Extensions to methods of sib-pair linkage analyses.

Sib-pair methods provide simple, robust, easily implemented ways to screen for linkage between a marker locus and a suspected disease susceptibility locus. The basic analysis reflects the idea that, in the presence of linkage, siblings who share more alleles at the marker locus should also tend to be concordant for disease. Available sib-pair methods do not lead directly to estimates of risk associated with nongenetic factors, may not account for a variable age-at-onset, or may require that the age-at-onset distribution be known. In this paper, we propose a method for sib-pair linkage analyses that allows for a variable age-at-onset using a logistic model, easily allows modelling of nongenetic factors, reflects the correlation of sibs within a sibship, and allows for nonzero risk in those without the susceptibility genotype. Based on a limited number of simulations, the method has as good or better power than another recently described method that also allows for a variable age-at-onset.     

Improved sib-pair linkage test for disease susceptibility loci

An improved sib-pair test for linkage is introduced which is superior to the previously proposed tests. The test is derived from the standard chi-squared goodness of fit statistic by restricting the alternative hypothesis to the genetically possible. Critical values are given and exact power comparisons are made with the previously proposed tests. The new test is shown to be more powerful for finite samples as well as being asymptotically uniformly most powerful. © 1993 Wiley-Liss, Inc.     

A comparison of sib-pair linkage tests for disease susceptibility loci

An analytical study is conducted of the properties of statistical tests to detect linkage between a disease locus and a very polymorphic marker locus when data on sib pairs are available. In most instances the most powerful test is the test based on the mean number of marker alleles shared identical by descent by the two members of a sib pair, and the most efficient sampling strategy is almost always to sample only pairs with both sibs affected. We show it is valid to use the information from all possible sib pairs as though they came from separate families when data on sibships of size three or larger are available, though more power may be obtained if different weights are given to the different sibship sizes.     

Methods to estimate genetic components of variance for quantitative traits in family studies

The aim of this paper was to compare several methods of estimating the genetic components of a quantitative trait in familial data. The Expectation and Maximization (E‐M) algorithm, the Newton‐Raphson method, and the scoring method were compared for estimating polygenic and environmental effects on nuclear families. We also compared scoring and quasilikelihood (QL) methods when a linked genetic marker was available to estimate effects from a major gene. Generally, all procedures performed similarly in estimating polygenic and environmental variance components. The E‐M algorithm yielded more precise estimators when heritability was low. The scoring method was much faster than the other methods and yielded slightly more precise estimates of mean effects but slightly less precise estimates of the variance components. Estimates of major gene effects were not affected by the number of alleles at the trait locus. For these relatively large sample sizes, QL and scoring had similar precision, but QL took 32 times longer than scoring. Finally, we compared the results of applying these methods to data from the Bogalusa Heart Study. Results showed larger imprecision when the QL method was applied, consistent with earlier studies that showed decreased precision of quasilikelihood compared with maximum likelihood in moderately small sample sizes. Genet. Epidemiol. 17:64–76, 1999. © 1999 Wiley‐Liss, Inc.     

Power and robustness of a score test for linkage analysis of quantitative traits using identity by descent data on sib pairs

Identification of genes involved in complex traits by traditional (lod score) linkage analysis is difficult due to many complicating factors. An unfortunate drawback of non-parametric procedures in general, though, is their low power to detect genetic effects. Recently, Dudoit and Speed [2000] proposed using a (likelihood-based) score test for detecting linkage with IBD data on sib pairs. This method uses the likelihood for theta, the recombination fraction between a trait locus and a marker locus, conditional on the phenotypes of the two sibs to test the null hypothesis of no linkage (theta = (1/2)). Although a genetic model must be specified, the approach offers several advantages. This paper presents results of simulation studies characterizing the power and robustness properties of this score test for linkage, and compares the power of the test to the Haseman-Elston and modified Haseman-Elston tests. The score test is seen to have impressively high power across a broad range of true and assumed models, particularly under multiple ascertainment. Assuming an additive model with a moderate allele frequency, in the range of p = 0.2 to 0.5, along with heritability H = 0.3 and a moderate residual correlation rho = 0.2 resulted in a very good overall performance across a wide range of trait-generating models. Generally, our results indicate that this score test for linkage offers a high degree of protection against wrong assumptions due to its strong robustness when used with the recommended additive model.     

