors) and ‘true recaptures’ being undetected (Type II errors). Here we present a new likelihood method that allows every pairwise genotype comparison to be evaluated independently. We apply this method to determine the total number of recaptures by estimating and optimising the balance between Type I errors and Type II errors. We show through simulation that the standard error of recapture estimates can be minimised through our algorithms. Interestingly, the precision of our recapture estimates actually improved when we included individuals with missing genotypes, as this increased the number of pairwise comparisons potentially uncovering more recaptures. Simulations suggest that the method is tolerant to per locus error rates of up to 5% per locus and can theoretically work in datasets with as little as 60% of loci genotyped. Our methods can be implemented in datasets where standard mismatch analyses fail to distinguish recaptures. Finally, we show that by assigning a low Type I error rate to our matching algorithms we can generate a dataset of individuals of known capture histories that is suitable for the downstream analysis with traditional mark-recapture methods.     

Computationally efficient sibship and parentage assignment from multilocus marker data

Quite a few methods have been proposed to infer sibship and parentage among individuals from their multilocus marker genotypes. They are all based on Mendelian laws either qualitatively (exclusion methods) or quantitatively (likelihood methods), have different optimization criteria, and use different algorithms in searching for the optimal solution. The full-likelihood method assigns sibship and parentage relationships among all sampled individuals jointly. It is by far the most accurate method, but is computationally prohibitive for large data sets with many individuals and many loci. In this article I propose a new likelihood-based method that is computationally efficient enough to handle large data sets. The method uses the sum of the log likelihoods of pairwise relationships in a configuration as the score to measure its plausibility, where log likelihoods of pairwise relationships are calculated only once and stored for repeated use. By analyzing several empirical and many simulated data sets, I show that the new method is more accurate than pairwise likelihood and exclusion-based methods, but is slightly less accurate than the full-likelihood method. However, the new method is computationally much more efficient than the full-likelihood method, and for the cases of both sexes polygamous and markers with genotyping errors, it can be several orders faster. The new method can handle a large sample with thousands of individuals and the number of markers limited only by the computer memory.     

Accurate partition of individuals into full-sib families from genetic data without parental information.

Two Markov chain Monte Carlo algorithms are proposed that allow the partitioning of individuals into full-sib groups using single-locus genetic marker data when no parental information is available. These algorithms present a method of moving through the sibship configuration space and locating the configuration that maximizes an overall score on the basis of pairwise likelihood ratios of being full-sib or unrelated or maximizes the full joint likelihood of the proposed family structure. Using these methods, up to 757 out of 759 Atlantic salmon were correctly classified into 12 full-sib families of unequal size using four microsatellite markers. Large-scale simulations were performed to assess the sensitivity of the procedures to the number of loci and number of alleles per locus, the allelic distribution type, the distribution of families, and the independent knowledge of population allelic frequencies. The number of loci and the number of alleles per locus had the most impact on accuracy. Very good accuracy can be obtained with as few as four loci when they have at least eight alleles. Accuracy decreases when using allelic frequencies estimated in small target samples with skewed family distributions with the pairwise likelihood approach. We present an iterative approach that partly corrects that problem. The full likelihood approach is less sensitive to the precision of allelic frequencies estimates but did not perform as well with the large data set or when little information was available (e.g., four loci with four alleles).     

Robustness and power of the maximum-likelihood-binomial and maximum-likelihood-score methods,in multipoint linkage analysis of affected-sibship data.

The maximum-likelihood-binomial (MLB) method, based on the binomial distribution of parental marker alleles among affected offspring, recently was shown to provide promising results by two-point linkage analysis of affected-sibship data. In this article, we extend the MLB method to multipoint linkage analysis, using the general framework of hidden Markov models. Furthermore, we perform a large simulation study to investigate the robustness and power of the MLB method, compared with those of the maximum-likelihood-score (MLS) method as implemented in MAPMAKER/SIBS, in the multipoint analysis of different affected-sibship samples. Analyses of multiple-affected sibships by means of the MLS were conducted by consideration of all possible sib pairs, with (weighted MLS [MLSw]) or without (unweighted MLS [MLSu]) application of a classic weighting procedure. In simulations under the null hypothesis, the MLB provided very consistent type I errors regardless of the type of family sample (sib pairs or multiple-affected sibships), as did the MLS for samples with sib pairs only. When samples included multiple-affected sibships, the MLSu led to inflation of low type I errors, whereas the MLSw yielded very conservative tests. Power comparisons showed that the MLB generally was more powerful than the MLS, except in recessive models with allele frequencies <.3. Missing parental marker data did not strongly influence type I error and power results in these multipoint analyses. The MLB approach, which in a natural way accounts for multiple-affected sibships and which provides a simple likelihood-ratio test for linkage, is an interesting alternative for multipoint analysis of sibships.     

