Multipoint linkage detection in the presence of heterogeneity

Linkage heterogeneity is common for complex diseases. It is well known that loss of statistical power for detecting linkage will result if one assumes complete homogeneity in the presence of linkage heterogeneity. To this end, Smith (1963, Annals of Human Genetics 27, 175-182) proposed an admixture model to account for linkage heterogeneity. It is well known that for this model, the conventional chi-squared approximation to the likelihood ratio test for no linkage does not apply even when the sample size is large. By dealing with nuclear families and one marker at a time for genetic diseases with simple modes of inheritance, score-based test statistics (Liang and Rathouz, 1999, Biometrics 55, 65-74) and likelihood-ratio-based test statistics (Lemdani and Pons, 1995, Biometrics 51, 1033-1041) have been proposed which have a simple large-sample distribution under the null hypothesis of linkage. In this paper, we extend their work to more practical situations that include information from multiple markers and multi-generational pedigrees while allowing for a class of general genetic models. Three different approaches are proposed to eliminate the nuisance parameters in these test statistics. We show that all three approaches lead to the same asymptotic distribution under the null hypothesis of no linkage. Simulation results show that the proposed test statistics have adequate power to detect linkage and that the performances of these two classes of test statistics are quite comparable. We have applied the proposed method to a family study of asthma (Barnes et al., 1996), in which the score-based test shows evidence of linkage with p-value <0.0001 in the region of interest on chromosome 12. Additionally, we have implemented this score-based test within the frequently used computer package GENEHUNTER.     

A new statistical test for linkage heterogeneity.

A new, statistical test for linkage heterogeneity is described. It is a likelihood-ratio test based on a beta distribution for the prior distribution of the recombination fraction among families (or individuals). The null distribution for this statistic (called the B-test) is derived under a broad range of circumstances. Two other heterogeneity test statistics--the admixture test or A-test first described by Smith and Morton's test (here referred to as the K-test)--are also examined. The probability distribution for the K-test statistic is very sensitive to family size, whereas the other two statistics are not. All three statistics are somewhat sensitive to the magnitude of the recombination fraction theta. Critical values for each of the test statistics are given. A conservative approximation for both the A-test and B-test is given by a chi 2 distribution when P/2 instead of P is used for the observed significance level. In terms of power, the B-test performs best among the three tests over a broad range of alternate heterogeneity hypotheses--except for the specific case of admixture with loose linkage, in which the A-test performs best. Overall, the difference in power among the three tests is not large. An application to some recently published data on the fragile-X syndrome and X-chromosome markers is given.     

Hypothesis testing under mixture models: application to genetic linkage analysis

In this paper we propose a new class of statistics to test a simple hypothesis against a family of alternatives characterized by a mixture model. Unlike the likelihood ratio statistic, whose large sample distribution is still unknown in this situation, these new statistics have a simple asymptotic distribution to which to refer under the null hypothesis. Simulation results suggest that it has adequate power in detecting the alternatives. Its application to genetic linkage analysis in the presence of the genetic heterogeneity that motivated this work is emphasized.     

A combined test of linkage heterogeneity.

This paper focuses on the problem of testing for heterogeneity once linkage is established. In an investigation of genetic linkage, Morton first proposed a general purpose test to detect heterogeneity in the recombination fraction. Two more commonly used tests of linkage heterogeneity are the admixture test (A-test) of Smith, Ott, and Risch and Baron, and the B-test of Risch. All are likelihood-ratio tests, but they differ in the models specifying the heterogeneity. A new test of heterogeneity in the presence of linkage is presented here. I propose a mixture model of heterogeneity, which allows the recombination fraction to vary among families, as does the B-model, yet also allows some families to be unlinked, as the A-model does. This model contains the A and B models as special cases and thus allows a direct test (D-test), which can provide justification for choosing one of these extremes.     