Score test for detecting linkage to quantitative traits

The two most popular methods to detect linkage of a quantitative trait to a marker are the Haseman-Elston regression method and the variance components likelihood-ratio test. In the literature, these methods are frequently compared and the relative advantages and disadvantages of each method are well known. In this article, we derive a score test for the variance component attributable to a specific quantitative trait locus and show that for sib-pairs it is mathematically equivalent to a recently proposed version of the Haseman-Elston method that optimally combines the sum squared and the difference squared of the centered phenotype values of the sibs. Because score tests and likelihood-ratio tetsts are equivalent for large sample sizes, the variance components likelihood-ratio test is also asymptotically equivalent to this optimal Haseman-Elston test. This fact gives a theoretical explanation of the empirical observation from simulation studies reporting similar power of the variance components likelihood-ratio test and the optimal Haseman-Elston method. Perhaps more importantly for practical purposes, the score test can also be extended in a natural way to support the simultaneous analysis of more than two subjects and multivariate phenotypes.     

The Maximum-Likelihood-Binomial method revisited: a robust approach for model-free linkage analysis of quantitative traits in large sibships

Model‐free linkage analysis methods, based on identity‐by‐descent allele sharing, are commonly used for complex trait analysis. The Maximum‐Likelihood‐Binomial (MLB) approach, which is based on the hypothesis that parental alleles are binomially distributed among affected sibs, is particularly popular. An extension of this method to quantitative traits (QT) has been proposed (MLB‐QTL), based on the introduction of a latent binary variable capturing information about the linkage between the QT and the marker. Interestingly, the MLB‐QTL method does not require the decomposition of sibships into constituent sibpairs and requires no prior assumption about the distribution of the QT. We propose a new formulation of the MLB method for quantitative traits (nMLB‐QTL) that explicitly takes advantage of the independence of paternal and maternal allele transmission under the null hypothesis of no linkage. Simulation studies under H₀ showed that the nMLB‐QTL method generated very consistent type I errors. Furthermore, simulations under the alternative hypothesis showed that the nMLB‐QTL method was slightly, but systematically more powerful than the MLB‐QTL method, whatever the genetic model, residual correlation, ascertainment strategy and sibship size considered. Finally, the power of the nMLB‐QTL method is illustrated by a chromosome‐wide linkage scan for a quantitative endophenotype of leprosy infection. Overall, the nMLB‐QTL method is a robust, powerful, and flexible approach for detecting linkage with quantitative phenotypes, particularly in studies of non Gaussian phenotypes in large sibships. Genet. Epidemiol. 35:46–56, 2011. © 2010 Wiley‐Liss, Inc.     

Statistical validity of the Haseman-Elston sib-pair test in small samples

Simulation experiments were used to determine the empiric type I error rate of the Haseman-Elston sib-pair test for linkage between a quantitative trait and a marker locus in samples with 60 or fewer sib pairs. The effect of different levels of marker-locus heterozygosity on statistical validity was also considered. The test was performed on the trait and each of five unlinked markers, and evaluated using two different degrees of freedom for the t-distribution. The number of degrees of freedom in the first evaluation was based on the number of sib pairs in each sibship, ∑ s_I(s_I - 1)/2 - 2. In the second evaluation, the number of degrees of freedom was based on the number of sibs minus 1 in each sibship, ∑ (s_i - 1) - 2. Empirically determined type I error rates using C s_I(s_I - 1)/2 - 2 degrees of freedom were slightly liberal. For ∑ (s_i - 1) - 2 degrees of freedom, the estimated empiric p-values were nearly identical to their respective nominal p-values. Decreasing levels of heterozygosity did not increase the empiric type I error rate when the sample size was small. © 1993 Wiley-Liss, Inc.     

A brute force dichotomization approach to quantitative trait linkage analysis

The effect of dichotomizing a continuous phenotype in linkage analysis of a simulated oligogenic trait is explored. We conclude that dichotomization does not in itself preclude the detection of loci which account for as little as 16% of the genetic variance of the disease. The effects of inclusion of known covariates and quantitative trait linkage analysis are also discussed. © 1995 Wiley-Liss, Inc.     

