A Family‐Based Joint Test for Mean and Variance Heterogeneity for Quantitative Traits

Traditional quantitative trait locus (QTL) analysis focuses on identifying loci associated with mean heterogeneity. Recent research has discovered loci associated with phenotype variance heterogeneity (vQTL), which is important in studying genetic association with complex traits, especially for identifying gene–gene and gene–environment interactions. While several tests have been proposed to detect vQTL for unrelated individuals, there are no tests for related individuals, commonly seen in family‐based genetic studies. Here we introduce a likelihood ratio test (LRT) for identifying mean and variance heterogeneity simultaneously or for either effect alone, adjusting for covariates and family relatedness using a linear mixed effect model approach. The LRT test statistic for normally distributed quantitative traits approximately follows χ²‐distributions. To correct for inflated Type I error for non‐normally distributed quantitative traits, we propose a parametric bootstrap‐based LRT that removes the best linear unbiased prediction (BLUP) of family random effect. Simulation studies show that our family‐based test controls Type I error and has good power, while Type I error inflation is observed when family relatedness is ignored. We demonstrate the utility and efficiency gains of the proposed method using data from the Framingham Heart Study to detect loci associated with body mass index (BMI) variability.     

A Comparison of the Likelihood Ratio Test and the Variance‐Stabilising Transformation‐Based Tests for Detecting Association of Rare Variants

In a recent paper in this journal, the use of variance‐stabilising transformation techniques was proposed to overcome the problem of inadequacy in normality approximation when testing association for a low‐frequency variant in a case‐control study. It was shown that tests based on the variance‐stabilising transformations are more powerful than Fisher's exact test while controlling for type I error rate. Earlier in the journal, another study had shown that the likelihood ratio test (LRT) is superior to Fisher's exact test, Wald's test, and Pearson's χ² test in testing association for low‐frequency variants. Thus, it is of interest to make a direct comparison between the LRT and the tests based on the variance‐stabilising transformations. In this commentary, we show that the LRT and the variance‐stabilising transformation‐based tests have comparable power greater than Fisher's exact test, Wald's test, and Pearson's χ² test.     

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.     

Linkage analysis in Spanish families with nonspecific X‐linked mental retardation: Significant linkage at Xq13–q21

Mental retardation (MR) is a genetically heterogeneous, clinically variable condition. Many cases of MR are linked to the X chromosome. The aim of this study was to identify candidate loci for nonspecific MR in Spanish samples. We selected seven families with nonspecific MR and a pattern of inheritance compatible with an X‐linked disorder and a group of 26 sib pairs of mentally retarded individuals. We performed linkage analysis with a panel of 15 markers evenly distributed along the X chromosome. The study showed linkage to marker DXS8076, located in Xq21.1, by the lod score method (z = 2.11 at θ = 0.155) and the nonparametric extended relative pair analysis method (χ² = 5.32; P < 0.03). Genetic heterogeneity was found, with an estimated 75% of the families linked at recombination fraction θ = 0.10 to the DXS8076 locus (χ² = 9.51; P < 0.009). Xq13–q21 is one of the critical regions for X‐linked MR previously reported, and our study supports the idea that this region may contain a locus for MR in Spanish patients. © 2001 Wiley‐Liss, Inc.     

Decomposing Pearson's χ2 test: A linear regression and its departure from linearity

In case–control genetic association studies, a standard practice is to perform the Cochran‐Armitage (CA) trend test under the assumption of the additive model because of its robustness. We could even identify situations in which it outperformed the analysis model consistent with the underlying inheritance mode. In this article, we analytically reveal the statistical basis that leads to the phenomenon. By elucidating the origin of the CA trend test as a linear regression model, we decompose Pearson's χ²‐test statistic into two components—one is the CA trend test statistic that measures the goodness of fit of the linear regression model, and the other measures the discrepancy between data and the linear regression model. Under this framework, we show that the additive coding scheme, as well as the multiplicative coding scheme, increases the coefficient of determination of the regression model by increasing the spread of data points. We also obtain the conditions under which the CA trend test statistic equals the MAX statistic and Pearson's χ²‐test statistic.     