Regression models for linkage heterogeneity applied to familial prostate cancer

Linkage heterogeneity frequently occurs for complex genetic diseases, and statistical methods must account for it to avoid severe loss in power to discover susceptibility genes. A common method to allow for only a fraction of linked pedigrees is to fit a mixture likelihood and then to test for linkage homogeneity, given linkage (admixture test), or to test for linkage while allowing for heterogeneity, using the heterogeneity LOD (HLOD) score. Furthermore, features of the families, such as mean age at diagnosis, may help to discriminate families that demonstrate linkage from those that do not. Pedigree features are often used to create homogeneous subsets, and LOD or HLOD scores are then computed within the subsets. However, this practice introduces several problems, including reduced power (which results from multiple testing and small sample sizes within subsets) and difficulty in interpretation of results. To address some of these limitations, we present a regression-based extension of the mixture likelihood for which pedigree features are used as covariates that determine the probability that a family is the linked type. Some advantages of this approach are that multiple covariates can be used (including quantitative covariates), covariates can be adjusted for each other, and interactions among covariates can be assessed. This new regression method is applied to linkage data for familial prostate cancer and provides new insights into the understanding of prostate cancer linkage heterogeneity.     

A goodness-of-fit test for capture-recapture model Mt under closure

A new, fully efficient goodness-of-fit test for the time-specific closed-population capture-recapture model Mt is presented. This test is based on the residual distribution of the capture history data given the maximum likelihood parameter estimates under model Mt, is partitioned into informative components, and is based on chi-square statistics. Comparison of this test with Leslie's test (Leslie, 1958, Journal of Animal Ecology 27, 84-86) for model Mt, using Monte Carlo simulations, shows the new test generally outperforms Leslie's test. The new test is frequently computable when Leslie's test is not, has Type I error rates that are closer to nominal error rates than Leslie's test, and is sensitive to behavioral variation and heterogeneity in capture probabilities. Leslie's test is not sensitive to behavioral variation in capture probabilities but, when computable, has greater power to detect heterogeneity than the new test.     

成人型多囊肾病的遗传研究

３＇ＨＶＲ是成人型多囊肾病基因诊断中最常用的探针。我们分析了５１个无亲缘关系健康学生和３个成人多囊肾病家系的３＇ＨＶＲ－ＰｖｕＩＩ ＲＦＬＰ,所得多态信息用计算机软件ＬＩＮＫ－ＡＧＥ和ＨＯＭＯＧ进行连锁分析和同质性检验,其中１个家系致病基因位点与３＇ＨＶＲ不连锁,因而判定为ｎｏｎ ＰＫＤ１。剩下的两个家系中１个有明显的重组,但还不能判定为ｎｏｎ ＰＫＤ１,另１个与３＇ＨＶＲ连锁,属于ＰＫＤ１。成人     

Further evidence for genetic heterogeneity in the fragile X syndrome

Summary The X-linked fragile X[fra(X)] syndrome, associated with a fragile site at Xq27.3, is the most common Mendeban inherited form of mental deficiency. Approximately 1 in 1060 males and 1 in 677 females carry the fra (X) chromosome. However, diagnosis of carrier status can be difficult since about 20% of males and 44% of females are nonpenetrant for mental impairment and/or expression of fra (X). We analyzed DNA from 327 individuals in 23 families segregating fra (X) for linkage to three flanking polymorphic probes: 52A, F9, and ST14. This allowed probable nonpenetrant, transmitting males and carrier females to be identified. A combined linkage analysis was conducted using these families and published probe information on F9 in 27 other families, 52A in six families, and ST14 in five families. The two-point recombination fraction for 52A-F9 was 0.13 (90% confidence interval, 0.10–0.16), for F9-fra(X) was 0.21 (0.17–0.24), and for fra(X)-ST14 was 0.12 (0.07–0.17). Tight linkage between F9 and fra(X) was observed in some families; in others loose linkage was seen suggesting genetic linkage heterogeneity. Risk analysis of carrier status using flanking DNA probes showed that probable nonpenetrant transmitting males were included in families showing both tight and loose linkage. Thus, in contrast to our previous conclusions, it appears that the presence or absence of nonpenetrant, transmitting males in a family is not an indicator of heterogeneity. To determine if heterogeneity was present, we employed the admixture test. Evidence for linkage heterogeneity between F9 and fra(X) was found, significant at P<0.0005. Nonsignificant heterogeneity was seen for 52A-F9 linkage. No heterogeneity was found for fra(X)-ST14. The frequency of fra(X) expression was significantly lower in families with tight F9-fra(X) linkage than in families with loose linkage. Cognition appeared to relate to linkage type: affected males in tight linkage families had higher IQs than those in loose linkage families. These findings of genetic heterogeneity can account in part for the high prevalence and apparent high new mutation rate of fra(X). They will affect genetic counseling using RFLPs. An understanding of the basis for genetic heterogeneity in fra(X) will help to clarify the nature of the unusual pattern of inheritance seen in this syndrome.     

