Further evidence for the increased power of LOD scores compared with nonparametric methods.

In genetic analysis of diseases in which the underlying model is unknown, "model free" methods-such as affected sib pair (ASP) tests-are often preferred over LOD-score methods, although LOD-score methods under the correct or even approximately correct model are more powerful than ASP tests. However, there might be circumstances in which nonparametric methods will outperform LOD-score methods. Recently, Dizier et al. reported that, in some complex two-locus (2L) models, LOD-score methods with segregation analysis-derived parameters had less power to detect linkage than ASP tests. We investigated whether these particular models, in fact, represent a situation that ASP tests are more powerful than LOD scores. We simulated data according to the parameters specified by Dizier et al. and analyzed the data by using a (a) single locus (SL) LOD-score analysis performed twice, under a simple dominant and a recessive mode of inheritance (MOI), (b) ASP methods, and (c) nonparametric linkage (NPL) analysis. We show that SL analysis performed twice and corrected for the type I-error increase due to multiple testing yields almost as much linkage information as does an analysis under the correct 2L model and is more powerful than either the ASP method or the NPL method. We demonstrate that, even for complex genetic models, the most important condition for linkage analysis is that the assumed MOI at the disease locus being tested is approximately correct, not that the inheritance of the disease per se is correctly specified. In the analysis by Dizier et al., segregation analysis led to estimates of dominance parameters that were grossly misspecified for the locus tested in those models in which ASP tests appeared to be more powerful than LOD-score analyses.     

Conclusions of segregation analysis for family data generated under two-locus models.

Susceptibility to a disease may involve the interactive effect of two genes. What conclusions will be drawn by segregation analysis in such a case? To answer this question, we considered a set of two-locus models and the corresponding exact distribution for 300 families. We investigated the conclusions and parameter estimations obtained for this sample, by comparing the likelihood expectations of the unified model and of more restricted models. In many cases, segregation analysis leads to the conclusion of a major gene effect, with or without a polygenic component--usually without a polygenic component in multiplicative models (i.e., where two genes have a multiplicative effect) and with such a component in nonmultiplicative models. For all the models considered, existence of a major gene effect is supported by transmission probability tests; there is evidence for transmission and agreement with the hypothesis of Mendelian transmission. Accordingly, there is no means of detecting that the effect of a major gene, with or without a polygenic component, does not correspond to the correct model. In addition, the parameter estimates for the major gene do not correspond to the characteristics of either of the two genes of the true model. This may substantially affect further linkage analysis.     

A new strategy for estimating two-locus recombination fractions under some natural inequality restrictions

Linkage analysis is now being widely used to map markers on each chromosome in the human genome, to map genetic diseases, and to identify genetic forms of common diseases. Two-locus linkage analysis and multi-locus analysis have been investigated comprehensively, and many computer programs have been developed to perform linkage analysis. Yet there exists a shortcoming in traditional methods, i.e., the parameter space of two-locus recombination fractions has not been emphasized sufficiently in the usual analyses. In this paper, we propose a new strategy for estimating the two-locus recombination fractions based on data of backcross family in the framework of some natural and necessary parameter restrictions. The new strategy is based on a restricted projection algorithm, which can provide fast reasonable estimates of recombination fraction, and can therefore serve as a superior alternative algorithm. Results obtained from both real and simulated data indicate that the new algorithm performs well in the estimation of recombination fractions and outperforms current methods.     

Evidence for two unlinked loci regulating total serum IgE levels.

Studies investigating the genetic control of total serum IgE levels are of major importance in understanding basic pathophysiologic mechanisms in atopy and asthma, since IgE levels predict onset and correlate with the clinical expression of these disorders. Previous analysis of data from 92 families, ascertained through a parent with asthma, showed evidence for recessive inheritance of high IgE levels with linkage to chromosome 5q. Since there was significant residual familial correlation in the one-locus segregation analysis, two-locus segregation and linkage analyses were performed. Segregation analyses provided evidence for a second major locus unlinked to the locus on 5q. Utilization of this two-locus model corroborates the previous evidence for linkage between this trait and markers on 5q31-q33. The LODs for the most informative marker D5S436 increased from 3.00 at 10% recombination to 4.67 at 9% recombination, when the two-locus model was used. Additional linkage studies are needed to map this second locus. These results demonstrate the importance of performing multilocus segregation and linkage analyses for quantitative traits that are related to the phenotype of a complex disorder. This approach has given further insight into the genetics of allergy and asthma by providing evidence for a two-locus model.     

