Multivariate Phenotype Association Analysis by Marker‐Set Kernel Machine Regression

Genetic studies of complex diseases often collect multiple phenotypes relevant to the disorders. As these phenotypes can be correlated and share common genetic mechanisms, jointly analyzing these traits may bring more power to detect genes influencing individual or multiple phenotypes. Given the advancement brought by the multivariate phenotype approaches and the multimarker kernel machine regression, we construct a multivariate regression based on kernel machine to facilitate the joint evaluation of multimarker effects on multiple phenotypes. The kernel machine serves as a powerful dimension‐reduction tool to capture complex effects among markers. The multivariate framework incorporates the potentially correlated multidimensional phenotypic information and accommodates common or different environmental covariates for each trait. We derive the multivariate kernel machine test based on a score‐like statistic, and conduct simulations to evaluate the validity and efficacy of the method. We also study the performance of the commonly adapted strategies for kernel machine analysis on multiple phenotypes, including the multiple univariate kernel machine tests with original phenotypes or with their principal components. Our results suggest that none of these approaches has the uniformly best power, and the optimal test depends on the magnitude of the phenotype correlation and the effect patterns. However, the multivariate test retains to be a reasonable approach when the multiple phenotypes have none or mild correlations, and gives the best power once the correlation becomes stronger or when there exist genes that affect more than one phenotype. We illustrate the utility of the multivariate kernel machine method through the Clinical Antipsychotic Trails of Intervention Effectiveness antibody study.     

An empirical test of the significance of an observed quantitative trait locus effect that preserves additive genetic variation

We propose a constrained permutation test that assesses the significance of an observed quantitative trait locus effect against a background of genetic and environmental variation. Permutations of phenotypes are not selected at random, but rather are chosen in a manner that attempts to maintain the additive genetic variability in phenotypes. Such a constraint maintains the nonindependence among observations under the null hypothesis of no linkage. The empirical distribution of the lod scores calculated using permuted phenotypes is compared to that obtained using phenotypes simulated from the assumed underlying multivariate normal model. We make comparisons of univariate analyses for both a quantitative phenotype that appears consistent with a multivariate normal model and a quantitative phenotype containing pronounced outliers. An example of a bivariate analysis is also presented.     

Multivariate segregation analysis using the mixed model

Most major genes involved in the etiology of complex diseases are likely to have pleiotropic effects on a number of intervening quantitative traits. Methods of segregation analysis that incorporate the additional information from such multiple traits will exhibit greater power for detecting the effects of major genes and allow explicit tests of major locus pleiotropy hypotheses. In this study, we present a new method for multivariate segregation analysis that utilizes a multivariate generalization of Hasstedt's [1982] technique for calculating approximate mixed model likelihoods on pedigrees. The method is based on a simplification of the multivariate conditional likelihood via a transformation that simultaneously orthogonalizes the residual additive genetic and environmental covariance matrices. This transformation allows the multivariate conditional likelihood to be factored into the product of independent univariate conditional likelihoods. Resulting computations are relatively fast, making it feasible to analyze multiple traits in extended pedigrees. We demonstrate our method with a bivariate analysis of high-density lipoprotein cholesterol (HDL-C) and apolipoprotein AI (apo AI) serum levels in 585 pedigreed baboons.     

Bivariate logistic Bayesian LASSO for detecting rare haplotype association with two correlated phenotypes

In genetic association studies, joint modeling of related traits/phenotypes can utilize the correlation between them and thereby provide more power and uncover additional information about genetic etiology. Moreover, detecting rare genetic variants are of current scientific interest as a key to missing heritability. Logistic Bayesian LASSO (LBL) has been proposed recently to detect rare haplotype variants using case-control data, that is, a single binary phenotype. As there is currently no haplotype association method that can handle multiple binary phenotypes, we extend LBL to fill this gap. We develop a bivariate model by using a latent variable to induce correlation between the two outcomes. We carry out extensive simulations to investigate the bivariate LBL and compare with the univariate LBL. The bivariate LBL performs better or similar to the univariate LBL in most settings. It has the highest gain in power when a haplotype is associated with both traits and it affects at least one trait in a direction opposite to the direction of the correlation between the traits. We analyze two data sets—Genetic Analysis Workshop 19 sequence data on systolic and diastolic blood pressures and a genome-wide association data set on lung cancer and smoking and detect several associated rare haplotypes.     