Detection of linkage for heterogeneous disorders by using multipoint linkage analysis.

We have compared the efficiency of the lod score test which assumes heterogeneity (lod2) to the standard lod score test which assumes homogeneity (lod1) when three-point linkage analysis is used in successive map intervals. If it is assumed that a gene located midway between two linked marker loci is responsible for a proportion of disease cases, then the lod1 test loses power relative to the lod2 test, as the proportion of linked families decreases, as the flanking markers are more closely linked, and as more map intervals are tested. Moreover, when multipoint analysis is used, linkage for a disease gene is more likely to be incorrectly excluded from a complete and dense linkage map if true genetic heterogeneity is ignored. We thus conclude that, in general, the lod2 linkage test is more efficient for detecting a true linkage when a complete genetic marker map is screened for a heterogeneous disorder.     

Detecting linkage disequilibrium in bacterial populations.

The distribution of the number of pairwise differences calculated from comparisons between n haploid genomes has frequently been used as a starting point for testing the hypothesis of linkage equilibrium. For this purpose the variance of the pairwise differences, VD, is used as a test statistic to evaluate the null hypothesis that all loci are in linkage equilibrium. The problem is to determine the critical value of the distribution of VD. This critical value can be estimated either by Monte Carlo simulation or by assuming that VD is distributed normally and calculating a one-tailed 95% critical value for VD, L, L = EVD + 1.645 sqrt(VarVD), where E(VD) is the expectation of VD, and Var(VD) is the variance of VD. If VD (observed) > L, the null hypothesis of linkage equilibrium is rejected. Using Monte Carlo simulation we show that the formula currently available for Var(VD) is incorrect, especially for genetically highly diverse data. This has implications for hypothesis testing in bacterial populations, which are often genetically highly diverse. For this reason we derive a new, exact formula for Var(VD). The distribution of VD is examined and shown to approach normality as the sample size increases. This makes the new formula a useful tool in the investigation of large data sets, where testing for linkage using Monte Carlo simulation can be very time consuming. Application of the new formula, in conjunction with Monte Carlo simulation, to populations of Bradyrhizobium japonicum, Rhizobium leguminosarum, and Bacillus subtilis reveals linkage disequilibrium where linkage equilibrium has previously been reported.     

Heterogeneity for multiple disease loci in linkage analysis.

Many mendelian traits show heterogeneity; that is, the disease phenotype in different families may be caused by genes at different locations. In linkage analysis, this admixture type of heterogeneity (locus heterogeneity) has often been accommodated with one of the HOMOG programs, which thus far have been restricted to at most two disease gene locations. Here, an extension to an arbitrary number of disease locations is described. It has been implemented in a computer program, HOMOGM. This approach is also suitable as an approximation to the situation of complex traits, in which multiple disease genes may occur in the same family.     

