Quantitative linkage: a statistical procedure for its detection and estimation

A new approach for detecting and estimating quantitative linkage which uses sibship data is presented. Using a nested analysis of variance design (with marker genotype nested within sibship), it is shown that under the null hypothesis of no linkage, the expected between marker genotype within sibship mean square (EMSbeta) is equal to the expected within marker genotype within sibship mean square (EMSe), while under the alternative hypothesis of linkage, the first is greater than the second. Thus the regular F-ratio, MSbeta/MSe, can be used to test for quantitative linkage. This is true for both backcross and intercross matings and whether or not there is dominance at the marker locus. A second test involving the comparison of the within marker genotype within sibship variances is available for intercross matings. A maximum likelihood procedure for the estimation for the recombination frequency is also presented.     

Family-based tests of association and/or linkage

The large literature on family-based tests of association and/or linkage is reviewed, concentrating on the underlying principles and on recent methodological developments. We explain the distinction between testing for association and testing for linkage, and give our views on the circumstances in which each is the appropriate null hypothesis.     

Robust TDT-type candidate-gene association tests

In studies of association between genetic markers and a disease, the transmission disequilibrium test (TDT) has become a standard procedure. It was introduced originally as a test for linkage in the presence of association and can be used as a test for association under appropriate assumptions. The power of the TDT test for association between a candidate gene and disease depends on the underlying genetic model and the TDT is the optimal test if the additive model holds. Related methods have been obtained for a given mode of inheritance (e.g. dominant or recessive). Quite often, however, the true model is unknown and selection of a single method of analysis is problematic, since use of a test optimal for one genetic model usually leads to a substantial loss of power if another genetic model is the true one. The general approach of efficiency robustness has suggested two types of robust procedures, which we apply to TDT-type association tests. When the plausible range of alternative models is wide (e.g. dominant through recessive) our results indicate that the maximum (MAX) of several test statistics, each of which is optimal for quite different models, has good power under all genetic models. In situations where the set of possible models can be narrowed (e.g. dominant through additive) a simple linear combination also performs well. In general, the MAX has better power properties than the TDT for the study of candidate genes when the mode of inheritance is unknown.     

A new method of linkage analysis using LOD scores for quantitative traits supports linkage of monoamine oxidase activity to D17S250 in the Collaborative Study on the Genetics of Alcoholism pedigrees

OBJECTIVE: Although LOD score methods have been applied to diseases with complex modes of inheritance, linkage analysis of quantitative traits has tended to rely on non-parametric methods based on regression or variance components analysis. Here, we describe a new method for LOD score analysis of quantitative traits which does not require specification of a mode of inheritance. METHODS: The technique is derived from the MFLINK method for dichotomous traits. A range of plausible transmission models is constructed, constrained to yield the correct population mean and variance for the trait but differing with respect to the contribution to the variance due to the locus under consideration. Maximized LOD scores under homogeneity and admixture are calculated, as is a model-free LOD score which compares the maximized likelihoods under admixture assuming linkage and no linkage. These LOD scores have known asymptotic distributions and hence can be used to provide a statistical test for linkage. The method has been implemented in a program called QMFLINK. It was applied to data sets simulated using a variety of transmission models and to a measure of monoamine oxidase activity in 105 pedigrees from the Collaborative Study on the Genetics of Alcoholism. RESULTS: With the simulated data, the results showed that the new method could detect linkage well if the true allele frequency for the trait was close to that specified. However, it performed poorly on models in which the true allele frequency was much rarer. For the Collaborative Study on the Genetics of Alcoholism data set only a modest overlap was observed between the results obtained from the new method and those obtained when the same data were analysed previously using regression and variance components analysis. Of interest is that D17S250 produced a maximized LOD score under homogeneity and admixture of 2.6 but did not indicate linkage using the previous methods. However, this region did produce evidence for linkage in a separate data set, suggesting that QMFLINK may have been able to detect a true linkage which was not picked up by the other methods. CONCLUSION: The application of model-free LOD score analysis to quantitative traits is novel and deserves further evaluation of its merits and disadvantages relative to other methods.     