Ascertainment: An Overwiew of the Classical Segregation Analysis Model for Independent Sibships

Several different methodologies for parameter estimation under various ascertainment sampling schemes have been proposed in the past. In this article, some of the methodologies that have been proposed for independent sibships under the classical segregation analysis model are synthesized, and the general likelihoods derived for single, multiple and complete ascertainment. The issue of incorporating the sibship size distribution into the analysis is addressed, and the effect of conditioning the likelihood on the observed sibship sizes is discussed. It is shown that when the number of probands in a sibship is not specified, the corresponding likelihood can be used for a broader class of ascertainment schemes than is subsumed by the classical model.     

A survey of affected-sibship statistics for nonparametric linkage analysis

We have compared the power of a large number of allele-sharing statistics for "nonparametric" linkage analysis with affected sibships. Our rationale was that there is an extensive literature comparing statistics for sibling pairs but that there has not been much guidance on how to choose statistics for studies that include sibships of various sizes. We concentrated on statistics that can be described as assigning scores to each identity-by-descent-sharing configuration that a pedigree might take on (Whittemore and Halpern 1994). We considered sibships of sizes two through five, 27 different genetic models, and varying recombination fractions between the marker and the trait locus. We tried to identify statistics whose power was robust over a wide variety of models. We found that the statistic that is probably used most often in such studies-S(all)-performs quite well, although it is not necessarily the best. We also found several other statistics (such as the R criterion, S(robdom), and the Sobel-and-Lange statistic C) that perform well in most situations, a few (such as S(-#geno) and the Feingold-and-Siegmund version of S(pairs)) that have high power only in very special situations, and a few (such as S(-#geno), the N criterion, and the Sobel-and-Lange statistic B) that seem to have low power for the majority of the trait models. For the most part, the same statistics performed well for all sibship sizes. We also used our results to give some suggestions regarding how to weight sibships of different sizes, in forming an overall statistic.     

Sibship reconstruction in hierarchical population structures using Markov chain Monte Carlo techniques

Markov chain Monte Carlo procedures allow the reconstruction of full-sibships using data from genetic marker loci only. In this study, these techniques are extended to allow the reconstruction of nested full- within half-sib families, and to present an efficient method for calculating the likelihood of the observed marker data in a nested family. Simulation is used to examine the properties of the reconstructed sibships, and of estimates of heritability and common environmental variance of quantitative traits obtained from those populations. Accuracy of reconstruction increases with increasing marker information and with increasing size of the nested full-sibships, but decreases with increasing population size. Estimates of variance component are biased, with the direction and magnitude of bias being dependent upon the underlying errors made during pedigree reconstruction.     

A Random Model Approach to Interval Mapping of Quantitative Trait Loci

Mapping quantitative trait loci in outbred populations is important because many populations of organisms are noninbred. Unfortunately, information about the genetic architecture of the trait may not be available in outbred populations. Thus, the allelic effects of genes can not be estimated with ease. In addition, under linkage equilibrium, marker genotypes provide no information about the genotype of a QTL (our terminology for a single quantitative trait locus is QTL while multiple loci are referred to as QTLs). To circumvent this problem, an interval mapping procedure based on a random model approach is described. Under a random model, instead of estimating the effects, segregating variances of QTLs are estimated by a maximum likelihood method. Estimation of the variance component of a QTL depends on the proportion of genes identical-by-descent (IBD) shared by relatives at the locus, which is predicted by the IBD of two markers flanking the QTL. The marker IBD shared by two relatives are inferred from the observed marker genotypes. The procedure offers an advantage over the regression interval mapping in terms of high power and small estimation errors and provides flexibility for large sibships, irregular pedigree relationships and incorporation of common environmental and fixed effects.     

A modified revisited Haseman-Elston method to further improve power

The original and revisited Haseman-Elston methods are simple robust methods to detect linkage, but neither is uniformly optimal in terms of power. In this report, we propose a simple modification of the revisited Haseman-Elston method that retains the simplicity and robustness properties, but increases its power. We demonstrate theoretically that the modification can be more powerful than the optimally weighted Haseman-Elston method when the sibship mean can be correctly specified. We then examine the properties of this modification by simulation when the sibship mean is replaced by its best linear unbiased predictor. The simulation results indicate that this modification maintains good control over type I error, even in the case of larger sibships, and that the empirical power of this modification is similar to that of the optimally weighted Haseman-Elston method in most cases.     

Efficiency of typing unaffected relatives in an affected-sib-pair linkage study with single-locus and multiple tightly linked markers.