The quantitative trait linkage disequilibrium test: a more powerful alternative to the quantitative transmission disequilibrium test for use in the absence of population stratification

Linkage analysis based on identity-by-descent allele-sharing can be used to identify a chromosomal region harboring a quantitative trait locus (QTL), but lacks the resolution required for gene identification. Consequently, linkage disequilibrium (association) analysis is often employed for fine-mapping. Variance-components based combined linkage and association analysis for quantitative traits in sib pairs, in which association is modeled as a mean effect and linkage is modeled in the covariance structure has been extended to general pedigrees (quantitative transmission disequilibrium test, QTDT). The QTDT approach accommodates data not only from parents and siblings, but also from all available relatives. QTDT is also robust to population stratification. However, when population stratification is absent, it is possible to utilize even more information, namely the additional information contained in the founder genotypes. In this paper, we introduce a simple modification of the allelic transmission scoring method used in the QTDT that results in a more powerful test of linkage disequilibrium, but is only applicable in the absence of population stratification. This test, the quantitative trait linkage disequilibrium (QTLD) test, has been incorporated into a new procedure in the statistical genetics computer package SOLAR. We apply this procedure in a linkage/association analysis of an electrophysiological measurement previously shown to be related to alcoholism. We also demonstrate by simulation the increase in power obtained with the QTLD test, relative to the QTDT, when a true association exists between a marker and a QTL.     

Increase in linkage information by stratification of pedigree data into gold-standard and standard diagnoses: application to the NIMH Alzheimer Disease Genetics Initiative Dataset

Patients diagnosed with a standard clinical method (subject to misclassification error) are often combined with patients diagnosed with a gold-standard method (with zero or very small misclassification error) in family-based studies of complex disease. For example, non-autopsied patients (NAP) are often included along with autopsy-proven (AP) patients in family-based studies of complex diseases, such as Alzheimer's disease (AD). Theoretical and simulation studies suggest that certain misclassification errors can result in severe reduction of power in genetic linkage and association analyses and that phenotype (or diagnostic) error can produce misleading results. Morton's test for heterogeneity can identify genomic regions where error may have led to loss in power. We applied this test to pedigree data from the NIMH Alzheimer's Disease Genetics Initiative Database separated into AP and NAP pedigrees. Morton's test identified one highly significant region of heterogeneity on chromosome 2. The source of the heterogeneity was due to significant indication of linkage in the AP pedigrees at position 109 cM (p value = 6.68 x 10(-5)) with no indication in the NAP pedigrees. Furthermore, Morton's test showed no evidence for heterogeneity on chromosome 19 in early-onset pedigrees that showed highly significant evidence for linkage in other published reports. These results suggest that supplementing linkage analysis with Morton's test can be usefully applied to genetic data sets that have AP and NAP samples, or other sample mixtures that include a 'gold standard' subgroup with reduced error rate, to increase power to detect linkage in the presence of diagnostic misclassification.     

Replication of linkage studies of complex traits: an examination of variation in location estimates.

In linkage studies, independent replication of positive findings is crucial in order to distinguish between true positives and false positives. Recently, the following question has arisen in linkage studies of complex traits: at what distance do we reject the hypothesis that two location estimates in a genomic region represent the same gene? Here we attempt to address this question. Sampling distributions for location estimates were constructed by computer simulation. The conditions for simulation were chosen to reflect features of "typical" complex traits, including incomplete penetrance, phenocopies, and genetic heterogeneity. Our findings, which bear on what is considered a replication in linkage studies of complex traits, suggest that, even with relatively large numbers of multiplex families, chance variation in the location estimate is substantial. In addition, we report evidence that, for the conditions studied here, the standard error of a location estimate is a function of the magnitude of the expected LOD score.     