Predicting the number and sizes of IBD regions among family members and evaluating the family size requirement for linkage studies

With genotyping of high-density single nucleotide polymorphisms (SNPs) replacing that of microsatellite markers in linkage studies, it becomes possible to accurately determine the genomic regions shared identity by descent (IBD) by family members. In addition to evaluating the likelihood of linkage for a region with the underlining disease (the LOD score approach), an appropriate question to ask is what would be the expected number and sizes of IBD regions among the affecteds, as there could be more than one region reaching the maximum achievable LOD score for a given family. Here, we introduce a computer program to allow the prediction of the total number of IBD regions among family members and their sizes. Reversely, it can be used to predict the portion of the genome that can be excluded from consideration according to the family size and user-defined inheritance mode and penetrance. Such information has implications on the feasibility of conducting linkage analysis on a given family of certain size and structure or on a few small families when interfamily homogeneity can be assumed. It can also help determine the most relevant members to be genotyped for such a study. Simulation results showed that the IBD regions containing true mutations are usually larger than regions IBD due to random chance. We have made use of this feature in our program to allow evaluation of the identified IBD regions based on Bayesian probability calculation and simulation results.     

The effect of genotyping error in sib-pair genomewide linkage scans depends crucially upon the method of analysis

In genomewide linkage scans for complex diseases involving many loci with small genetic effects, it may be the case that no loci reach conventional statistical significance. A complementary method of evaluating linkage results, locus counting, may provide evidence for the existence of a number of genetic loci in these cases. Sib-pair study designs are often used in genomewide linkage scans, but because all genotype configurations are consistent with Mendelian inheritance, genotyping error will go largely undetected. Previous work on the effect of genotyping error has focused on a single disease locus. We considered the effect of two levels of genotyping error on genomewide evidence for linkage by using the simulated GAW 13 data. For affected sib-pair and non-parametric quantitative trait study designs, a 0.5% genotyping error rate reduced the number of independent linkage regions towards that expected under the null hypothesis of no linkage. A 2% genotyping error rate yielded less independent linkage regions than expected under the null hypothesis of no linkage. For a quantitative trait analysed using a parametric regression-based method, there was very little erosion of the linkage signal, even for error rates as high as 2%.     

Assessing the statistical power to detect linkage in a sample of 51 bipolar affective disorder pedigrees

We used computer simulation method to address the question of power in an initial collaborative sample of 51 bipolar affective disorder pedigrees. Simulations were performed for all possible combinations using (1) two levels of diagnostic stringency, (2) three transmission models, (3) locus heterogeneity, and (4) different assumed phenocopy rates. Some of the factors that affect the power to detect linkage are (1) the specification of the correct genetic model, (2) the degree of locus heterogeneity, and (3) the frequency of phenocopies. The first two assertions were supported by our simulation results, but varying the rates of phenocopy did not substantially alter the power of the sample until a critical point. However, it is important to point out that these results are dependent on the genetic models under study and on the use of the “correct” model (i.e., the one used to simulate the data). If we assume a dominant mode of inheritance and locus homogeneity, the power to detect linkage is 97.5% at a θ of .01. However, the power declines dramatically, to 60.5 and 14.7%, if only 75 and 50% of the families are linked, respectively. Locus heterogeneity has a similar effect on the power of the sample to exclude linkage. The relative lack of power in our data, in the presence of significant locus heterogeneity, and for an intermediate mode of inheritance, underscores the need for multicenter collaboration.     

A genome-wide linkage screen in the Amish with Parkinson disease points to chromosome 6

Parkinson disease (PD) is a common complex neurodegenerative disorder with an underlying genetic etiology that has been difficult to dissect. Although some PD risk genes have been discovered, most of the underlying genetic etiology remains unknown. To further elucidate the genetic component, we have undertaken a genome-wide linkage screen in an isolated founder population of Amish living in the Midwestern United States. We performed tests for linkage and for association using a marker set of nearly 6000 single-nucleotide polymorphisms. Parametric multipoint linkage analysis generated a logarithm of the odds of linkage (LOD) score of 2.44 on chromosome 6 in the SYNE1 gene, approximately 8 Mbp from the PARK2 gene. In a different region on chromosome 6 (～67 Mbp from PARK2) an association was found for rs4302647 (p < 4.0 × 10(-6) ), which is not within 300 kb of any gene. While the association exceeds Bonferroni correction, it may yet represent a false positive due to the small sample size and the low minor allele frequency. The minor allele frequency in affecteds is 0.07 compared to 0.01 in unaffecteds. Taken together, these results support involvement of loci on chromosome 6 in the genetic etiology of PD.     