Bivariate combined linkage and association mapping of quantitative trait loci

In this paper, bivariate/multivariate variance component models are proposed for high-resolution combined linkage and association mapping of quantitative trait loci (QTL), based on combinations of pedigree and population data. Suppose that a quantitative trait locus is located in a chromosome region that exerts pleiotropic effects on multiple quantitative traits. In the region, multiple markers such as single nucleotide polymorphisms are typed. Two regression models, "genotype effect model" and "additive effect model", are proposed to model the association between the markers and the trait locus. The linkage information, i.e., recombination fractions between the QTL and the markers, is modeled in the variance and covariance matrix. By analytical formulae, we show that the "genotype effect model" can be used to model the additive and dominant effects simultaneously; the "additive effect model" only takes care of additive effect. Based on the two models, F-test statistics are proposed to test association between the QTL and markers. By analytical power analysis, we show that bivariate models can be more powerful than univariate models. For moderate-sized samples, the proposed models lead to correct type I error rates; and so the models are reasonably robust. As a practical example, the method is applied to analyze the genetic inheritance of rheumatoid arthritis for the data of The North American Rheumatoid Arthritis Consortium, Problem 2, Genetic Analysis Workshop 15, which confirms the advantage of the proposed bivariate models.     

A univariate perspective of multivariate genome‐wide association analysis

Multiple correlated phenotypes are frequently collected in genome‐wide association studies (GWASs), and a systematic, simultaneous analysis of multiple phenotypes can integrate the signals from single phenotypes, therefore increasing the power of detecting genetic signals. However, fundamental questions remain open, including the conditions and reasons under which the multivariate analysis is beneficial, how a highly significant signal arises in the multivariate analysis. To understand these issues, we propose to decompose the multivariate model into a series of simple univariate models. This transformation offers a clearer quantitative analysis of the circumstances under which a multivariate approach can be beneficial for the bivariate phenotypes case. A real data analysis is employed to illustrate how to interpret how the signals arising from multivariate GWASs.     

Testing cross‐phenotype effects of rare variants in longitudinal studies of complex traits

Many gene mapping studies of complex traits have identified genes or variants that influence multiple phenotypes. With the advent of next‐generation sequencing technology, there has been substantial interest in identifying rare variants in genes that possess cross‐phenotype effects. In the presence of such effects, modeling both the phenotypes and rare variants collectively using multivariate models can achieve higher statistical power compared to univariate methods that either model each phenotype separately or perform separate tests for each variant. Several studies collect phenotypic data over time and using such longitudinal data can further increase the power to detect genetic associations. Although rare‐variant approaches exist for testing cross‐phenotype effects at a single time point, there is no analogous method for performing such analyses using longitudinal outcomes. In order to fill this important gap, we propose an extension of Gene Association with Multiple Traits (GAMuT) test, a method for cross‐phenotype analysis of rare variants using a framework based on the distance covariance. The approach allows for both binary and continuous phenotypes and can also adjust for covariates. Our simple adjustment to the GAMuT test allows it to handle longitudinal data and to gain power by exploiting temporal correlation. The approach is computationally efficient and applicable on a genome‐wide scale due to the use of a closed‐form test whose significance can be evaluated analytically. We use simulated data to demonstrate that our method has favorable power over competing approaches and also apply our approach to exome chip data from the Genetic Epidemiology Network of Arteriopathy.     