Removing phenotypic distribution assumptions from tests of linkage disequilibrium for quantitative traits

The transmission disequilibrium test (TDT) has become a family-based method of reference to search for linkage disequilibrium (LD). Although it was first developed for dichotomous traits, numerous approaches have extended the TDT to quantitative phenotypes that either rely on regression or variance component techniques. Both of these approaches are based on some phenotypic distribution assumptions, the violation of which can lead to inflation of type I error rates, and derive information from phenotypic variability, so that their power is very low under some selection schemes (e.g., one-tailed selection). We propose a new family-based test of association for quantitative traits, denoted maximum-likelihood-binomial (MLB)-QTDT, which addresses the two previous issues by incorporating a latent binary variable that captures the LD information between the marker allele and the quantitative phenotype. The method can be understood as a classical TDT for binary traits that would include pure affected and pure unaffected children, and the probability for a child to be affected or unaffected depends on his/her quantitative phenotypic value. Simulation studies under the null hypothesis show that the MLB-QTDT provides very consistent type I errors even in small and/or selected samples. Under the alternative hypothesis, the MLB-QTDT has good power to analyze one-tailed selected samples, and performs as well as a classical approach in other designs. The MLB-QTDT is a flexible distribution-free method to test for LD with quantitative phenotypes in nuclear families, and can easily incorporate previous extensions developed in the context of family-based association studies with binary traits.     

Unified sampling approach for multipoint linkage disequilibrium mapping of qualitative and quantitative traits

Rapid development in biotechnology has enhanced the opportunity to deal with multipoint gene mapping for complex diseases, and association studies using quantitative traits have recently generated much attention. Unlike the conventional hypothesis-testing approach for fine mapping, we propose a unified multipoint method to localize a gene controlling a quantitative trait. We first calculate the sample size needed to detect linkage and linkage disequilibrium (LD) for a quantitative trait, categorized by decile, under three different modes of inheritance. Our results show that sampling trios of offspring and their parents from either extremely low (EL) or extremely high (EH) probands provides greater statistical power than sampling in the intermediate range. We next propose a unified sampling approach for multipoint LD mapping, where the goal is to estimate the map position (tau) of a trait locus and to calculate a confidence interval along with its sampling uncertainty. Our method builds upon a model for an expected preferential transmission statistic at an arbitrary locus conditional on the sampling scheme, such as sampling from EL and EH probands. This approach is valid regardless of the underlying genetic model. The one major assumption for this model is that no more than one quantitative trait locus (QTL) is linked to the region being mapped. Finally we illustrate the proposed method using family data on total serum IgE levels collected in multiplex asthmatic families from Barbados. An unobserved QTL appears to be located at tau; = 41.93 cM with 95% confidence interval of (40.84, 43.02) through the 20-cM region framed by markers D12S1052 and D12S1064 on chromosome 12. The test statistic shows strong evidence of linkage and LD (chi-square statistic = 18.39 with 2 df, P-value = 0.0001).     

The power of two-locus affected sib-pair linkage analysis to detect interacting disease loci

It has been shown that two-locus linkage analysis can, for some two-locus disease models, be used to detect effects at disease loci that do not reach significance in a genome scan. However, few examples exist where two-locus linkage has been successfully used to map genes. We study the possible gain in power of affected sib-pair nonparametric two-locus linkage analysis for two-locus models which fulfil the two-locus triangle constraints. Using a new parameterization of the two-locus joint identity-by-descent sharing probabilities we can, for fixed marginal sharing at both of two unlinked disease loci, derive a two-locus distribution such that the power of a two-locus analysis is maximized. In a simulation study we look at two test statistics, the two-locus maximum likelihood score and the correlation between nonparametric linkage scores, and study power as a function of marginal sharing. We show that in a best-case scenario two-locus linkage can have considerable power to detect pairs of interacting loci if there is a moderate increase in allele sharing at one of the two loci, even if there is a very small increase in allele sharing at the other locus. But we also show that the power to detect interacting loci in a two-locus analysis decreases as the marginal sharing at the two loci decreases and for any distribution with a small increase in allele sharing at both loci the power of a two-locus analysis is always low.     

A novel polylocus method for linkage analysis using the lod-score or affected sib-pair method

A novel approach to combining data from multiple linked loci is proposed that can provide substantial increases in power over normal two-point linkage analysis or sib-pair analysis, with a substantial saving in computing time over traditional multipoint methods. © 1993 Wiley-Liss, Inc.     

Ordered subset analysis in genetic linkage mapping of complex traits

Etiologic heterogeneity is a fundamental feature of complex disease etiology; genetic linkage analysis methods to map genes for complex traits that acknowledge the presence of genetic heterogeneity are likely to have greater power to identify subtle changes in complex biologic systems. We investigate the use of trait-related covariates to examine evidence for linkage in the presence of heterogeneity. Ordered-subset analysis (OSA) identifies subsets of families defined by the level of a trait-related covariate that provide maximal evidence for linkage, without requiring a priori specification of the subset. We propose that examining evidence for linkage in the subset directly may result in a more etiologically homogeneous sample. In turn, the reduced impact of heterogeneity will result in increased overall evidence for linkage to a specific region and a more distinct lod score peak. In addition, identification of a subset defined by a specific trait-related covariate showing increased evidence for linkage may help refine the list of candidate genes in a given region and suggest a useful sample in which to begin searching for trait-associated polymorphisms. This method provides a means to begin to bridge the gap between initial identification of linkage and identification of the disease predisposing variant(s) within a region when mapping genes for complex diseases. We illustrate this method by analyzing data on breast cancer age of onset and chromosome 17q [Hall et al., 1990, Science 250:1684-1689]. We evaluate OSA using simulation studies under a variety of genetic models.     

