Robust multipoint simultaneous identical-by-descent mapping for two linked loci

A challenging issue in genetic mapping of complex human diseases is localizing disease susceptibility genes when the genetic effects are small to moderate. There are greater complexities when multiple loci are linked to a chromosomal region. Liang et al. [Hum Hered 2001;51:64-78] proposed a robust multipoint method that can simultaneously estimate both the position of a trait locus and its effect on disease status by using affected sib pairs (ASPs). Based on the framework of generalized estimating equations (GEEs), the estimate and standard error of the position of a trait locus are robust to different genetic models. To utilize other relative pairs collected in pedigree data, Schaid et al. [Am J Hum Genet 2005;76:128-138] extended Liang's method to various types of affected relative pairs (ARPs) by two approaches: unconstrained and constrained methods. However, the above methods are limited to situations in which only one trait locus exists on the chromosome of interest. The mean functions are no longer correctly specified when there are multiple causative loci linked to a chromosomal region. To overcome this, Biernacka et al. [Genet Epidemiol 2005;28:33-47] considered the multipoint methods for ASPs to allow for two linked disease genes. We further generalize the approach to cover other types of ARPs. To reflect realistic situations for complex human diseases, we set modest sizes of genetic effects in our simulation. Our results suggest that several hundred independent pedigrees are needed, and markers with high information, to provide reliable estimates of trait locus positions and their confidence intervals. Bootstrap resampling can correct the downward bias of the robust variance for location estimates. These methods are applied to a prostate cancer linkage study on chromosome 20 and compared with the results for the one-locus model [Am J Hum Genet 2005;76:128-138]. We have implemented the multipoint IBD mapping for one and two linked loci in our software GEEARP, which allows analyses for five general types of ARPs.     

Multipoint quantitative-trait linkage analysis in general pedigrees.

Multipoint linkage analysis of quantitative-trait loci (QTLs) has previously been restricted to sibships and small pedigrees. In this article, we show how variance-component linkage methods can be used in pedigrees of arbitrary size and complexity, and we develop a general framework for multipoint identity-by-descent (IBD) probability calculations. We extend the sib-pair multipoint mapping approach of Fulker et al. to general relative pairs. This multipoint IBD method uses the proportion of alleles shared identical by descent at genotyped loci to estimate IBD sharing at arbitrary points along a chromosome for each relative pair. We have derived correlations in IBD sharing as a function of chromosomal distance for relative pairs in general pedigrees and provide a simple framework whereby these correlations can be easily obtained for any relative pair related by a single line of descent or by multiple independent lines of descent. Once calculated, the multipoint relative-pair IBDs can be utilized in variance-component linkage analysis, which considers the likelihood of the entire pedigree jointly. Examples are given that use simulated data, demonstrating both the accuracy of QTL localization and the increase in power provided by multipoint analysis with 5-, 10-, and 20-cM marker maps. The general pedigree variance component and IBD estimation methods have been implemented in the SOLAR (Sequential Oligogenic Linkage Analysis Routines) computer package.     

A two-stage variable-stringency semiparametric method for mapping quantitative-trait loci with the use of genomewide-scan data on sib pairs

Genomewide scans for mapping loci have proved to be extremely powerful and popular. We present a semiparametric method of mapping a quantitative-trait locus (QTL) or QTLs with the use of sib-pair data generated from a two-stage genomic scan. In a two-stage genomic scan, either the entire genome or a large portion of the genome is saturated with low-density markers at the first stage. At the second stage, the intervals that are identified as probable locations of the trait loci, by means of analysis of data from the first stage, are then saturated with higher-density markers. These data are then analyzed for fine mapping of the loci. Our statistical strategy for analysis of data from the first stage is a low-stringency method based on the rank correlation of squared trait-difference values of the sib pairs and the estimated identity-by-descent scores at the marker loci. We suggest the use of a low-stringency method at the first stage, to save on computational time and to avoid missing any marker interval that may contain the trait loci. For analysis of data from the second stage, we have developed a high-stringency nonparametric-regression approach, using the kernel-smoothing technique. Through extensive simulations, we show that this approach is more powerful than is a currently used method for mapping QTLs by use of sib pairs, particularly in the presence of dominance and epistatic effects at the trait loci.     

A general statistical model for detecting complex-trait loci by using affected relative pairs in a genome search.