A Bivariate Haseman‐Elston Method and Application to the Analysis of Asthma‐Related Phenotypes on Chromosome 5q

We propose a bivariate combination of different Haseman‐Elston (HE) methods for model free linkage analysis of quantitative traits. Adjustments for correlations of phenotypes and sibship sizes > 2 are performed using generalized estimating equations (GEE). All calculations are carried out with freely available software packages. We illustrate the application of standard HE methods, the unified HE method, and our novel approach to asthma‐associated quantitative traits from the COAG‐Perth data set [Palmer et al., 1998, Am J Respir Crit Care Med 158:1825–30]. Our multipoint analyses provide evidence for linkage between log IgE levels adjusted for age, gender and antigen‐specific IgE titers. Our results are consistent with previous findings that suggest the existence of loci regulating asthma‐associated quantitative traits in the 5q31–33 chromosomal region. Simulation studies are required to compare the power of our novel bivariate HE with other HE approaches and the variance component method. © 2001 Wiley‐Liss, Inc.     

Estimating and Testing Pleiotropy of Single Genetic Variant for Two Quantitative Traits

Along with the accumulated data of genetic variants and biomedical phenotypes in the genome era, statistical identification of pleiotropy is of growing interest for dissecting and understanding genetic correlations between complex traits. We proposed a novel method for estimating and testing pleiotropic effect of a genetic variant on two quantitative traits. Based on a covariance decomposition and estimation, our method quantifies pleiotropy as the portion of between‐trait correlation explained by the same genetic variant. Unlike most multiple‐trait methods that assess potential pleiotropy (i.e., whether a variant contributes to at least one trait), our method formulates a statistic that tests exact pleiotropy (i.e., whether a variant contributes to both of two traits). We developed two approaches (a regression approach and a bootstrapping approach) for such test and investigated their statistical properties, in comparison with other potential pleiotropy test methods. Our simulation shows that the regression approach produces correct P‐values under both the complete null (i.e., a variant has no effect on both two traits) and the incomplete null (i.e., a variant has effect on only one of two traits), but requires large sample sizes to achieve a good power, when the bootstrapping approach has a better power and produces conservative P‐values under the complete null. We demonstrate our method for detecting exact pleiotropy using a real GWAS dataset. Our method provides an easy‐to‐implement tool for measuring, testing, and understanding the pleiotropic effect of a single variant on the correlation architecture of two complex traits.     

Comparison of Heritability Estimation and Linkage Analysis for Multiple Traits Using Principal Component Analyses

A disease trait often can be characterized by multiple phenotypic measurements that can provide complementary information on disease etiology, physiology, or clinical manifestations. Given that multiple phenotypes may be correlated and reflect common underlying genetic mechanisms, the use of multivariate analysis of multiple traits may improve statistical power to detect genes and variants underlying complex traits. The literature, however, has been unclear as to the optimal approach for analyzing multiple correlated traits. In this study, heritability and linkage analysis was performed for six obstructive sleep apnea hypopnea syndrome (OSAHS) related phenotypes, as well as principal components of the phenotypes and principal components of the heritability (PCHs) using the data from Cleveland Family Study, which include both African and European American families. Our study demonstrates that principal components generally result in higher heritability and linkage evidence than individual traits. Furthermore, the PCHs can be transferred across populations, strongly suggesting that these PCHs reflect traits with common underlying genetic mechanisms for OSAHS across populations. Thus, PCHs can provide useful traits for using data on multiple phenotypes and for genetic studies of trans‐ethnic populations.     

An evaluation of statistical approaches to rare variant analysis in genetic association studies

Genome‐wide association (GWA) studies have proved to be extremely successful in identifying novel common polymorphisms contributing effects to the genetic component underlying complex traits. Nevertheless, one source of, as yet, undiscovered genetic determinants of complex traits are those mediated through the effects of rare variants. With the increasing availability of large‐scale re‐sequencing data for rare variant discovery, we have developed a novel statistical method for the detection of complex trait associations with these loci, based on searching for accumulations of minor alleles within the same functional unit. We have undertaken simulations to evaluate strategies for the identification of rare variant associations in population‐based genetic studies when data are available from re‐sequencing discovery efforts or from commercially available GWA chips. Our results demonstrate that methods based on accumulations of rare variants discovered through re‐sequencing offer substantially greater power than conventional analysis of GWA data, and thus provide an exciting opportunity for future discovery of genetic determinants of complex traits. Genet. Epidemiol. 34: 188–193, 2010. © 2009 Wiley‐Liss, Inc.     