The consistency of the posterior probability of linkage

When searching for trait loci along the genome, properly incorporating prior genomic information into the analysis will almost certainly increase the chance of success. Recently, we devised a method that utilizes such prior information in the mapping of trait genes for complex disorders (Vieland, 1998; Wang et al . 1999; Vieland et al . 2000). This method uses the posterior probability of linkage (PPL) based on the admixture model as a measure of linkage information. In this paper, we study the consistency of the PPL. It is shown that, as the number of pedigrees increases, the PPL converges in probability to 1 when there is linkage between the marker and a trait locus, and converges to 0 otherwise. This conclusion is shown to be true for general pedigrees and trait models, even when the likelihood functions are based on misspecified trait models. As part of the effort to prove this conclusion, it is shown that when there is no linkage, the maximum likelihood estimator of the recombination fraction in the admixture model is asymptotically 0.5, even when the admixture model misrepresents the true model.     

The additive genetic gamma frailty model for linkage analysis of age-of-onset variation

Age of onset is a key factor in the linkage analysis of many complex diseases. Current methods in nonparametric linkage analysis are mainly concentrated on the affected relative pairs or affected family members with age of onset information either ignored or taken into account by specifying age-dependent penetrances for liability classes. On the other hand, gamma frailty models were developed in the biostatistics literature to model familial aggregation of age of onset. However, these frailty models cannot be used directly for linkage analysis. This paper extends the gamma frailty model by incorporating inheritance vector information and provides a semiparametric approach for linkage testing. For a given inheritance vector at the putative disease locus, we construct an additive genetic gamma frailty for each individual within a nuclear family and use the Cox proportional hazard model to model age of onset. We derive the conditional hazard ratio parameter for sib pairs and define a likelihood ratio based LOD score statistic under our model. The EM algorithm is used for estimating the parameters and the maximum likelihood functions. Simulated data sets are used to illustrate these new statistical methods.     

Affected Sib Pair Tests in Inbred Populations

The affected‐sib‐pair (ASP) method for detecting linkage between a disease locus and marker loci was first established 50 years ago, and since then numerous modifications have been made. We modify two identity‐by‐state (IBS) test statistics of Lange ( Lange, 1986a, 1986b ) to allow for inbreeding in the population. We evaluate the power and false positive rates of the modified tests under three disease models, using simulated data. Before estimating false positive rates, we demonstrate that IBS tests are tests of both linkage and linkage disequilibrium between marker and disease loci. Therefore, the null hypothesis of IBS tests should be no linkage and no LD. When the population inbreeding coefficient is large, the false positive rates of Lange's tests become much larger than the nominal value, while those of our modified tests remain close to the nominal value. To estimate power with a controlled false positive rate, we choose the cutoff values based on simulated datasets under the null hypothesis, so that both Lange's tests and the modified tests generate same false positive rate. The powers of Lange's z‐test and our modified z‐test are very close and do not change much with increasing inbreeding. The power of the modified chi‐square test also stays stable when the inbreeding coefficient increases. However, the power of Lange's chi‐square test increases with increasing inbreeding, and is larger than that of our modified chi‐square test for large inbreeding coefficients. The power is high under a recessive disease model for both Lange's tests and the modified tests, though the power is low for additive and dominant disease models. Allowing for inbreeding is therefore appropriate, at least for diseases known to be recessive.     

Genome-wide linkage scan for spontaneous DZ twinning

In humans, spontaneous DZ twinning is known to have a genetic basis. A prior investigation in the Flemish and Dutch population showed that the phenotype of 'having DZ twins' was consistent with an autosomal monogenic dominant model, with a gene frequency of 3.5% and a female-specific lifetime penetrance of 10%. Recessive, X-linked, polygenic and sporadic models were rejected. This study reports on a genome-wide scan of 14 Flemish families containing 57 mothers of spontaneous DZ twins. Two-point linkage analysis using the autosomal dominant model showed nine chromosomal regions with a LOD score around 1. After multipoint linkage analysis, including heterogeneity, three chromosomes continued to give high LOD scores. These regions were further haplotyped with additional markers at 1 cM distance. The multipoint analysis was not in favour of linkage of the DZ twinning trait in most candidate genes and other regions (LOD score < -2) under the genetic model of autosomal dominance. To further evaluate the evidence for linkage given some uncertainty about the correct mode of inheritance of twinning susceptibility other models of inheritance were tested. Results of this analysis showed all models gave highest LOD scores under dominant models. If heterogeneity among the families is taken into account, the peaks that were observed on chromosome 2, 7 and 18 could well contain a potential gene contributing to DZ twinning. These results give suggestive evidence that the mode of inheritance of DZ twinning is probably more complex than was originally expected.     