Scanning of the human genome by use of affected relative pairs and dense sets of highly polymorphic markers or by emerging techniques such as genomic mismatch scanning. (GMS) is making it possible to identify the genetic etiology of a disease through detection of susceptibility loci. We present a general statistical model and test to detect disease genes, using affected relative pairs and either markers or GMS technologies in a genome search. There are an exact test and large-sample normal approximation that control for the elevated probability of false detection of linkage in a genome search. The approach can be used to determine the sample size needed to obtain a prespecified power to detect a disease gene in the presence of etiologic heterogeneity for a single class or mixture of relative classes, with any number of markers, or clones, markers PIC values, or mapping function. The approach is used to examine differences in performance of markers and GMS technologies in a common statistical framework and to provide practical information for designing studies of complex traits.     

A comprehensive method for genome scans

In applications involving the use of genome scans the problem of correcting for multiple testing figures prominently. A frequently used approach is the Bonferroni adjustment, but this is known to be often severely conservative. As an alternative we use the method of importance sampling to accurately and efficiently obtain required exceedance probabilities. This method is comprehensive in the sense that it has application to exceedance probabilities for other classes of test statistics, such as those for linkage disequilibrium or Hardy-Weinberg equilibrium at multiple loci. We illustrate the importance sampling technique by focusing on affected sib pair tests done at a large number of fully informative markers. We demonstrate how our approach can be used to obtain exceedance probabilities for arbitrary marker spacings, and we compare our approach with that of Feingold et al. [1993], which uses the method of large deviations and does not provide the means for adjusting for unequal marker spacing.     

Testing the robustness of the new Haseman-Elston quantitative-trait loci-mapping procedure

Variance components (VC) techniques have emerged as among the more powerful methods for detection of quantitative-trait loci (QTL) in linkage analysis. Allison et al. found that, with particularly marked leptokurtosis in the phenotypic distribution and moderate-to-high residual sibling correlation, maximum likelihood (ML) VC methods may produce a severe excess of type I errors. The new Haseman-Elston (NHE) method is a least-squares-based VC method for mapping of QTL in sib pairs (Elston et al.). Using simulation, we investigate the robustness of the NHE to marked nonnormality, by means of the same distributions and worst-case conditions identified by Allison et al. for the ML approach (i.e., 100 pairs; high residual sibling correlation). Results showed that, when marked nonnormality is present, the NHE can be used without severe type I error-rate inflation, even at very small alpha levels.     

A joint model for longitudinal measurements and survival data in the presence of multiple failure types

Summary .   In this article we study a joint model for longitudinal measurements and competing risks survival data. Our joint model provides a flexible approach to handle possible nonignorable missing data in the longitudinal measurements due to dropout. It is also an extension of previous joint models with a single failure type, offering a possible way to model informatively censored events as a competing risk. Our model consists of a linear mixed effects submodel for the longitudinal outcome and a proportional cause-specific hazards frailty submodel ( Prentice et al., 1978 , Biometrics 34, 541–554) for the competing risks survival data, linked together by some latent random effects. We propose to obtain the maximum likelihood estimates of the parameters by an expectation maximization (EM) algorithm and estimate their standard errors using a profile likelihood method. The developed method works well in our simulation studies and is applied to a clinical trial for the scleroderma lung disease.     

Robust multipoint identical-by-descent mapping for affected relative pairs

The genetic mapping of complex traits has been challenging and has required new statistical methods that are robust to misspecified models. Liang et al. proposed a robust multipoint method that can be used to simultaneously estimate, on the basis of sib-pair linkage data, both the position of a trait locus on a chromosome and its effect on disease status. The advantage of their method is that it does not require specification of an underlying genetic model, so estimation of the position of a trait locus on a specified chromosome and of its standard error is robust to a wide variety of genetic mechanisms. If multiple loci influence the trait, the method models the marginal effect of a locus on a specified chromosome. The main critical assumption is that there is only one trait locus on the chromosome of interest. We extend this method to different types of affected relative pairs (ARPs) by two approaches. One approach is to estimate the position of a trait locus yet allow unconstrained trait-locus effects across different types of ARPs. This robust approach allows for differences in sharing alleles identical-by-descent across different types of ARPs. Some examples for which an unconstrained model would apply are differences due to secular changes in diagnostic methods that can change the frequency of phenocopies among different types of relative pairs, environmental factors that modify the genetic effect, epistasis, and variation in marker-information content. However, this unconstrained model requires a parameter for each type of relative pair. To reduce the number of parameters, we propose a second approach that models the marginal effect of a susceptibility locus. This constrained model is robust for a trait caused by either a single locus or by multiple loci without epistasis. To evaluate the adequacy of the constrained model, we developed a robust score statistic. These methods are applied to a prostate cancer-linkage study, which emphasizes their potential advantages and limitations.     

