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.     

Inter- and intrafamilial heterogeneity: effective sampling strategies and comparison of analysis methods.

Heterogeneity, both inter- and intrafamilial, represents a serious problem in linkage studies of common complex diseases. In this study we simulated different scenarios with families who phenotypically have identical diseases but who genotypically have two different forms of the disease (both forms genetic). We examined the proportion of families displaying intrafamilial heterogeneity, as a function of mode of inheritance, gene frequency, penetrance, and sampling strategies. Furthermore, we compared two different ways of analyzing linkage in these data sets: a two-locus (2L) analysis versus a one-locus (SL) analysis combined with an admixture test. Data were simulated with tight linkage between one disease locus and a marker locus; the other disease locus was not linked to a marker. Our findings are as follows: (1) In contrast to what has been proposed elsewhere to minimize heterogeneity, sampling only "high-density" pedigrees will increase the proportion of families with intrafamilial heterogeneity, especially when the two forms are relatively close in frequency. (2) When one form is dominant and one is recessive, this sampling strategy will greatly decrease the proportions of families with a recessive form and may therefore make it more difficult to detect linkage to the recessive form. (3) An SL analysis combined with an admixture test achieves about the same lod scores and estimate of the recombination fraction as does a 2L analysis. Also, a 2L analysis of a sample of families with intrafamilial heterogeneity does not perform significantly better than an SL analysis. (4) Bilineal pedigrees have little effect on the mean maximum lod score and mean maximum recombination fraction, and therefore there is little danger that including these families will lead to a false exclusion of linkage.     

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.     

Evidence for heterogeneity in facioscapulohumeral muscular dystrophy (FSHD)

Facioscapulohumeral muscular dystrophy (FSHD) is a slowly progressive primary disease of muscle which is usually inherited as an autosomal dominant disorder. FSHD has been localized to the long arm of chromosome 4, specifically to the 4q3.5-qter region. Initially published linkage studies showed no evidence for heterogeneity in FSHD. In the present study we have examined individuals in seven FSHD families. Two-point lod scores show significant evidence for linkage for D4S163 (lod score 3.04 at recombination fraction .21) and D4S139 (lod score 3.84 at recombination fraction .20). D4S171 also gave a positive score (lod score 2.56 at recombination fraction .24). Significant evidence for heterogeneity was found for each of the three markers. Multipoint linkage analysis in this region resulted in a peak multipoint lod score of 6.47. The multipoint analysis supported the two-point studies with odds of 20:1 showing linkage and heterogeneity over linkage and homogeneity. Five of the seven families gave a posterior probability of >95% of being of the linked type, while two families appeared unlinked to this region of 4q (P < .01%). Individuals in the two unlinked families met the clinical criteria for the diagnosis of FSHD, including facial weakness, clavicular flattening, scapula winging, proximal muscle weakness, and myopathic changes on muscle biopsies without inflammatory or mitochondrial pathology. This study demonstrates genetic heterogeneity in FSHD and has important implications for both genetic counseling and the elucidation of the etiology of FSHD.     

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.     

Two-locus maximum lod score analysis of a multifactorial trait: joint consideration of IDDM2 and IDDM4 with IDDM1 in type 1 diabetes.

To investigate the genetic component of multifactorial diseases such as type 1 (insulin-dependent) diabetes mellitus (IDDM), models involving the joint action of several disease loci are important. These models can give increased power to detect an effect and a greater understanding of etiological mechanisms. Here, we present an extension of the maximum lod score method of N. Risch, which allows the simultaneous detection and modeling of two unlinked disease loci. Genetic constraints on the identical-by-descent sharing probabilities, analogous to the "triangle" restrictions in the single-locus method, are derived, and the size and power of the test statistics are investigated. The method is applied to affected-sib-pair data, and the joint effects of IDDM1 (HLA) and IDDM2 (the INS VNTR) and of IDDM1 and IDDM4 (FGF3-linked) are assessed with relation to the development of IDDM. In the presence of genetic heterogeneity, there is seen to be a significant advantage in analyzing more than one locus simultaneously. Analysis of these families indicates that the effects at IDDM1 and IDDM2 are well described by a multiplicative genetic model, while those at IDDM1 and IDDM4 follow a heterogeneity model.     