Multivariate genetic analysis of an oligogenic disease

Joint multivariate segregation and linkage analysis provides a method for simultaneously analyzing data on affection status, correlated phenotypic traits, environmental risk factors, and other covariates. The power of this approach for mapping disease susceptibility loci of small effect (oligogenes) was evaluated by analyzing the GAW9 Problem 2 data set. The program REGRESS, which assumes a pleiotropy model in which one locus influences both affection status (AF) and a quantitative trait, was used to conduct joint segregation and linkage analysis of bivariate phenotypes, each comprising AF and one quantitative trait (Q2,Q3,Q4). A genome-wide search using markers spaced approximately 10 cM apart was conducted and regions on chromosomes 1, 2, and 5 were identified as demonstrating linkage with three respective bivariate phenotypes at the following markers: AF/Q2 - D1G2; AF/Q3 - D2G10; and AF/Q4 - D5G18. The effects of other loci were included in a general model by specifying the quantitative traits they influenced as covariates along with age, sex, and an environmental effect. Use of covariate and quantitative trait data in each analysis resulted in respective χ² values with 1 df of 38.4, 65.4, and 22.0 to reject the no linkage hypothesis at $ {\rm \hat \theta } $ = 0, with respective equivalent lod scores of 8.3, 14.2, and 4.8. Rejection at p < 0.0002 occurred using markers as far away as 20 cM. These loci were not detected when AF alone was analyzed. © 1995 Wiley-Liss, Inc.     

Multivariate linkage analysis of blood pressure and body mass index

Multivariate linkage analyses of correlated traits provide greater statistical power to identify genetic loci with effects too small to be detected in single trait analyses. We conducted genomewide multivariate analyses of systolic BP, diastolic BP, and body mass index (BMI) in 1,848 non-Hispanic white subjects (968 females, 880 males) from 279 multigenerational pedigrees from Rochester, Minnesota. Blood pressure was measured by random zero sphygmomanometer; body mass index was calculated from measurements of height and weight; and genotypes were measured at 520 microsatellite marker loci distributed across the 22 autosomes. Univariate linkage analyses demonstrated tentative evidence of linkage (defined by univariate LOD scores of 1.30-1.99) for diastolic BP on chromosome 18 and for BMI on chromosomes 3, 10, and 18. Bivariate linkage analyses showed tentative evidence of linkage (defined by bivariate LOD scores of 2.06-2.86) for systolic and diastolic BP on chromosome 14 and for either measure of BP and BMI on chromosomes 2, 3, 10, and 18; and suggestive evidence of linkage (defined by bivariate LOD scores of 2.87-3.99) for either measure of BP and BMI on chromosomes 10 and chromosome 15. Trivariate linkage analyses of systolic and diastolic BP and BMI provided evidence of a region influencing all three traits on chromosome 10, where the trivariate LOD score rose to a maximum value of 4.09 (at 144 cM, P=0.0007), and possibly on chromosome 2, where it rose to a maximum value of 2.80 (at 77 cM, P=0.0075). For genomewide linkage analyses to succeed in localizing genes influencing BP, it may be advantageous to exploit the greater statistical power of multivariate linkage analyses to identify loci with pleiotropic effects on correlated traits.     

Bayesian modelling of multivariate quantitative traits using seemingly unrelated regressions

We investigate a Bayesian approach to modelling the statistical association between markers at multiple loci and multivariate quantitative traits. In particular, we describe the use of Bayesian Seemingly Unrelated Regressions (SUR) whereby genotypes at the different loci are allowed to have non-simultaneous effects on the phenotypes considered with residuals from each regression assumed correlated. We present results from simulations showing that, under rather general conditions that are likely to hold in real situations, the Bayesian SUR approach has increased probability of selecting the true model compared to univariate analyses. Finally, we apply our methods to data from subjects genotyped for 12 SNPs in the apolipoprotein E (APOE) gene. Phenotypes relate to response to treatment with atorvastatin and include changes in total cholesterol, low-density lipoprotein cholesterol, and triglycerides. Missing genotype data are naturally accommodated in our Bayesian framework by imputing them using a nested haplotype phasing algorithm.     