A genome-wide search for genes predisposing to familial psoriasis by using a stratification approach

We have performed a genome scan, using markers spaced by 10 cM, in the search for psoriasis-susceptibility loci. The family material of 134 affected sibling pairs was ascertained on the basis of a population genetic study in which 65% of the probands had two healthy parents. Genotyping results were analyzed for non-random excessive allele-sharing between sib pairs by using GENEHUNTER ver 1.1. A stratification approach was applied to increase the homogeneity of the material by means of an operational definition of joint complaints among affected individuals. Significant linkage to the human leukocyte antigen region on chromosome 6p in a cohort including 42 families without joint complaints (nonparametric linkage score of 2.83, P=0.002) strongly supported the validity of this operational definition as it replicated results from an earlier linkage report with similar stratification criteria. New candidate regions on chromosomes 3 and 15 were identified. The highest non-parametric linkage values in this study, 2.96 (P=0.0017) and 2.89 (P=0.0020), were reached on chromosome 15 in a subgroup with joint complaints and on chromosome 3 in a subgroup without joint complaints. In addition, confirmation of previously reported loci was established on chromosomes 4q, 6p, and 17q. This study indicates that distinct disease loci might be involved in psoriasis etiology for various phenotypes.     

Comparison of family-based association tests in chromosome regions selected by linkage-based confidence intervals

We use the Genetic Analysis Workshop 14 simulated data to explore the effectiveness of a two-stage strategy for mapping complex disease loci consisting of an initial genome scan with confidence interval construction for gene location, followed by fine mapping with family-based tests of association on a dense set of single-nucleotide polymorphisms. We considered four types of intervals: the 1-LOD interval, a basic percentile bootstrap confidence interval based on the position of the maximum Zlr score, and asymptotic and bootstrap confidence intervals based on a generalized estimating equations method. For fine mapping we considered two family-based tests of association: a test based on a likelihood ratio statistic and a transmission-disequilibrium-type test implemented in the software FBAT. In two of the simulation replicates, we found that the bootstrap confidence intervals based on the peak Zlr and the 1-LOD support interval always contained the true disease loci and that the likelihood ratio test provided further strong confirmatory evidence of the presence of disease loci in these regions.     

Multilocus linkage analysis of affected sib pairs

OBJECTIVE: The conventional affected sib pair methods evaluate the linkage information at a locus by considering only marginal information. We describe a multilocus linkage method that uses both the marginal information and information derived from the possible interactions among several disease loci, thereby increasing the significance of loci with modest effects. METHODS: Our method is based on a statistic that quantifies the linkage information contained in a set of markers. By a marker selection-reduction process, we screen a set of polymorphisms and select a few that seem linked to disease. RESULTS: We test our approach on genome scan data for inflammatory bowel disease (InfBD) and on simulated data. On real data we detect 6 of the 8 known InfBD loci; on simulated data we obtain improvements in power of up to 40% compared to a conventional single-locus method. CONCLUSION: Our extensive simulations and the results on real data show that our method is in general more powerful than single-locus methods in detecting disease loci responsible for complex traits. A further advantage of our approach is that it can be extended to make use of both the linkage and the linkage disequilibrium between disease loci and nearby markers.     

Model-free linkage analysis with covariates confirms linkage of prostate cancer to chromosomes 1 and 4

As with many complex genetic diseases, genome scans for prostate cancer have given conflicting results, often failing to provide replication of previous findings. One factor contributing to the lack of consistency across studies is locus heterogeneity, which can weaken or even eliminate evidence for linkage that is present only in a subset of families. Currently, most analyses either fail to account for locus heterogeneity or attempt to account for it only by partitioning data sets into smaller and smaller portions. In the present study, we model locus heterogeneity among affected sib pairs with prostate cancer by including covariates in the linkage analysis that serve as surrogate measures of between-family linkage differences. The model is a modification of the Olson conditional logistic model for affected relative pairs. By including Gleason score, age at onset, male-to-male transmission, and/or number of affected first-degree family members as covariates, we detected linkage near three locations that were previously identified by linkage (1q24-25 [HPC1; LOD score 3.25, P=.00012], 1q42.2-43 [PCAP; LOD score 2.84, P=.0030], and 4q [LOD score 2.80, P=.00038]), near the androgen-receptor locus on Xq12-13 (AR; LOD score 3.06, P=.00053), and at five new locations (LOD score > 2.5). Without covariates, only a few weak-to-moderate linkage signals were found, none of which replicate findings of previous genome scans. We conclude that covariate-based linkage analysis greatly improves the likelihood that linked regions will be found by incorporation of information about heterogeneity within the sample.     