Efficient Calculation of P‐value and Power for Quadratic Form Statistics in Multilocus Association Testing

We address the asymptotic and approximate distributions of a large class of test statistics with quadratic forms used in association studies. The statistics of interest take the general form D=X^TA X , where A is a general similarity matrix which may or may not be positive semi‐definite, and X follows the multivariate normal distribution with mean μ and variance matrix Σ, where Σ may or may not be singular. We show that D can be written as a linear combination of independent χ² random variables with a shift. Furthermore, its distribution can be approximated by a χ² or the difference of two χ² distributions. In the setting of association testing, our methods are especially useful in two situations. First, when the required significance level is much smaller than 0.05 such as in a genome scan, the estimation of p‐values using permutation procedures can be challenging. Second, when an EM algorithm is required to infer haplotype frequencies from un‐phased genotype data, the computation can be intensive for a permutation procedure. In either situation, an efficient and accurate estimation procedure would be useful. Our method can be applied to any quadratic form statistic and therefore should be of general interest.     

The prior probability of autosomal linkage

An expression is derived for the prior probability of linkage between a random trait locus and any one of m random marker loci, and this probability is computed form=1, 10, 20, 30, 50 and 100. A similar expression is derived for two trait loci, and computed for m=1, 10, 20 and 30. When one trait locus and 30 marker loci are being studied, a priori there is over a three-quarter probability that the trait locus should be syntenic with at least one of the markers, and about a one-half probability that there should be a linkage mappable from recombination frequencies. If two traits are studied, then the prior probability that at least one should be syntenic with one of the 30 markers is 0-94, and there is a three-quarter probability that such a linkage should be mappable.     

Design and Analysis of Genetic Association Studies to Finely Map a Locus Identified by Linkage Analysis: Sample Size and Power Calculations

Association (e.g. case‐control) studies are often used to finely map loci identified by linkage analysis. We investigated the influence of various parameters on power and sample size requirements for such a study. Calculations were performed for various values of a high‐risk functional allele (f_A), frequency of a marker allele associated with the high risk allele (f₁), degree of linkage disquilibrium between functional and marker alleles (D′) and trait heritability attributable to the functional locus (h²). The calculations show that if cases and controls are selected from equal but opposite extreme quantiles of a quantitative trait, the primary determinants of power are h² and the specific quantiles selected. For a dichotomous trait, power also depends on population prevalence. Power is optimal if functional alleles are studied (f_A= f₁ and D′= 1.0) and can decrease substantially as D′ diverges from 1.0 or as f₁ diverges from f_A. These analyses suggest that association studies to finely map loci are most powerful if potential functional polymorphisms are identified a priori or if markers are typed to maximize haplotypic diversity. In the absence of such information, expected minimum power at a given location for a given sample size can be calculated by specifying a range of potential frequencies for f_A (e.g. 0.1‐0.9) and determining power for all markers within the region with specification of the expected D′ between the markers and the functional locus. This method is illustrated for a fine‐mapping project with 662 single nucleotide polymorphisms in 24 Mb. Regions differed by marker density and allele frequencies. Thus, in some, power was near its theoretical maximum and little additional information is expected from additional markers, while in others, additional markers appear to be necessary. These methods may be useful in the analysis and interpretation of fine‐mapping studies.     

Posterior probability of linkage and maximal lod score

To detect linkage between a trait and a marker, Morton (1955) proposed to calculate the lod score z(θ₁) at a given value θ₁ of the recombination fraction. If z(θ₁) reaches +3 then linkage is concluded. However, in practice, lod scores are calculated for different values of the recombination fraction between 0 and 0·5 and the test is based on the maximum value of the lod score Zmax. The impact of this deviation of the test on the probability that in fact linkage does not exist, when linkage was concluded, is documented here. This posterior probability of no linkage can be derived by using Bayes' theorem. It is less than 5% when the lod score at a predetermined θ₁ is used for the test. But, for a Zmax of +3, we showed that it can reach 16·4%. Thus, considering a composite alternative hypothesis instead of a single one decreases the reliability of the test. The reliability decreases rapidly when Zmax is less than + 3. Given a Zmax of +2·5, there is a 33% chance that linkage does not exist. Moreover, the posterior probability depends not only on the value of Zmax but also jointly on the family structures and on the genetic model. For a given Zmax, the chance that linkage exists may then vary.     