TLINKAGE-IMPRINT: a model-based approach to performing two-locus genetic imprinting analysis

OBJECTIVES: Imprinting refers to the expression of only one copy of a gene pair, which is determined by the parental origin of the copy. Imprinted genes play a role in the development of several complex diseases, including cancers and mental disorders. In certain situations, two-trait-loci models are shown to be more powerful than one-trait-locus models. However, no current methods use pedigree structure efficiently and perform two-locus imprinting analyses. In this paper, we apply the Elston-Stewart algorithm to the parametric two-trait-loci imprinting model used by Strauch et al. [2000] to obtain a method for qualitative trait linkage analyses that explicitly models imprinting and can be applied to large pedigrees. METHODS: We considered a parametric approach based on 4 x 4 penetrance matrix to account for imprinting and modified TLINKAGE software to implement this approach. We performed simulation studies using a small and a large pedigree under dominant and imprinted and dominant or imprinted scenarios. Furthermore, we developed a likelihood ratio-based test for imprinting that compares the logarithm of odds (LOD) score obtained using the two-locus imprinting model with that obtained using the standard two-locus model that does not allow for imprinting. RESULTS: In simulation studies of three scenarios where the true mode of inheritance included imprinting, accurate modeling through the proposed approach yielded higher LOD scores and better recombination fraction estimates than the traditional two-locus model that does not allow for imprinting. CONCLUSIONS: This imprinting model will be useful in identifying the genes responsible for several complex disorders that are potentially caused by a combination of imprinted and non-imprinted genes.     

Effect of genetic heterogeneity and assortative mating on linkage analysis: a simulation study.

Linkage studies of complex genetic traits raise questions about the effects of genetic heterogeneity and assortative mating on linkage analysis. To further understand these problems, I have simulated and analyzed family data for a complex genetic disease in which disease phenotype is determined by two unlinked disease loci. Two models were studied, a two-locus threshold model and a two-locus heterogeneity model. Information was generated for a marker locus linked to one of the disease-defining loci. Random-mating and assortative-mating samples were generated. Linkage analysis was then carried out by use of standard methods, under the assumptions of a single-locus disease trait and a random-mating population. Results were compared with those from analysis of a single-locus homogeneous trait in samples with the same levels of assortative mating as those considered for the two-locus traits. The results show that (1) introduction of assortative mating does not, in itself, markedly affect the estimate of the recombination fraction; (2) the power of the analysis, reflected in the LOD scores, is somewhat lower with assortative rather than random mating. Loss of power is greater with increasing levels of assortative mating; and (3) for a heterogeneous genetic disease, regardless of mating type, heterogeneity analysis permits more accurate estimate of the recombination fraction but may be of limited use in distinguishing which families belong to each homogeneous subset. These simulations also confirmed earlier observations that linkage to a disease "locus" can be detected even if the disease is incorrectly defined as a single-locus (homogeneous) trait, although the estimated recombination fraction will be significantly greater than the true recombination fraction between the linked disease-defining locus and the marker locus.     

How to model a complex trait. 2. Analysis with two disease loci

Complex traits are often governed by more than one trait locus. The first step towards an adequate model for such diseases is a linkage analysis with two trait loci. Such an analysis can be expected to have higher power to detect linkage than a standard single-trait-locus linkage analysis. However, it is crucial to accurately specify the parameters of the two-locus model. Here, we recapitulate the general two-locus model with and without genomic imprinting. We relate heterogeneity, multiplicative, and additive two-locus models to biological or pathophysiological mechanisms, and give the corresponding averaged ("best-fitting") single-trait-locus models for each of the two loci. Furthermore, we derive the two-locus penetrances from the averaged single-locus models, under the assumption of one of the three model classes mentioned above. Using these formulae, if the best-fitting single-locus models are available, investigators may perform a two-trait-locus linkage analysis under a realistic model. This procedure will maximize the power to detect linkage for traits which are governed by two or more loci, and lead to more accurate estimates of the disease-locus positions.     

Combined segregation and linkage analysis of Graves disease with a thyroid autoantibody diathesis.

Combined segregation and linkage analysis is a powerful technique for modeling linkage to diseases whose etiology is more complex than the effect of a well-described single genetic locus and for investigating the influence of single genes on various aspects of the disease phenotype. Graves disease is familial and is associated with human leukocyte antigen (HLA) allele DR3. Probands with Graves disease, as well as close relatives, have raised levels of thyroid autoantibodies. This phenotypic information additional to affection status may be considered by the computer program COMDS for combined segregation and linkage analysis, when normals are classified into diathesis classes of increasing thyroid autoantibody titer. The ordinal model considers the cumulative odds of lying in successive classes, and a single additional parameter is introduced for each gene modeled. Distributional assumptions are avoided by providing estimates of the population frequencies of each class. Evidence for linkage was increased by considering the thyroid autoantibody diathesis and by testing two-locus models. The analysis revealed evidence for linkage to HLA-DR when the strong coupling of the linked locus to allele DR3 was considered (lod score of 6.6). Linkage analysis of the residual variation revealed no evidence of linkage to Gm, but a suggestion of linkage to Km.     