Age-related macular degeneration: a high-resolution genome scan for susceptibility loci in a population enriched for late-stage disease

Age-related macular degeneration (AMD) is a complex multifactorial disease that affects the central region of the retina. AMD is clinically heterogeneous, leading to geographic atrophy (GA) and/or choroidal neovascularization (CNV) at advanced stages. Considerable data exists in support of a genetic predisposition for AMD. Recent linkage studies have provided evidence in favor of several AMD susceptibility loci. We have performed a high-resolution (5-cM) genome scan of 412 affected relative pairs that were enriched for late-stage disease (GA and/or CNV). Nonparametric linkage analysis was performed using two different diagnostic criteria and also by dividing the affected individuals according to GA or CNV phenotype. Our results demonstrate evidence of linkage in regions that were suggested in at least one previous study at chromosomes 1q (236-240 cM in the Marshfield genetic map), 5p (40-50 cM), and 9q (111 cM). Multipoint analysis of affected relatives with CNV provided evidence of additional susceptibility loci on chromosomes 2p (10 cM) and 22q (25 cM). A recently identified Gln5345Arg change in HEMICENTIN-1 on chromosome 1q25 was not detected in 274 affected members in the restricted group with AMD, 346 additional patients with AMD, and 237 unaffected controls. Our results consolidate the chromosomal locations of several AMD susceptibility loci and, together with previous reports, should facilitate the search for disease-associated sequence variants.     

A Maximum Likelihood Method for Estimating Genome Length Using Genetic Linkage Data

The genetic length of a genome, in units of Morgans or centimorgans, is a fundamental characteristic of an organism. We propose a maximum likelihood method for estimating this quantity from counts of recombinants and nonrecombinants between marker locus pairs studied from a backcross linkage experiment, assuming no interference and equal chromosome lengths. This method allows the calculation of the standard deviation of the estimate and a confidence interval containing the estimate. Computer simulations have been performed to evaluate and compare the accuracy of the maximum likelihood method and a previously suggested method-of-moments estimator. Specifically, we have investigated the effects of the number of meioses, the number of marker loci, and variation in the genetic lengths of individual chromosomes on the estimate. The effect of missing data, obtained when the results of two separate linkage studies with a fraction of marker loci in common are pooled, is also investigated. The maximum likelihood estimator, in contrast to the method-of-moments estimator, is relatively insensitive to violation of the assumptions made during analysis and is the method of choice. The various methods are compared by application to partial linkage data from Xiphophorus.     

Positional cloning by linkage disequilibrium

Recently, metric linkage disequilibrium (LD) maps that assign an LD unit (LDU) location for each marker have been developed (Maniatis et al. 2002). Here we present a multiple pairwise method for positional cloning by LD within a composite likelihood framework and investigate the operating characteristics of maps in physical units (kb) and LDU for two bodies of data (Daly et al. 2001; Jeffreys et al. 2001) on which current ideas of blocks are based. False-negative indications of a disease locus (type II error) were examined by selecting one single-nucleotide polymorphism (SNP) at a time as causal and taking its allelic count (0, 1, or 2, for the three genotypes) as a pseudophenotype, Y. By use of regression and correlation, association between every pseudophenotype and the allelic count of each SNP locus (X) was based on an adaptation of the Malecot model, which includes a parameter for location of the putative gene. By expressing locations in kb or LDU, greater power for localization was observed when the LDU map was fitted. The efficiency of the kb map, relative to the LDU map, to describe LD varied from a maximum of 0.87 to a minimum of 0.36, with a mean of 0.62. False-positive indications of a disease locus (type I error) were examined by simulating an unlinked causal SNP and the allele count was used as a pseudophenotype. The type I error was in good agreement with Wald's likelihood theorem for both metrics and all models that were tested. Unlike tests that select only the most significant marker, haplotype, or haploset, these methods are robust to large numbers of markers in a candidate region. Contrary to predictions from tagging SNPs that retain haplotype diversity, the sample with smaller size but greater SNP density gave less error. The locations of causal SNPs were estimated with the same precision in blocks and steps, suggesting that block definition may be less useful than anticipated for mapping a causal SNP. These results provide a guide to efficient positional cloning by SNPs and a benchmark against which the power of positional cloning by haplotype-based alternatives may be measured.     