Mapping quantitative traits in unselected families: algorithms and examples

Linkage analysis has been widely used to identify from family data genetic variants influencing quantitative traits. Common approaches have both strengths and limitations. Likelihood ratio tests typically computed in variance component analysis can accommodate large families but are highly sensitive to departure from normality assumptions. Regression‐based approaches are more robust but their use has primarily been restricted to nuclear families. In this paper, we develop methods for mapping quantitative traits in moderately large pedigrees. Our methods are based on the score statistic, which in contrast to the likelihood ratio statistic can use nonparametric estimators of variability to achieve robustness of the false‐positive rate against departures from the hypothesized phenotypic model. Because the score statistic is easier to calculate than the likelihood ratio statistic, our basic mapping methods utilize relatively simple computer code that performs statistical analysis on output from any program that computes estimates of identity by descent. This simplicity also permits development and evaluation of methods to deal with multivariate and ordinal phenotypes, and with gene‐gene and gene‐environment interaction. We demonstrate our methods on simulated data and on fasting insulin, a quantitative trait measured in the Framingham Heart Study. Genet. Epidemiol. 33:617–627, 2009. © 2009 Wiley‐Liss, Inc.     

WPC test based on randomization for analyzing quantitative traits on simulated pedigrees

The weighted pairwise correlation (WPC) approach provides simple and flexible tests for genetic linkage which may be adapted to qualitative, quantitative or age-dependent traits. These tests also seem to have good power. However, when working with large pedigrees, a disease susceptibility gene not linked to the marker studied induces correlations of the trait values, leading to inflated type I errors for these tests. We propose here a new approach for inference based on the randomization of the alleles following the Mendelian laws and conditioning on the alleles of the founders. This approach is applied to the analysis of the quantitative traits in a set of simulated pedigrees. The a posteriori comparison of the findings to the true model indicates directions for future work. © 1997 Wiley-Liss, Inc.     

Multipoint linkage analysis of quantitative traits on sex-chromosomes

Variance component models form a powerful and flexible tool for multipoint linkage analysis of quantitative traits. Estimates of genetic similarity are needed for the variance component model to detect linkage and to locate genes, and two methods are commonly used to calculate multipoint identity-by-descent (IBD) estimates for autosomes. Fulker et al. ([1995] Am. J. Hum. Genet. 56: 1229-1233) and Almasy and Blangero ([1998] Am. J. Hum. Genet. 62: 119-121) used multiple regression to estimate the IBD sharing along a chromosome, while the approach of Kruglyak and Lander ([1995] Am. J. Hum. Genet. 57: 439-454) is based on a hidden Markov model. In this paper, we modify the variance component model to accommodate sex-chromosomes, and we extend both multipoint IBD estimation methods to accommodate sex-linked loci. Simulation studies demonstrate the power and precision of the variance component model to detect QTLs located on the sex-chromosome. The two multipoint IBD estimation methods have the same accuracy to identify QTL position, but the hidden Markov model yields a larger average maximum LOD score to detect linkage than the regression model. The extension of the multipoint IBD estimation methods and the variance component model to the X chromosome shows that the variance component model is a powerful and flexible tool for linkage analysis of quantitative traits on both autosomes and sex-chromosomes.     

Detecting case-control expression quantitative trait loci using locally most powerful or maximin robust rank tests

In testing genome-wide gene expression quantitative trait loci, efficiency robust statistical methods and their computational convenience are most relevant. For this purpose, we propose to use a modified locally most powerful rank test for the analysis of case-control expression data. This modified rank test statistic is computationally simple, robust for non-normally distributed expression data, and asymptotically locally most powerful. It depends on the specification of a location distribution form for data but is not sensitive to misspecifications. When such a location distribution form cannot be specified, we apply Gastwirth's maximin efficiency robust rank test to gene expression data to maximize the worst Pitman asymptotic relative efficiency among a family of location distributions. We conduct simulation studies to assess their performance and use an application to real data for illustration.