When to conduct probabilistic linkage vs. deterministic linkage? A simulation study

IntroductionWhen unique identifiers are unavailable, successful record linkage depends greatly on data quality and types of variables available. While probabilistic linkage theoretically captures more true matches than deterministic linkage by allowing imperfection in identifiers, studies have shown inconclusive results likely due to variations in data quality, implementation of linkage methodology and validation method. The simulation study aimed to understand data characteristics that affect the performance of probabilistic vs. deterministic linkage.MethodsWe created ninety-six scenarios that represent real-life situations using non-unique identifiers. We systematically introduced a range of discriminative power, rate of missing and error, and file size to increase linkage patterns and difficulties. We assessed the performance difference of linkage methods using standard validity measures and computation time.ResultsAcross scenarios, deterministic linkage showed advantage in PPV while probabilistic linkage showed advantage in sensitivity. Probabilistic linkage uniformly outperformed deterministic linkage as the former generated linkages with better trade-off between sensitivity and PPV regardless of data quality. However, with low rate of missing and error in data, deterministic linkage performed not significantly worse. The implementation of deterministic linkage in SAS took less than 1 min, and probabilistic linkage took 2 min to 2 h depending on file size.DiscussionOur simulation study demonstrated that the intrinsic rate of missing and error of linkage variables was key to choosing between linkage methods. In general, probabilistic linkage was a better choice, but for exceptionally good quality data (<5% error), deterministic linkage was a more resource efficient choice.     

Effect of misspecification of gene frequency on the two-point LOD score.

In this study, we used computer simulation of simple and complex models to ask: (1) What is the penalty in evidence for linkage when the assumed gene frequency is far from the true gene frequency? (2) If the assumed model for gene frequency and inheritance are misspecified in the analysis, can this lead to a higher maximum LOD score than that obtained under the true parameters? Linkage data simulated under simple dominant, recessive, dominant and recessive with reduced penetrance, and additive models, were analysed assuming a single locus with both the correct and incorrect dominance model and assuming a range of different gene frequencies. We found that misspecifying the analysis gene frequency led to little penalty in maximum LOD score in all models examined, especially if the assumed gene frequency was lower than the generating one. Analysing linkage data assuming a gene frequency of the order of 0.01 for a dominant gene, and 0.1 for a recessive gene, appears to be a reasonable tactic in the majority of realistic situations because underestimating the gene frequency, even when the true gene frequency is high, leads to little penalty in the LOD score.     

Idiopathic haemochromatosis in the Australian population: HLA linkage and recessivity

Studies of 73 unrelated patients and 19 families with idiopathic haemochromatosis are reported. The unrelated patients showed a highly significant association between the disease and HLA-A3. There was a less strong association with HLA-B7 and HLA-DRw2 attributed to the linkage disequilibrium between HLA-A3, B7, and DRw2. Lod scores and haplotytype analysis of the families indicated a recessive mode of inheritance for an idiopathic haemochromatosis susceptibility factor in close linkage with the HLA region. These results, for Australian caucasoid patients, are not in total agreement with those reported in studies of other populations.     

Systematic search for major genes in schizophrenia: Methodological issues and results from chromosome 12

We describe a method of systematically searching for major genes in disorders of unknown mode of inheritance, using linkage analysis. Our method is designed to minimize the probability of missing linkage due to inadequate exploration of data. We illustrate this method with the results of a search for a locus for schizophrenia on chromosome 12 using 22 highly polymorphic markers in 23 high density pedigrees. The markers span approximately 85–90% of the chromosome and are on average 9.35 cM apart. We have analysed the data using the most plausible current genetic models and allowing for the presence of genetic heterogeneity. None of the markers was supportive of linkage and the distribution of the heterogeneity statistics was in accordance with the null hypothesis. © 1995 Wiley-Liss, Inc.     