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.     

Analysis of chromosome 1q42.2-43 in 152 families with high risk of prostate cancer

One hundred fifty-two families with prostate cancer were analyzed for linkage to markers spanning a 20-cM region of 1q42.2-43, the location of a putative prostate cancer-susceptibility locus (PCAP). No significant evidence for linkage was found, by use of both parametric and nonparametric tests, in our total data set, which included 522 genotyped affected men. Rejection of linkage may reflect locus heterogeneity or the confounding effects of sporadic disease in older-onset cases; therefore, pedigrees were stratified into homogeneous subsets based on mean age at diagnosis of prostate cancer and number of affected men. Analyses of these subsets also detected no significant evidence for linkage, although LOD scores were positive at higher recombination fractions, which is consistent with the presence of a small proportion of families with linkage. The most suggestive evidence of linkage was in families with at least five affected men (nonparametric linkage score of 1.2; P=.1). If heterogeneity is assumed, an estimated 4%-9% of these 152 families may show linkage in this region. We conclude that the putative PCAP locus does not account for a large proportion of these families with prostate cancer, although the linkage of a small subset is compatible with these data.     

Segregation analysis incorporating linkage markers. I. Single-locus models with an application to type I diabetes.

A method is described for segregation analysis that incorporates linkage markers. The model allows for segregation (penetrance), linkage (recombination fraction), and association (linkage disequilibrium) parameters. A single-locus-multiple-allele model underlying the trait phenotype is assumed. When families have been ascertained in a systematic fashion, a joint (markers, phenotypes) likelihood with ascertainment is advocated. When ascertainment correction is not feasible, a conditional (markers given phenotypes) approach is recommended, which is also valid in the presence of reduced fertility and assortative mating. This approach, oriented toward determining mode of inheritance, differs from conventional linkage analysis, which is oriented toward detection of linkage. Therefore, it is more appropriately considered an extension of the affected sib-pair method to arbitrary pedigrees, including association information and allowing for multiple alleles. Incorporation of coupling parameters allows for discrimination between pleiotropy and linkage disequilibrium. The method is demonstrated through a reanalysis of four recently published family studies on type 1 diabetes and HLA. Recessive inheritance is rejected in all four data sets. For three of them, dominant inheritance is not rejected, while in the fourth, all two-allele models are rejected in favor of three alleles. Although association with the DR3 and DR4 alleles is quite strong, pleiotropy with regard to these alleles is unlikely. The results also suggest an additional familial factor(s) (e.g., locus).     

Linkage of familial breast cancer to chromosome 17q21 may not be restricted to early-onset disease.

Lod scores for linkage between familial breast and ovarian cancer and markers on chromosome 17q21 are more frequently positive among families with disease diagnosed at younger ages than they are among older-onset families, suggesting that linkage is restricted to early-onset disease. However, for late-onset cases, the relative probability of sporadic rather than inherited disease is higher than previously suggested. If this correction is made, then later-onset families are much less informative; linkage heterogeneity based on age at onset is no longer significant; and for the sample of families as a whole, linkage is significant at a recombination fraction since demonstrated to be close to the correct local. There is probably more than one gene for inherited breast cancer, but heterogeneity may not be due to age at disease onset.     

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.     

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.     

Linkage data on MN and the Hb beta locus.

A sample of 28 informative families was studied for linkage between Hb beta and MN. Values of the neuterized recombination fraction from these and other families from the literature excluded a recombination fraction of less than .30 between these loci. Our results support different recombination values for males and females (theta equals .34 abd .50, respectively). A simple approach to estimate the sample size required as well as a study of the relationship between sibship size and sample size under conditions of loose linkage are also presented.     

Nonsyndromic cleft lip with or without cleft palate: evidence of linkage to BCL3 in 17 multigenerational families.