Affected sibling pair linkage analysis of qualitative and quantitative traits for schizophrenia on chromosome 22 in a Chinese population

We performed nonparametric linkage analysis on 136 families with two or more siblings with schizophrenia from Sichuan, southwestern China. In addition to categorical diagnosis, we used quantitative trait information from the Positive and Negative Symptom Scale and the modified Overt Aggression Scale. Categorical analysis using the diagnosis of schizophrenia and a maximum likelihood identity-by-descent method produced scores of close to 0 throughout the whole region tested. Multipoint analysis allowed exclusion of most markers with a relative risk of > 2, but did not exclude the possibility of a relative risk of < 1.5 for four of the markers. Our results provide no significant evidence for a locus for schizophrenia on chromosome 22. Quantitative linkage analysis using the PANSS-G scale score produced a maximum LOD score of approximately 1.2 with the marker D22S310, using either the Haseman-Elston method or maximum likelihood variance estimation with or without dominance. PANSS-N produced a maximum LOD score of 1.2 at the D22S283 locus. LOD score of about 1 are easily produced by chance. Thus, we conclude that under quantitative trait we also find no evidence of linkage between schizophrenia and markers on chromosome 22 in our Chinese sibling pair sample.     

Power of Linkage Disequilibrium Mapping to Detect a Quantitative Trait Locus (QTL) in Selected Samples of Unrelated Individuals

We considered a strategy to map quantitative trait loci (QTLs) using linkage disequilibrium (LD) when the QTL and marker locus were multiallelic. The strategy involved phenotyping a large number of unrelated individuals and genotyping only selected individuals from the two tails of the trait distribution. Power to detect trait‐marker association was assessed as a function of the number of QTL and marker alleles. Two patterns of LD were used to study their influence on power. When the frequency of the QTL allele with the largest effect and that of the marker allele linked in coupling were equal, power was maximum. In this case, increasing the number of QTL alleles reduced the power. The maximum difference in power between the two LD patterns studied was ～30%. For low QTL heritabilities (h²_QTL < 0.1) and single trait studies we recommend selecting around 5% of the upper and lower tails of the trait distribution.     

Combined Segregation and Linkage Analysis of Fibrinogen Variability in Israeli Families: Evidence for Two Quantitative-trait Loci, One of Which is Linked to a Functional Variant (−58G > A) in the Promoter of the α-fibrinogen Gene

The association of α‐ and β‐fibrinogen polymorphisms with plasma fibrinogen levels was examined in a sample of 452 family members from 80 Israeli kindreds. The measured genotype analysis indicated that the β‐fibrinogen ?455G > A polymorphism was not associated with fibrinogen levels, while the α‐fibrinogen ?58G > A locus showed a significant association with fibrinogen levels (χ²= 17.7; df = 3; p < 0.001) , with the ?58A allele being associated with higher levels. Segregation analysis in this sample suggested a recessive quantitative‐trait locus (QTL) with a major effect that controlled the sex‐ and age‐adjusted fibrinogen levels. Results from a combined segregation/linkage analysis indicated that a single QTL influencing plasma fibrinogen is in gametic equilibrium with the β‐fibrinogen ?455G > A and α‐fibrinogen ?58G > A polymorphisms. An extended analysis with a two‐QTL model significantly improved the fit of the model (p ≤ 0.001) , and gave support for linkage between the fibrinogen QTL and the α‐fibrinogen polymorphism. In vitro analysis with a DNA fragment containing this variant, linked to a reporter gene, showed 2‐fold higher expression of the A allele compared to the G allele in the liver cell line HepG2, both under basal conditions and after stimulation with interleukin 6. These results demonstrate that two QTLs are jointly involved in determining plasma fibrinogen levels in this sample of families, one of which is located close to a functional variant in the α‐fibrinogen locus.     

A simple method to detect linkage for rare recessive diseases: An application to juvenile diabetes

A simple procedure designed specifically to detect linkage for rare recessive diseases is described. The method uses information on identity by descent scores for a pair of sibs at a marker locus conditioned on the number of affected sibs in the pair. A procedure for estimating the recombination fraction is described, and a table facilitating the likelihood ratio test of linkage is provided. The method, when applied to a collection of multiplex families segregating for juvenile diabetes mellitus, suggests the possibility that this disease is linked to the HLA complex. The method is found to compare favorably to the maximum likelihood approach, for which the computer program LIPED gives a maximum lod score of 2.48 at a male and female recombination fraction of theta = 0.20.     

Linkage detection under heterogeneity and the mixture problem

Linkage analysis has contributed to the localization of many human disease genes. The presence of locus heterogeneity reduces statistical power and can prejudice the detection of linkage if the analysis assumes homogeneity. Nevertheless, mixed genetic models are not routinely used in gene searches. The null distribution of the test statistic is not uniquely defined. In this paper, a transformation is used to determine an approximate asymptotic distribution of the test statistic under a mixture model. The equivalent critical values of the test are computed and the performance of the test under various levels of heterogeneity and family size is investigated. For gene searches, we recommend the routine use of an admixture model with a critical lod score of 3·44.     