Multiple-trait quantitative trait locus mapping with incomplete phenotypic data

Background

The goal of linkage analysis is to determine the chromosomal location of the gene(s) for a trait of interest such as a common disease. Three-locus linkage analysis is an important case of multi-locus problems. Solutions can be found analytically for the case of triple backcross mating. However, in the present study of linkage analysis and gene mapping some natural inequality restrictions on parameters have not been considered sufficiently, when the maximum likelihood estimates (MLEs) of the two-locus recombination fractions are calculated.

Results

In this paper, we present a study of estimating the two-locus recombination fractions for the phase-unknown triple backcross with two offspring in each family in the framework of some natural and necessary parameter restrictions. A restricted expectation-maximization (EM) algorithm, called REM is developed. We also consider some extensions in which the proposed REM can be taken as a unified method.

Conclusion

Our simulation work suggests that the REM performs well in the estimation of recombination fractions and outperforms current method. We apply the proposed method to a published data set of mouse backcross families.     

Model misspecification and multipoint linkage analysis.

Pairwise linkage analysis is robust to genetic model misspecification provided dominance is correctly specified, the primary effect being inflation of the recombination fraction. By contrast, we show that multipoint analysis under misspecified models is not robust when a putative disease locus is placed between close flanking markers, with potentially spuriously negative multipoint lod scores being produced. The problem is due to incorrect attribution of segregation of a disease allele and the consequent conclusion of (unlikely) double crossovers between flanking markers. As a possible solution, we propose the use of high disease allele frequencies, as this allows probabilistically for nonsegregation (through parental homozygosity or dual matings). We show analytically and through analysis of pedigree data simulated under a two-locus heterogeneity model that using a disease allele frequency of 0.05 in the dominant case and 0.25 in the recessive case is quite robust in producing positive multipoint lod scores with close flanking markers across a broad range of conditions including varying allele frequencies, epistasis, genetic heterogeneity and phenocopies.     

The search for heterogeneity in insulin-dependent diabetes mellitus (IDDM): linkage studies, two-locus models, and genetic heterogeneity

One hundred families with insulin-dependent diabetes mellitus (IDDM) were analyzed for linkage with 27 genetic markers, including HLA, properdin factor B (BF), and glyoxalase 1(GLO) on chromosome 6, and Kidd blood group (Jk) on chromosome 2. The linkage analyses were performed under several different genetic models. An approximate correction for two-locus linkage analysis was developed and applied to four markers. Two different heterogeneity tests were implemented and applied to all the markers. One, the Predivided-Sample Test, utilizes various criteria thought to be relevant to genetic heterogeneity in IDDM. The other, the Admixture Test, looks for heterogeneity without specifying a prior how the sample should be divided. Results continued to support linkage of IDDM with three chromosome 6 markers: HLA, BF, and GLO. The total lod score for Kidd blood group, under the recessive model with 20% penetrance, is 1.63--down 1.2 from the 2.83 reported by us earlier. The only other marker whose lod score exceeded 1.0 under any model was pancreatic amylase (AMY2). The two-locus correction, which involved lowering the penetrance values used in the analysis, affected estimates of theta (recombination fraction) but did not markedly change the lod scores themselves. There was little evidence for heterogeneity within any of the lod scores, under either the Predivided-Sample Test or the Admixture Test.     

Linkage analysis of 49 high-risk families does not support a common familial prostate cancer-susceptibility gene at 1q24-25.

Linkage of a putative prostate cancer-susceptibility locus (HPC1) to chromosome 1q24-25 has recently been reported. Confirmation of this linkage in independent data sets is essential because of the complex nature of this disease. Here we report the results of a linkage analysis using 10 polymorphic markers spanning approximately 37 cM in the region of the putative HPC1 locus in 49 high-risk prostate cancer families. Data were analyzed by use of two parametric models and a nonparametric method. For the parametric LOD-score method, the first model was identical to the original report by Smith and coworkers ("Hopkins"), and the second was based on a segregation analysis previously reported by Carter and coworkers ("Seattle"). In both cases, our results do not confirm the linkage reported for this region. Calculated LOD scores from the two-point analysis for each marker were highly negative at small recombination fractions. Multipoint LOD scores for this linkage group were also highly negative. Additionally, we were unable to demonstrate heterogeneity within the data set, using HOMOG. Although these data do not formally exclude linkage of a prostate cancer-susceptibility locus at HPC1, it is likely that other prostate cancer-susceptibility loci play a more critical role in the families that we studied.     