Nonsyndromic cleft lip with or without cleft palate (CL/P) is a common craniofacial developmental defect. Recent segregation analyses have suggested that major genes play a role in the etiology of CL/P. Linkage to 22 candidate genes was tested in 11 multigenerational families with CL/P, and 21 of these candidates were excluded. APOC2, 19q13.1, which is linked to the proto-oncogene BCL3, gave suggestive evidence for linkage to CL/P. The study was expanded to include a total of 39 multigenerational CL/P families. Linkage was tested in all families, using an anonymous marker, D19S178, and intragenic markers in BCL3 and APOC2. Linkage was tested under two models, autosomal dominant with reduced penetrance and affecteds only. Homogeneity testing on the two-point data gave evidence of heterogeneity at APOC2 under the affecteds-only model. Both models showed evidence of heterogeneity, with 43% of families linked at zero recombination to BCL3 when marker data from BCL3 and APOC2 were included. A maximum multipoint LOD score of 7.00 at BCL3 was found among the 17 families that had posterior probabilities > = 50% in favor of linkage. The transmission disequilibrium test provided additional evidence for linkage with the 3 allele of BCL3 more often transmitted to affected children. These results suggest that BCL3, or a nearby gene, plays a role in the etiology of CL/P in some families.     

In Swedish families with hereditary prostate cancer, linkage to the HPC1 locus on chromosome 1q24-25 is restricted to families with early-onset prostate cancer.

Prostate cancer clusters in some families, and an estimated 5%-10% of all cases are estimated to result from inheritance of prostate cancer-susceptibility genes. We previously reported evidence of linkage to the 1q24-25 region (HPC1) in 91 North American and Swedish families each with multiple cases of prostate cancer (Smith et al. 1996). In the present report we analyze 40 (12 original and 28 newly identified) Swedish families with hereditary prostate cancer (HPC) that, on the basis of 40 markers spanning a 25-cM interval within 1q24-25, have evidence of linkage. In the complete set of families, a maximum two-point LOD score of 1.10 was observed at D1S413 (at a recombination fraction [theta] of.1), with a maximum NPL (nonparametric linkage) Z score of 1.64 at D1S202 (P=.05). The evidence of linkage to this region originated almost exclusively from the subset of 12 early-onset (age <65 years) families, which yielded a maximum LOD score of 2.38 at D1S413 (straight theta=0) and an NPL Z score of 1.95 at D1S422 (P=.03). Estimates from heterogeneity tests suggest that, within Sweden, as many as 50% of early-onset families had evidence of linkage to the HPC1 region. These results are consistent with the hypothesis of linkage to HPC1 in a subset of families with prostate cancer, particularly those with an early age at diagnosis.     

Effect of recombination on the accuracy of the likelihood method for detecting positive selection at amino acid sites

Maximum-likelihood methods based on models of codon substitution accounting for heterogeneous selective pressures across sites have proved to be powerful in detecting positive selection in protein-coding DNA sequences. Those methods are phylogeny based and do not account for the effects of recombination. When recombination occurs, such as in population data, no unique tree topology can describe the evolutionary history of the whole sequence. This violation of assumptions raises serious concerns about the likelihood method for detecting positive selection. Here we use computer simulation to evaluate the reliability of the likelihood-ratio test (LRT) for positive selection in the presence of recombination. We examine three tests based on different models of variable selective pressures among sites. Sequences are simulated using a coalescent model with recombination and analyzed using codon-based likelihood models ignoring recombination. We find that the LRT is robust to low levels of recombination (with fewer than three recombination events in the history of a sample of 10 sequences). However, at higher levels of recombination, the type I error rate can be as high as 90%, especially when the null model in the LRT is unrealistic, and the test often mistakes recombination as evidence for positive selection. The test that compares the more realistic models M7 (beta) against M8 (beta and omega) is more robust to recombination, where the null model M7 allows the positive selection pressure to vary between 0 and 1 (and so does not account for positive selection), and the alternative model M8 allows an additional discrete class with omega = d(N)/d(S) that could be estimated to be >1 (and thus accounts for positive selection). Identification of sites under positive selection by the empirical Bayes method appears to be less affected than the LRT by recombination.     