In an affected-sib-pair study, the parents are often unavailable for typing, particularly for diseases of late onset. In many cases, however, it is possible to sample unaffected siblings. It is therefore desirable to assess the contribution of such siblings to the power of such a study. The likelihood ratio introduced by Risch and improved by Holmans was extended to incorporate data from unaffected siblings. Tests based on two likelihoods were considered: the full likelihood of the data, based on the identity-by-descent (IBD) sharing states of the entire sibship, and a pseudolikelihood based on the IBD sharing states of the affected pair only, using the unaffected siblings to infer parental genotypes. The latter approach was found to be more powerful, except when penetrance was high. Typing an unaffected sibling, or just one parent, was found to give only a small increase in power except when the PIC of the marker was low. Even then, typing an unaffected relative increased the overall number of individuals that had to be typed to achieve a given power. If there is no highly informative marker locus in the area under study, it may be possible to "build" one by combining the alleles from two or more neighboring tightly linked loci into haplotypes. Typing two loci gave a sizeable power increase over a single locus, but typing further loci gave much smaller gains. Building haplotypes will introduce phase uncertainties, with the result that such a system will yield less power than will a single locus with the same number of alleles. This power loss was small, however, and did not affect the conclusions regarding the worth of typing unaffected relatives.     

Susceptibility of biallelic haplotype and genotype frequencies to genotyping error

With the availability of fast genotyping methods and genomic databases, the search for statistical association of single nucleotide polymorphisms with a complex trait has become an important methodology in medical genetics. However, even fairly rare errors occurring during the genotyping process can lead to spurious association results and decrease in statistical power. We develop a systematic approach to study how genotyping errors change the genotype distribution in a sample. The general M-marker case is reduced to that of a single-marker locus by recognizing the underlying tensor-product structure of the error matrix. Both method and general conclusions apply to the general error model; we give detailed results for allele-based errors of size depending both on the marker locus and the allele present. Multiple errors are treated in terms of the associated diffusion process on the space of genotype distributions. We find that certain genotype and haplotype distributions remain unchanged under genotyping errors, and that genotyping errors generally render the distribution more similar to the stable one. In case-control association studies, this will lead to loss of statistical power for nondifferential genotyping errors and increase in type I error for differential genotyping errors. Moreover, we show that allele-based genotyping errors do not disturb Hardy-Weinberg equilibrium in the genotype distribution. In this setting we also identify maximally affected distributions. As they correspond to situations with rare alleles and marker loci in high linkage disequilibrium, careful checking for genotyping errors is advisable when significant association based on such alleles/haplotypes is observed in association studies.     

Association tests in nuclear families

We present a conditional likelihood approach for testing linkage disequilibrium in nuclear families having multiple affected offspring. The likelihood, conditioned on the identity-by-descent (IBD) structure of the sibling genotypes, is unaffected by familial correlation in disease status that arises from linkage between a marker locus and the unobserved trait locus. Two such conditional likelihoods are compared: one that conditions on IBD and phase of the transmitted alleles and a second which conditions only on IBD of the transmitted alleles. Under the log-additive model, the first likelihood is equivalent to the allele-counting methods proposed in the literature. The second likelihood is valid under the added assumption of equal male and female recombination fractions. In a simulation study, we demonstrated that in sibships having two or three affected siblings the score test from each likelihood had the correct test size for testing disequilibrium. They also led to equivalent power to detect linkage disequilibrium at the 5% significance level.     

Score tests of genetic association in the presence of linkage based on the additive genetic gamma frailty model

Nuclear families with multiple affected sibs are often collected for genetic linkage analysis of complex diseases. Once linkage evidence is established, dense markers are often typed in the linked region for genetic association analysis based on linkage disequilibrium (LD). Detection of association in the presence of linkage localizes disease genes more accurately than the methods that rely on linkage alone. However, test of association due to LD in the linked region needs to account for dependency of the allele transmissions to different sibs within a family. In this paper, we define a joint model for genetic linkage and association and derive the corresponding joint survival function of age of onset for the sibs within a sibship. The joint survival function is a function of both the inheritance vector and the genotypes at the candidate marker locus. Based on this joint survival function, we derive score tests for genetic association. The proposed methods utilize the phenotype data of all the sibs and have the advantages of family-based designs which can avoid the potential spurious association caused by population admixture. In addition, the methods can account for variable age of onset or age at censoring and possible covariate effects, and therefore provide important tools for modelling disease heterogeneity. Simulation studies and application to the data sets from the 12th Genetic Analysis Workshop indicate that the proposed methods have correct type 1 error rates and increased power over other existing methods for testing allelic association.