Joint regression analysis of correlated data using Gaussian copulas

Summary .  This article concerns a new joint modeling approach for correlated data analysis. Utilizing Gaussian copulas, we present a unified and flexible machinery to integrate separate one-dimensional generalized linear models (GLMs) into a joint regression analysis of continuous, discrete, and mixed correlated outcomes. This essentially leads to a multivariate analogue of the univariate GLM theory and hence an efficiency gain in the estimation of regression coefficients. The availability of joint probability models enables us to develop a full maximum likelihood inference. Numerical illustrations are focused on regression models for discrete correlated data, including multidimensional logistic regression models and a joint model for mixed normal and binary outcomes. In the simulation studies, the proposed copula-based joint model is compared to the popular generalized estimating equations, which is a moment-based estimating equation method to join univariate GLMs. Two real-world data examples are used in the illustration.     

Quantitative trait linkage analysis of longitudinal change in body weight

One of the great strengths of the Framingham Heart Study data, provided for the Genetic Analysis Workshop 13, is the long-term survey of phenotypic data. We used this unique data to create new phenotypes representing the pattern of longitudinal change of the provided phenotypes, especially systolic blood pressure and body weight. We performed a linear regression of body weight and systolic blood pressure on age and took the slopes as new phenotypes for quantitative trait linkage analysis using the SOLAR package. There was no evidence for heritability of systolic blood pressure change. Heritability was estimated as 0.15 for adult life "body weight change", measured as the regression slope, and "body weight gain" (including only individuals with a positive regression slope), and as 0.22 for body weight "change up to 50" (regression slope of weight on age up to an age of 50). With multipoint analysis, two regions on the long arm of chromosome 8 showed the highest LOD scores of 1.6 at 152 cM for "body weight change" and of >1.9 around location 102 cM for "body weight gain" and "change up to 50". The latter two LOD scores almost reach the threshold for suggestive linkage. We conclude that the chromosome 8 region may harbor a gene acting on long-term body weight regulation, thereby contributing to the development of the metabolic syndrome.     

Quantitative trait nucleotide analysis using Bayesian model selection

Although much attention has been given to statistical genetic methods for the initial localization and fine mapping of quantitative trait loci (QTLs), little methodological work has been done to date on the problem of statistically identifying the most likely functional polymorphisms using sequence data. In this paper we provide a general statistical genetic framework, called Bayesian quantitative trait nucleotide (BQTN) analysis, for assessing the likely functional status of genetic variants. The approach requires the initial enumeration of all genetic variants in a set of resequenced individuals. These polymorphisms are then typed in a large number of individuals (potentially in families), and marker variation is related to quantitative phenotypic variation using Bayesian model selection and averaging. For each sequence variant a posterior probability of effect is obtained and can be used to prioritize additional molecular functional experiments. An example of this quantitative nucleotide analysis is provided using the GAW12 simulated data. The results show that the BQTN method may be useful for choosing the most likely functional variants within a gene (or set of genes). We also include instructions on how to use our computer program, SOLAR, for association analysis and BQTN analysis.     

Multitrait fine mapping of quantitative trait loci using combined linkage disequilibria and linkage analysis

A novel multitrait fine-mapping method is presented. The method is implemented by a model that treats QTL effects as random variables. The covariance matrix of allelic effects is proportional to the IBD matrix, where each element is the probability that a pair of alleles is identical by descent, given marker information and QTL position. These probabilities are calculated on the basis of similarities of marker haplotypes of individuals of the first generation of genotyped individuals, using "gene dropping" (linkage disequilibrium) and transmission of markers from genotyped parents to genotyped offspring (linkage). A small simulation study based on a granddaughter design was carried out to illustrate that the method provides accurate estimates of QTL position. Results from the simulation also indicate that it is possible to distinguish between a model postulating one pleiotropic QTL affecting two traits vs. one postulating two closely linked loci, each affecting one of the traits.     

Quantitative trait Loci analysis using the false discovery rate

False discovery rate control has become an essential tool in any study that has a very large multiplicity problem. False discovery rate-controlling procedures have also been found to be very effective in QTL analysis, ensuring reproducible results with few falsely discovered linkages and offering increased power to discover QTL, although their acceptance has been slower than in microarray analysis, for example. The reason is partly because the methodological aspects of applying the false discovery rate to QTL mapping are not well developed. Our aim in this work is to lay a solid foundation for the use of the false discovery rate in QTL mapping. We review the false discovery rate criterion, the appropriate interpretation of the FDR, and alternative formulations of the FDR that appeared in the statistical and genetics literature. We discuss important features of the FDR approach, some stemming from new developments in FDR theory and methodology, which deem it especially useful in linkage analysis. We review false discovery rate-controlling procedures--the BH, the resampling procedure, and the adaptive two-stage procedure-and discuss the validity of these procedures in single- and multiple-trait QTL mapping. Finally we argue that the control of the false discovery rate has an important role in suggesting, indicating the significance of, and confirming QTL and present guidelines for its use.     