A Likelihood Ratio Test for Genome‐Wide Association under Genetic Heterogeneity

Most existing association tests for genome‐wide association studies (GWASs) fail to account for genetic heterogeneity. Zhou and Pan proposed a binomial‐mixture‐model‐based association test to account for the possible genetic heterogeneity in case‐control studies. The idea is elegant, however, the proposed test requires an expectation‐maximization (EM)‐type iterative algorithm to identify the penalised maximum likelihood estimates and a permutation method to assess p‐values. The intensive computational burden induced by the EM‐algorithm and the permutation becomes prohibitive for direct applications to GWASs. This paper develops a likelihood ratio test (LRT) for GWASs under genetic heterogeneity based on a more general alternative mixture model. In particular, a closed‐form formula for the LRT statistic is derived to avoid the EM‐type iterative numerical evaluation. Moreover, an explicit asymptotic null distribution is also obtained, which avoids using the permutation to obtain p‐values. Thus, the proposed LRT is easy to implement for GWASs. Furthermore, numerical studies demonstrate that the LRT has power advantages over the commonly used Armitage trend test and other existing association tests under genetic heterogeneity. A breast cancer GWAS dataset is used to illustrate the newly proposed LRT.     

Randomization tests of disease-marker associations

A powerful test for population association of a disease with alleles at a bi-allelic marker locus is the transmission/disequilibrium test ( TDT ). A generalization of the test to multi-allelic marker locus is proposed which utilizes the maximal association of individual alleles with the disease, given by the maximum TDT statistic, TDT _(max). To overcome the multiple testing problem encountered when using the maximal association to test the null hypothesis of no disease-marker association, a randomization procedure is developed. An investigation of the power of the test suggests that the randomization procedure performs almost as well as a recently proposed likelihood based test of linkage disequilibrium. The advantage of the new test is that it can be applied sequentially, based on a one-sided version of the TDT statistic, for investigating patterns of association of several individual alleles with the disease.     

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.     

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%.     

Mapping of the HLA locus controlling C2 structural variants and linkage disequilibrium between alleles C22 and Bw15

The newly discovered polymorphism of human C2 protein analyzed by isoelectric focusing is controlled by a locus closely linked to HLA. An informative cross-over family (HLA: A×C) indicates that this locus maps on the C side of the region. Population and family studies have shown that the C2² variant is significantly associated on the haplotype with alleles Bw15 and Cw3. The fact that the “deficient” C2 mutant is in linkage disequilibrium with other HLA antigens, suggests that C2² and C2⁰, here tentatively considered alleles of the same locus, are not directly related by descent.     

Power of regression and maximum likelihood methods to map QTL from sib-pair and DZ twin data

A common study design to map quantitative trait loci (QTL) is to compare the phenotypes and marker genotypes of two or more siblings in a sample of unrelated sib groups, and to test for linkage between chromosome location and quantitative trait values. The simplest case is sib pairs only, in particular dizygotic twin pairs, and a simple and elegant regression method was proposed by Haseman & Elston in 1972 to test for linkage. Since then, several other methods have been proposed to test for linkage. In this study, we derived the statistical power of linear regression and maximum likelihood methods to map QTL from sib pair data analytically, and determined which methods are superior under which set of population parameters. In particular, we considered four regression-based and three maximum likelihood-based approaches, and derived asymptotic approximations of the mean test statistic and statistical power for each method. It was found, both analytically and by computer simulation, that the revisited or new Haseman–Elston method (based upon the mean-corrected crossproduct of the observations on sib-pairs) is less powerful than a full maximum likelihood approach and is also inferior to the Haseman–Elston method under a realistic range of values for the population parameters. We found that a simple regression method, based upon both the squared difference and the mean-corrected squared sum of the observations on sib-pairs, is as powerful as a full maximum likelihood approach. Our derivations of statistical power for regression and maximum likelihood methods provide a simple way to compare alternative methods and obviate the need to perform elaborate computer simulations. DZ twin pairs are likely to be more powerful for linkage analysis than ordinary siblings because they may share more common environmental effects, thereby increasing the proportion of within-family variance that is explained by a QTL.