Dissection of heterogeneous phenotypes for quantitative trait mapping

We discuss analyses of Genetic Analysis Workshop 14 data from the Collaborative Study on the Genetics of Alcoholism (COGA) as well as from a simulated complex disease, Kofendrerd personality disorder (KPD), with both genetic and phenotypic heterogeneity. Both data sets included numerous related phenotypes in addition to disease definitions. All analyses either chose from the given selection of phenotypes or defined new ones, including traits that may not have been related to alcoholism or KPD. Some contributors evaluated the genetic components of the trait. Many investigated genome-wide linkage and/or association, using microsatellites and/or single-nucleotide polymorphism (SNP) chip data. Here we will focus on methodological issues that the investigators faced. Their results depended on phenotype selection, whether continuous or discrete, the covariates included, and ethnicity of the study population. For SNP chip data, members of our group detected no difference in results for Affymetrix or Illumina chips, although higher marker density for association studies appeared to be advantageous. Overall, there were some observations that different chromosomal segments, i.e., physical locations on the p-arm, q-arm, or middle segment, might lead to possible differences in type I error rates. This finding and others highlight the importance of empirical determination of P-values to determine significance.     

A flexible copula-based approach for the analysis of secondary phenotypes in ascertained samples

Data collected for a genome-wide association study of a primary phenotype are often used for additional genome-wide association analyses of secondary phenotypes. However, when the primary and secondary traits are dependent, naïve analyses of secondary phenotypes may induce spurious associations in non-randomly ascertained samples. Previously, retrospective likelihood-based methods have been proposed to correct for sampling biases arising in secondary trait association analyses. However, most methods have been introduced to handle studies featuring a case-control design based on a binary primary phenotype. As such, these methods are not directly applicable to more complicated study designs such as multiple-trait studies, where the sampling mechanism also depends on the secondary phenotype, or extreme-trait studies, where individuals with extreme primary phenotype values are selected. To accommodate these more complicated sampling mechanisms, only a few prospective likelihood approaches have been proposed. These approaches assume a normal distribution for the secondary phenotype (or the latent secondary phenotype) and a bivariate normal distribution for the primary-secondary phenotype dependence. In this paper, we propose a unified copula-based approach to appropriately detect genetic variant/secondary phenotype association in the presence of selected samples. Primary phenotype is either binary or continuous and the secondary phenotype is continuous although not necessary normal. We use both prospective and retrospective likelihoods to account for the sampling mechanism and use a copula model to allow for potentially different dependence structures between the primary and secondary phenotypes. We demonstrate the effectiveness of our approach through simulation studies and by analyzing data from the Avon Longitudinal Study of Parents and Children cohort.     

A hierarchical clustering method for dimension reduction in joint analysis of multiple phenotypes

Genome‐wide association studies (GWAS) have become a very effective research tool to identify genetic variants of underlying various complex diseases. In spite of the success of GWAS in identifying thousands of reproducible associations between genetic variants and complex disease, in general, the association between genetic variants and a single phenotype is usually weak. It is increasingly recognized that joint analysis of multiple phenotypes can be potentially more powerful than the univariate analysis, and can shed new light on underlying biological mechanisms of complex diseases. In this paper, we develop a novel variable reduction method using hierarchical clustering method (HCM) for joint analysis of multiple phenotypes in association studies. The proposed method involves two steps. The first step applies a dimension reduction technique by using a representative phenotype for each cluster of phenotypes. Then, existing methods are used in the second step to test the association between genetic variants and the representative phenotypes rather than the individual phenotypes. We perform extensive simulation studies to compare the powers of multivariate analysis of variance (MANOVA), joint model of multiple phenotypes (MultiPhen), and trait‐based association test that uses extended simes procedure (TATES) using HCM with those of without using HCM. Our simulation studies show that using HCM is more powerful than without using HCM in most scenarios. We also illustrate the usefulness of using HCM by analyzing a whole‐genome genotyping data from a lung function study.     