Evaluating genetic heterogeneity in complex disorders

OBJECTIVES: The Admixture test is routinely used in linkage analysis to take account of genetic heterogeneity, and yields an estimate of the proportion of families (alpha) segregating the linked disease gene. In complex disorders, the assumptions of the Admixture test are violated. We therefore explore how the estimate of alpha relates to the true proportion of linked families with a complex disorder in a population or dataset. METHODS: We simulated a two-locus heterogeneity model and varied genetic parameters, ascertainment scheme and phenocopy frequency. RESULTS: In this model, alpha is almost always overestimated, by as little as 5% to as much as 60%. The bias is largely attributable to (1). intrafamilial heterogeneity arising from ascertainment of families with many affected members or from analysis of dense pedigrees; (2). low informativeness, which occurs in the presence of reduced penetrance; and (3). differences in the evidence for linkage in linked and unlinked families. This bias is also affected by the analysis phenocopy frequency, but only if the linked locus is dominant and the unlinked locus is recessive. CONCLUSIONS: We conclude that, in complex diseases, the Admixture test has greater value in detecting linkage than in estimating the proportion of linked families in a dataset.     

Linkage analysis of quantitative trait loci in the presence of heterogeneity

Variance component modeling for linkage analysis of quantitative traits is a powerful tool for detecting and locating genes affecting a trait of interest, but the presence of genetic heterogeneity will decrease the power of a linkage study and may even give biased estimates of the location of the quantitative trait loci. Many complex diseases are believed to be influenced by multiple genes and therefore genetic heterogeneity is likely to be present for many real applications of linkage analysis. We consider a mixture of multivariate normals to model locus heterogeneity by allowing only a proportion of the sampled pedigrees to segregate trait-influencing allele(s) at a specific locus. However, for mixtures of normals the classical asymptotic distribution theory of the maximum likelihood estimates does not hold, so tests of linkage and/or heterogeneity are evaluated using resampling methods. It is shown that allowing for genetic heterogeneity leads to an increase in power to detect linkage. This increase is more prominent when the genetic effect of the locus is small or when the percentage of pedigrees not segregating trait-influencing allele(s) at the locus is high.     

Transmission test for linkage disequilibrium: the insulin gene region and insulin-dependent diabetes mellitus (IDDM).

A population association has consistently been observed between insulin-dependent diabetes mellitus (IDDM) and the "class 1" alleles of the region of tandem-repeat DNA (5'' flanking polymorphism [5''FP]) adjacent to the insulin gene on chromosome 11p. This finding suggests that the insulin gene region contains a gene or genes contributing to IDDM susceptibility. However, several studies that have sought to show linkage with IDDM by testing for cosegregation in affected sib pairs have failed to find evidence for linkage. As means for identifying genes for complex diseases, both the association and the affected-sib-pairs approaches have limitations. It is well known that population association between a disease and a genetic marker can arise as an artifact of population structure, even in the absence of linkage. On the other hand, linkage studies with modest numbers of affected sib pairs may fail to detect linkage, especially if there is linkage heterogeneity. We consider an alternative method to test for linkage with a genetic marker when population association has been found. Using data from families with at least one affected child, we evaluate the transmission of the associated marker allele from a heterozygous parent to an affected offspring. This approach has been used by several investigators, but the statistical properties of the method as a test for linkage have not been investigated. In the present paper we describe the statistical basis for this "transmission test for linkage disequilibrium" (transmission/disequilibrium test [TDT]). We then show the relationship of this test to tests of cosegregation that are based on the proportion of haplotypes or genes identical by descent in affected sibs. The TDT provides strong evidence for linkage between the 5''FP and susceptibility to IDDM. The conclusions from this analysis apply in general to the study of disease associations, where genetic markers are usually closely linked to candidate genes. When a disease is found to be associated with such a marker, the TDT may detect linkage even when haplotype-sharing tests do not.     