Linkage homogeneity near the fragile X locus in normal and fragile X families

The fragile X syndrome locus, FRAXA, is located at Xq27. Until recently, few polymorphic loci had been genetically mapped close to FRAXA. This has been attributed to an increased frequency of recombination at Xq27, possibly associated with the fragile X mutation. In addition, the frequency of recombination around FRAXA has been reported to vary among fragile X families. These observations suggested that the genetic map at Xq27 in normal populations was different from that in fragile X populations and that the genetic map also varied within the fragile X population. Such variability would reduce the reliability of carrier risk estimates based on DNA studies in fragile X families. Five polymorphic loci have now been mapped to within 4 cM of FRAXA--DXS369, DXS297, DXS296, IDS, and DXS304. The frequency of recombination at Xq26-q28 was evaluated using data at these loci and at more distant loci from 112 families with the fragile X syndrome. Two-point and multipoint linkage analyses failed to detect any difference in the recombination fractions in fragile X versus normal families. Two-point and multipoint tests of linkage homogeneity failed to detect any evidence of linkage heterogeneity in the fragile X families. On the basis of this analysis, genetic maps derived from large samples of normal families and those derived from fragile X families are equally valid as the basis for calculating carrier risk estimates in a particular family.     

Linkage of a gene regulating dopamine-beta-hydroxylase activity and the ABO blood group locus.

Previous studies have presented evidence suggesting that levels of dopamine-beta-hydroxylase (DBH) activity are controlled by a gene linked to the ABO blood group locus. In this study, linkage analyses in four large families of whites and one family of blacks were performed on the untransformed and on the square root--and natural log--transformed DBH activity. In the families of white individuals, the results of both the sib-pair and lod-score linkage analyses strongly indicate that a gene regulating DBH activity is linked to the ABO blood group locus on chromosome 9q (i.e., lod score 5.88 at a recombination fraction of .0). However, the transformation used has a large effect on the maximum lod score and estimated recombination fraction. This putative gene does not appear to be polymorphic in the family of blacks.     

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 score test for the linkage analysis of qualitative and quantitative traits based on identity by descent data from sib-pairs

We propose a general likelihood-based approach to the linkage analysis of qualitative and quantitative traits using identity by descent (IBD) data from sib-pairs. We consider the likelihood of IBD data conditional on phenotypes and test the null hypothesis of no linkage between a marker locus and a gene influencing the trait using a score test in the recombination fraction theta between the two loci. This method unifies the linkage analysis of qualitative and quantitative traits into a single inferential framework, yielding a simple and intuitive test statistic. Conditioning on phenotypes avoids unrealistic random sampling assumptions and allows sib-pairs from differing ascertainment mechanisms to be incorporated into a single likelihood analysis. In particular, it allows the selection of sib-pairs based on their trait values and the analysis of only those pairs having the most informative phenotypes. The score test is based on the full likelihood, i.e. the likelihood based on all phenotype data rather than just differences of sib-pair phenotypes. Considering only phenotype differences, as in Haseman and Elston (1972) and Kruglyak and Lander (1995), may result in important losses in power. The linkage score test is derived under general genetic models for the trait, which may include multiple unlinked genes. Population genetic assumptions, such as random mating or linkage equilibrium at the trait loci, are not required. This score test is thus particularly promising for the analysis of complex human traits. The score statistic readily extends to accommodate incomplete IBD data at the test locus, by using the hidden Markov model implemented in the programs MAPMAKER/SIBS and GENEHUNTER (Kruglyak and Lander, 1995; Kruglyak et al., 1996). Preliminary simulation studies indicate that the linkage score test generally matches or outperforms the Haseman-Elston test, the largest gains in power being for selected samples of sib-pairs with extreme phenotypes.