Estimation of segregation and linkage parameters in simulated data. II. Simultaneous estimation with one linked marker.

This study examined the method of simultaneous estimation of recombination frequency and parameters for a qualitative trait locus and compared the results with those of standard methods of linkage analysis. With both approaches we were able to detect linkage of an incompletely penetrant qualitative trait to highly polymorphic markers with recombination frequencies in the range of .00-.05. Our results suggest that detecting linkage at larger recombination frequencies may require larger data sets or large high-density families. When applied to all families without regard to informativeness of the family structure for linkage, analyses of simulated data could detect no advantage of simultaneous estimation over more traditional and much less time-consuming methods, either in detecting linkage, estimating frequency, refining estimates of parameters for the qualitative trait locus, or avoiding false evidence for linkage. However, the method of sampling affected results.     

A semi-symmetric two-locus model

The two-locus symmetric viability model characterized by its invariance with respect to the exchange of alleles at each locus, is a well-studied model of classical two-locus theory. The symmetric model introduced by Lewontin and Kojima is among the few multi-locus models with epistatic interactions between loci for which a polymorphism with linkage equilibrium can be stable and this happens when recombination is sufficiently large. We show that an analogous property holds true for a different model, in which symmetry need exist at only one locus. The properties of this new semi-symmetric model are compared with those of the classical symmetric model. For tight linkage, two classes of polymorphisms are possible, depending on the magnitude of additive epistasis. The recombination rate above which linkage equilibrium becomes stable is derived analytically. As in the symmetric model, intervals of recombination in which no polymorphism is stable are possible, and stable polymorphisms can coexist with stable fixations.     

HLODs, trait models, and ascertainment: implications of admixture for parameter estimation and linkage detection

Maximizing the homogeneity lod is known to be an appropriate procedure for estimating parameters of the trait model in an approximately 'ascertainment assumption free' (AAF) manner. We have investigated whether this same property also holds for the heterogeneity lod (HLOD). We show that, when the genetic models at linked and unlinked loci differ, HLODs are not AAF, and maximizing the HLOD yields parameter estimates that are for all practical purposes meaningless; indeed, the admixture parameter alpha does not even measure the proportion of linked families within the sample, as is commonly supposed. In spite of this, our results confirm a large body of evidence supporting the use of HLODs as robust tools for linkage detection, and suggest further that maximizing the HLOD over both alpha and parameters of the trait model can improve accuracy in estimation of the recombination fraction theta;. These findings have important implications for the optimal handling of nuisance parameters in linkage analysis, particularly when evaluating the evidence for or against linkage based on multiple independent heterogeneous sets of data.     

Detecting gene-gene interactions using affected sib pair analysis with covariates

Interest has recently focussed on allowing for interactions between loci as a way to increase power to detect linkage. In this paper, a simplified logistic regression method was used to perform affected sib pair analyses allowing for the inclusion of data from other loci. A systematic search of two-locus disease models was carried out to determine the situations in which this was advantageous. If IBD information is available (e.g. from a genome scan), it is unlikely that allowing for interactions will give a large lod score in the absence of linkage evidence from sinlge-locus analysis. Furthermore, allowing for interactions rarely gave a significant increase in power to detect linkage over a single-locus analysis, except for heterogeneity models with low K(P). Conversely, the availability of disease-associated genotypes may greatly increase the power both to detect linkage to a second locus and interaction between the loci. These results indicate that when only IBD information is available, two-locus analysis of genome scan data should be restricted to regions giving peaks under single-locus analysis. If disease-associated genotypes are available, it may be worth re-analysing the whole genome.     

Direct power comparisons between simple LOD scores and NPL scores for linkage analysis in complex diseases.