Lack of genetic linkage evidence for a trans-acting factor having a large effect on plasma lipoprotein[a] levels in African Americans

The distribution of plasma lipoprotein[a] (Lp[a]) concentrations, a risk factor for cardiovascular disease, varies greatly among racial groups, with African Americans having values that are shifted toward higher levels than those of whites. The underlying cause of this heterogeneity is unknown, but a role for "trans-acting" factors has been hypothesized. This study used genetic linkage analysis to localize genetic factors influencing Lp[a] levels in African Americans that were absent in other populations; linkage results were analyzed separately in non-Hispanic whites, Hispanic whites, and African Americans. As expected, all three samples showed highly significant linkage at the approximate location of the lysophosphatidic acid locus. The white populations also independently had regions of significant linkage on chromosome 19 (LOD 3.80) and suggestive linkage on chromosomes 12 (LOD 1.60), 14 (LOD 2.56), and 19 (LOD 2.52).No linkage evidence was found to support the hypothesis of another single gene with large effects specifically segregating in African Americans that may account for their elevated Lp[a] levels.     

Correlation-based inference for linkage disequilibrium with multiple alleles

The correlation between alleles at a pair of genetic loci is a measure of linkage disequilibrium. The square of the sample correlation multiplied by sample size provides the usual test statistic for the hypothesis of no disequilibrium for loci with two alleles and this relation has proved useful for study design and marker selection. Nevertheless, this relation holds only in a diallelic case, and an extension to multiple alleles has not been made. Here we introduce a similar statistic, R(2), which leads to a correlation-based test for loci with multiple alleles: for a pair of loci with k and m alleles, and a sample of n individuals, the approximate distribution of n(k - 1)(m - 1)/(km)R(2) under independence between loci is chi((k-1)(m-1))(2). One advantage of this statistic is that it can be interpreted as the total correlation between a pair of loci. When the phase of two-locus genotypes is known, the approach is equivalent to a test for the overall correlation between rows and columns in a contingency table. In the phase-known case, R(2) is the sum of the squared sample correlations for all km 2 x 2 subtables formed by collapsing to one allele vs. the rest at each locus. We examine the approximate distribution under the null of independence for R(2) and report its close agreement with the exact distribution obtained by permutation. The test for independence using R(2) is a strong competitor to approaches such as Pearson's chi square, Fisher's exact test, and a test based on Cressie and Read's power divergence statistic. We combine this approach with our previous composite-disequilibrium measures to address the case when the genotypic phase is unknown. Calculation of the new multiallele test statistic and its P-value is very simple and utilizes the approximate distribution of R(2). We provide a computer program that evaluates approximate as well as "exact" permutational P-values.     

A comparison of different linkage statistics in small to moderate sized pedigrees with complex diseases

ABSTRACT: BACKGROUND: In the last years GWA studies have successfully identified common SNPs associated with complex diseases. However, most of the variants found this way account for only a small portion of the trait variance. This fact leads researchers to focus on rare-variant mapping with large scale sequencing, which can be facilitated by using linkage information. The question arises why linkage analysis often fails to identify genes when analyzing complex diseases. Using simulations we have investigated the power of parametric and nonparametric linkage statistics (KC-LOD, NPL, LOD and MOD scores), to detect the effect of genes responsible for complex diseases using different pedigree structures. RESULTS: As expected, a small number of pedigrees with less than three affected individuals has low power to map disease genes with modest effect. Interestingly, the power decreases when unaffected individuals are included in the analysis, irrespective of the true mode of inheritance. Furthermore, we found that the best performing statistic depends not only on the type of pedigrees but also on the true mode of inheritance. CONCLUSIONS: When applied in a sensible way linkage is an appropriate and robust technique to map genes for complex disease. Unlike association analysis, linkage analysis is not hampered by allelic heterogeneity. So, why does linkage analysis often fail with complex diseases? Evidently, when using an insufficient number of small pedigrees, one might miss a true genetic linkage when actually a real effect exists. Furthermore, we show that the test statistic has an important effect on the power to detect linkage as well. Therefore, a linkage analysis might fail if an inadequate test statistic is employed. We provide recommendations regarding the most favorable test statistics, in terms of power, for a given mode of inheritance and type of pedigrees under study, in order to reduce the probability to miss a true linkage.