Transmission-ratio distortion and allele sharing in affected sib pairs: a new linkage statistic with reduced bias, with application to chromosome 6q25.3

We studied the effect of transmission-ratio distortion (TRD) on tests of linkage based on allele sharing in affected sib pairs. We developed and implemented a discrete-trait allele-sharing test statistic, Sad, analogous to the Spairs test statistic of Whittemore and Halpern, that evaluates an excess sharing of alleles at autosomal loci in pairs of affected siblings, as well as a lack of sharing in phenotypically discordant relative pairs, where available. Under the null hypothesis of no linkage, nuclear families with at least two affected siblings and one unaffected sibling have a contribution to Sad that is unbiased, with respect to the effects of TRD independent of the disease under study. If more distantly related unaffected individuals are studied, the bias of Sad is generally reduced compared with that of Spairs, but not completely. Moreover, Sad has higher power, in some circumstances, because of the availability of unaffected relatives, who are ignored in affected-only analyses. We discuss situations in which it may be an efficient use of resources to genotype unaffected relatives, which would give insights for promising study designs. The method is applied to a sample of pedigrees ascertained for asthma in a chromosomal region in which TRD has been reported. Results are consistent with the presence of transmission distortion in that region.     

Linkage and association mapping of the LRP5 locus on chromosome 11q13 in type 1 diabetes

Linkage of chromosome 11q13 to type 1 diabetes (T1D) was first reported from genome scans (Davies et al. 1994; Hashimoto et al. 1994) resulting in P <2.2 x 10(-5) (Luo et al. 1996) and designated IDDM4 ( insulin dependent diabetes mellitus 4). Association mapping under the linkage peak using 12 polymorphic microsatellite markers suggested some evidence of association with a two-marker haplotype, D11S1917*03-H0570POLYA*02, which was under-transmitted to affected siblings and over-transmitted to unaffected siblings ( P=1.5 x 10(-6)) (Nakagawa et al. 1998). Others have reported evidence for T1D association of the microsatellite marker D11S987, which is approximately 100 kb proximal to D11S1917 (Eckenrode et al. 2000). We have sequenced a 400-kb interval surrounding these loci and identified four genes, including the low-density lipoprotein receptor related protein (LRP5) gene, which has been considered as a functional candidate gene for T1D (Hey et al. 1998; Twells et al. 2001). Consequently, we have developed a comprehensive SNP map of the LRP5 gene region, and identified 95 SNPs encompassing 269 kb of genomic DNA, characterised the LD in the region and haplotypes (Twells et al. 2003). Here, we present our refined linkage curve of the IDDM4 region, comprising 32 microsatellite markers and 12 SNPs, providing a peak MLS=2.58, P=5 x 10(-4), at LRP5 g.17646G>T. The disease association data, largely focused in the LRP5 region with 1,106 T1D families, provided no further evidence for disease association at LRP5 or at D11S987. A second dataset, comprising 1,569 families from Finland, failed to replicate our previous findings at LRP5. The continued search for the variants of the putative IDDM4 locus will greatly benefit from the future development of a haplotype map of the genome.     

Optimal strategies for mapping complex diseases in the presence of multiple loci.

Recent advances in genome technology have led to mapping and subsequent isolation, by positional cloning, of a number of genes for common and/or complex human diseases. It therefore will be possible to utilize information about a known locus in the search for additional, perhaps less penetrant, genes for a particular disease. It is also unclear, under these situations, what the optimal sampling strategy should be. To address these questions, we have calculated the expected LOD score for localizing one locus in a variety of two-locus models of disease, for four different pedigree structures, and under three different scenarios regarding knowledge/testing of one of the two loci. These design considerations are evaluated by use of a cost function that incorporates the costs of ascertaining different family structures, the relative costs of genotyping and mutation testing family members, and the amount of information provided by each family structure and testing scenario. The results indicate that, in most cases, affected sib pairs are a particularly poor strategy, especially when linkage or mutation data are available at the known locus. We also demonstrate that prescreening the sample of families for mutations at known susceptibility loci is, in general, a cost-effective strategy.     

Likelihood analysis of ongoing gene flow and historical association

Abstract.— We develop a Monte Carlo-based likelihood method for estimating migration rates and population divergence times from data at unlinked loci at which mutation rates are sufficiently low that, in the recent past, the effects of mutation can be ignored. The method is applicable to restriction fragment length polymorphisms (RFLPs) and single nucleotide polymorphisms (SNPs) sampled from a subdivided population. The method produces joint maximum-likelihood estimates of the migration rate and the time of population divergence, both scaled by population size, and provides a framework in which to test either for no ongoing gene flow or for population divergence in the distant past. We show the method performs well and provides reasonably accurate estimates of parameters even when the assumptions under which those estimates are obtained are not completely satisfied. Furthermore, we show that, provided that the number of polymorphic loci is sufficiently large, there is some power to distinguish between ongoing gene flow and historical association as causes of genetic similarity between pairs of populations.