Genetic variance components analysis for binary phenotypes using generalized linear mixed models (GLMMs) and Gibbs sampling

The common complex diseases such as asthma are an important focus of genetic research, and studies based on large numbers of simple pedigrees ascertained from population‐based sampling frames are becoming commonplace. Many of the genetic and environmental factors causing these diseases are unknown and there is often a strong residual covariance between relatives even after all known determinants are taken into account. This must be modelled correctly whether scientific interest is focused on fixed effects, as in an association analysis, or on the covariances themselves. Analysis is straightforward for multivariate Normal phenotypes, but difficulties arise with other types of trait. Generalized linear mixed models (GLMMs) offer a potentially unifying approach to analysis for many classes of phenotype including multivariate Normal traits, binary traits, and censored survival times. Markov Chain Monte Carlo methods, including Gibbs sampling, provide a convenient framework within which such models may be fitted. In this paper, Bayesian inference Using Gibbs Sampling (a generic Gibbs sampler; BUGS) is used to fit GLMMs for multivariate Normal and binary phenotypes in nuclear families. BUGS is easy to use and readily available. We motivate a suitable model structure for Normal phenotypes and show how the model extends to binary traits. We discuss parameter interpretation and statistical inference and show how to circumvent a number of important theoretical and practical problems that we encountered. Using simulated data we show that model parameters seem consistent and appear unbiased in smaller data sets. We illustrate our methods using data from an ongoing cohort study. Genet. Epidemiol. 17:118–140, 1999. © 1999 Wiley‐Liss, Inc.     

Genetic association studies with bivariate mixed responses subject to measurement error and misclassification

In genetic association studies, mixed effects models have been widely used in detecting the pleiotropy effects which occur when one gene affects multiple phenotype traits. In particular, bivariate mixed effects models are useful for describing the association of a gene with a continuous trait and a binary trait. However, such models are inadequate to feature the data with response mismeasurement, a characteristic that is often overlooked. It has been well studied that in univariate settings, ignorance of mismeasurement in variables usually results in biased estimation. In this paper, we consider the setting with a bivariate outcome vector which contains a continuous component and a binary component both subject to mismeasurement. We propose an induced likelihood approach and an EM algorithm method to handle measurement error in continuous response and misclassification in binary response simultaneously. Simulation studies confirm that the proposed methods successfully remove the bias induced from the response mismeasurement.     

Pleiotropy Analysis of Quantitative Traits at Gene Level by Multivariate Functional Linear Models

In genetics, pleiotropy describes the genetic effect of a single gene on multiple phenotypic traits. A common approach is to analyze the phenotypic traits separately using univariate analyses and combine the test results through multiple comparisons. This approach may lead to low power. Multivariate functional linear models are developed to connect genetic variant data to multiple quantitative traits adjusting for covariates for a unified analysis. Three types of approximate F‐distribution tests based on Pillai–Bartlett trace, Hotelling–Lawley trace, and Wilks's Lambda are introduced to test for association between multiple quantitative traits and multiple genetic variants in one genetic region. The approximate F‐distribution tests provide much more significant results than those of F‐tests of univariate analysis and optimal sequence kernel association test (SKAT‐O). Extensive simulations were performed to evaluate the false positive rates and power performance of the proposed models and tests. We show that the approximate F‐distribution tests control the type I error rates very well. Overall, simultaneous analysis of multiple traits can increase power performance compared to an individual test of each trait. The proposed methods were applied to analyze (1) four lipid traits in eight European cohorts, and (2) three biochemical traits in the Trinity Students Study. The approximate F‐distribution tests provide much more significant results than those of F‐tests of univariate analysis and SKAT‐O for the three biochemical traits. The approximate F‐distribution tests of the proposed functional linear models are more sensitive than those of the traditional multivariate linear models that in turn are more sensitive than SKAT‐O in the univariate case. The analysis of the four lipid traits and the three biochemical traits detects more association than SKAT‐O in the univariate case.