Power and Related Statistical Properties of Conditional Likelihood Score Tests for Association Studies in Nuclear Families with Parental Genotypes

Both population based and family based case control studies are used to test whether particular genotypes are associated with disease. While population based studies have more power, cryptic population stratification can produce false‐positive results. Family‐based methods have been introduced to control for this problem. This paper presents the full likelihood function for family‐based association studies for nuclear families ascertained on the basis of their number of affected and unaffected children. The likelihood of a family factors into the probability of parental mating type, conditional on offspring phenotypes, times the probability of offspring genotypes given their phenotypes and the parental mating type. The first factor can be influenced by population stratification, whereas the latter factor, called the conditional likelihood, is not. The conditional likelihood is used to obtain score tests with proper size in the presence of population stratification (see also Clayton (1999) and Whittemore & Tu (2000) ). Under either the additive or multiplicative model, the TDT is known to be the optimal score test when the family has only one affected child. Thus, the class of score tests explored can be considered as a general family of TDT‐like procedures. The relative informativeness of the various mating types is assessed using the Fisher information, which depends on the number of affected and unaffected offspring and the penetrances. When the additive model is true, families with parental mating type Aa×Aa are most informative. Under the dominant (recessive) model, however, a family with mating type Aa×aa (AA×Aa) is more informative than a family with doubly heterozygous (Aa×Aa) parents. Because we derive explicit formulae for all components of the likelihood, we are able to present tables giving required sample sizes for dominant, additive and recessive inheritance models.     

A likelihood-based extended admixture model of oligogenic inheritance in 'model-based' and 'model-free' analysis

The admixture test of linkage heterogeneity is the most often and most successfully applied oligogenic-model linkage and/or LD analysis method. Full two-locus model linkage analysis is possible, but can be computationally intensive and difficult to interpret because of the need to specify so many indeterminate parameters. A novel, computationally efficient method is proposed for combining single locus lod scores which can allow for varying degrees of epistatic interaction. This method can be applied to two-point or multipoint (using complex-valued recombination fractions) linkage and/or linkage disequilibrium analysis to jointly test for multiple unlinked disease loci. Unlike the traditional admixture test, this algorithm permits joint analysis of multiple disease loci with different modes of inheritance for each, and can be applied to 'model-free' analysis as well through the use of 'pseudomarkers'. Software is available for computation of the various likelihood ratio tests described, for comparison of a variety of possible hypotheses regarding locus homogeneity, locus heterogeneity, and epistasis.     

Combined high resolution linkage and association mapping of quantitative trait loci

In this paper, we investigate variance component models of both linkage analysis and high resolution linkage disequilibrium (LD) mapping for quantitative trait loci (QTL). The models are based on both family pedigree and population data. We consider likelihoods which utilize flanking marker information, and carry out an analysis of model building and parameter estimations. The likelihoods jointly include recombination fractions, LD coefficients, the average allele substitution effect and allele dominant effect as parameters. Hence, the model simultaneously takes care of the linkage, LD or association and the effects of the putative trait locus. The models clearly demonstrate that linkage analysis and LD mapping are complementary, not exclusive, methods for QTL mapping. By power calculations and comparisons, we show the advantages of the proposed method: (1) population data can provide information for LD mapping, and family pedigree data can provide information for both linkage analysis and LD mapping; (2) using family pedigree data and a sparse marker map, one may investigate the prior suggestive linkage between trait locus and markers to obtain low resolution of the trait loci, because linkage analysis can locate a broad candidate region; (3) with the prior knowledge of suggestive linkage from linkage analysis, both population and family pedigree data can be used simultaneously in high resolution LD mapping based on a dense marker map, since LD mapping can increase the resolution for candidate regions; (4) models of high resolution LD mappings using two flanking markers have higher power than that of models of using only one marker in the analysis; (5) excluding the dominant variance from the analysis when it does exist would lose power; (6) by performing linkage interval mappings, one may get higher power than by using only one marker in the analysis.     

Assessing genetic linkage and association with robust components of variance approaches

Simulation studies are used to explore the properties of procedures for estimating components of variance and constructing test statistics in genetic linkage studies of quantitative traits. We evaluated the bias and median squared error of estimates of the linked additive genetic variance obtained by regression, maximum likelihood and quasilikelihood estimation procedures. The quasilikelihood and regression procedures provided unbiased estimates of the additive component of variance. Maximum likelihood procedures that assumed multivariate normality were biased for most sample sizes considered but had more precision for most generating models than regression or quasilikelihood methods did. Wald tests derived from quasilikelihood procedures had similar or greater power than Wald tests based upon estimators from maximum likelihood analyses. Quasilikelihood estimation may therefore be preferable whenever there is uncertainty about the generating distribution for the error variance, but the robustness of this approach is offset by its required computational complexity.