Several methods have been proposed for linkage analysis of complex traits with unknown mode of inheritance. These methods include the LOD score maximized over disease models (MMLS) and the "nonparametric" linkage (NPL) statistic. In previous work, we evaluated the increase of type I error when maximizing over two or more genetic models, and we compared the power of MMLS to detect linkage, in a number of complex modes of inheritance, with analysis assuming the true model. In the present study, we compare MMLS and NPL directly. We simulated 100 data sets with 20 families each, using 26 generating models: (1) 4 intermediate models (penetrance of heterozygote between that of the two homozygotes); (2) 6 two-locus additive models; and (3) 16 two-locus heterogeneity models (admixture alpha = 1.0,.7,.5, and.3; alpha = 1.0 replicates simple Mendelian models). For LOD scores, we assumed dominant and recessive inheritance with 50% penetrance. We took the higher of the two maximum LOD scores and subtracted 0.3 to correct for multiple tests (MMLS-C). We compared expected maximum LOD scores and power, using MMLS-C and NPL as well as the true model. Since NPL uses only the affected family members, we also performed an affecteds-only analysis using MMLS-C. The MMLS-C was both uniformly more powerful than NPL for most cases we examined, except when linkage information was low, and close to the results for the true model under locus heterogeneity. We still found better power for the MMLS-C compared with NPL in affecteds-only analysis. The results show that use of two simple modes of inheritance at a fixed penetrance can have more power than NPL when the trait mode of inheritance is complex and when there is heterogeneity in the data set.     

Detection of gene-environment interactions in joint segregation and linkage analysis.

We compare approaches for analysis of gene-environment (G x E) interaction, using segregation and joint segregation and linkage analyses of a quantitative trait. Analyses of triglyceride levels in a single large pedigree demonstrate the two methods and show evidence for a significant interaction (P=.015 when segregation analysis is used; P=.006 when joint analysis is used) between a codominant major gene and body-mass index. Genotype-specific correlation coefficients, between triglyceride levels and body-mass index, estimated from the joint model are rAA=.72, rAa=.49, and raa=. 20. Several simulation studies indicate that joint segregation and linkage analysis leads to less-biased and more-efficient estimates of a G x E-interaction effect, compared with segregation analysis alone. Depending on the heterozygosity of the marker locus and its proximity to the trait locus, we found joint analysis to be as much as 70% more efficient than segregation analysis, for estimation of a G x E-interaction effect. Over a variety of parameter combinations, joint analysis also led to moderate (5%-10%) increases in power to detect the interaction. On the basis of these results, we suggest the use of combined segregation and linkage analysis for improved estimation of G x E-interaction effects when the underlying trait gene is unmeasured.     

Multilocus linkage tests based on affected relative pairs

For complex diseases, recent interest has focused on methods that take into account joint effects at interacting loci. Conditioning on effects of disease loci at known locations can lead to increased power to detect effects at other loci. Moreover, use of joint models allows investigation of the etiologic mechanisms that may be involved in the disease. Here we present a method for simultaneous analysis of the joint genetic effects at several loci that uses affected relative pairs. The method is a generalization of the two-locus LOD-score analysis for affected sib pairs proposed by Cordell et al. We derive expressions for the relative risk, lambdaR, to a relative of an affected individual, in terms of the additive and epistatic components of variance at an arbitrary number of disease loci, and we show how these can be used to fit a likelihood model to the identity-by-descent sharing among pairs of affected relatives in extended pedigrees. We implement the method by use of a stepwise strategy in which, given evidence of linkage to disease at m-1 locations on the genome, we calculate the conditional likelihood curve across the genome for an mth disease locus, using multipoint methods similar to those proposed by Kruglyak et al. We evaluate the properties of our method by use of simulated data and present an application to real data from families with insulin-dependent diabetes mellitus.     

A hidden Markov model approach to multilocus linkage analysis in a full-sib family

Statistical packages for constructing genetic linkage maps in inbred lines are well developed and applied extensively, while linkage analysis in outcrossing species faces some statistical challenges because of their complicated genetic structures. In this article, we present a multilocus linkage analysis via hidden Markov models for a linkage group of markers in a full-sib family. The advantage of this method is the simultaneous estimation of the recombination fractions between adjacent markers that possibly segregate in different ratios, and the calculation of likelihood for a certain order of the markers. When the number of markers decreases to two or three, the multilocus linkage analysis becomes traditional two-point or three-point linkage analysis, respectively. Monte Carlo simulations are performed to show that the recombination fraction estimates of multilocus linkage analysis are more accurate than those just using two-point linkage analysis and that the likelihood as an objective function for ordering maker loci is the most powerful method compared with other methods. By incorporating this multilocus linkage analysis, we have developed a Windows software, FsLinkageMap, for constructing genetic maps in a full-sib family. A real example is presented for illustrating linkage maps constructed by using mixed segregation markers. Our multilocus linkage analysis provides a powerful method for constructing high-density genetic linkage maps in some outcrossing plant species, especially in forest trees.