Improved power of sib-pair linkage analysis using measures of complex trait dynamics

The influence of epistasis on a quantitative trait can reduce the power of linkage analysis to identify the underlying loci. In the present study, we simulated a complex trait derived from a dynamic one-locus gene expression system with epistasis arising from feedback regulation and tested the power of sib-pair linkage analysis methods for detecting the underlying quantitative trait locus (QTL). Using this simple genetic architecture, we demonstrate that the power of sib-pair linkage analysis can be greatly improved if measures of complex trait dynamics are considered.     

Mapping quantitative trait loci using molecular marker linkage maps

Summary High-density restriction fragment length polymorphism (RFLP) and allozyme linkage maps have been developed in several plant species. These maps make it technically feasible to map quantitative trait loci (QTL) using methods based on flanking marker genetic models. In this paper, we describe flanking marker models for doubled haploid (DH), recombinant inbred (RI), backcross (BC), F₁ testcross (F₁TC), DH testcross (DHTC), recombinant inbred testcross (RITC), F₂, and F₃ progeny. These models are functions of the means of quantitative trait locus genotypes and recombination frequencies between marker and quantitative trait loci. In addition to the genetic models, we describe maximum likelihood methods for estimating these parameters using linear, nonlinear, and univariate or multivariate normal distribution mixture models. We defined recombination frequency estimators for backcross and F₂ progeny group genetic models using the parameters of linear models. In addition, we found a genetically unbiased estimator of the QTL heterozygote mean using a linear function of marker means. In nonlinear models, recombination frequencies are estimated less efficiently than the means of quantitative trait locus genotypes. Recombination frequency estimation efficiency decreases as the distance between markers decreases, because the number of progeny in recombinant marker classes decreases. Mean estimation efficiency is nearly equal for these methods.     

Quantitative trait locus mapping using human pedigrees

In the past decade phenomenal progress has been made in molecular and statistical genetic methods for localizing quantitative trait loci. Because of these advances, we can anticipate a long period of active genetic research in which the genes influencing human quantitative variability will be mapped and their effects accurately evaluated. Here, we review the current state of the science in statistical genetic methods for quantitative trait linkage analysis. In particular, we detail a variance component-based framework for localizing quantitative trait loci and for accurately estimating their relative effect sizes. Attention is paid to the optimal design of human family studies for localizing genes of small to moderate effect. In addition, methods and strategies are described for dealing with the most important complications of quantitative variation, including the assessment of genotype x environment interaction and epistasis.     

Quantitative trait locus analysis using recombinant inbred intercrosses: theoretical and empirical considerations

We describe a new approach, called recombinant inbred intercross (RIX) mapping, that extends the power of recombinant inbred (RI) lines to provide sensitive detection of quantitative trait loci (QTL) responsible for complex genetic and nongenetic interactions. RIXs are generated by producing F1 hybrids between all or a subset of parental RI lines. By dramatically extending the number of unique, reproducible genomes, RIXs share some of the best properties of both the parental RI and F2 mapping panels. These attributes make the RIX method ideally suited for experiments requiring analysis of multiple parameters, under different environmental conditions and/or temporal sampling. However, since any pair of RIX genomes shares either one or no parental RIs, this cross introduces an unusual population structure requiring special computational approaches for analysis. Herein, we propose an efficient statistical procedure for QTL mapping with RIXs and describe a novel empirical permutation procedure to assess genome-wide significance. This procedure will also be applicable to diallel crosses. Extensive simulations using strain distribution patterns from CXB, AXB/BXA, and BXD mouse RI lines show the theoretical power of the RIX approach and the analysis of CXB RIXs demonstrates the limitations of this procedure when using small RI panels.     

Quantitative trait locus-specific genotype x alcoholism interaction on linkage for evoked electroencephalogram oscillations

We explored the evidence for a quantitative trait locus (QTL)-specific genotype x alcoholism interaction for an evoked electroencephalogram theta band oscillation (ERP) phenotype on a region of chromosome 7 in participants of the US Collaborative Study on the Genetics of Alcoholism. Among 901 participants with both genotype and phenotype data available, we performed variance component linkage analysis (SOLAR version 2.1.2) in the full sample and stratified by DSM-III-R and Feighner-definite alcoholism categories. The heritability of the ERP phenotype after adjusting for age and sex effects in the combined sample and in the alcoholism classification sub-groups ranged from 40% to 66%. Linkage on chromosome 7 was identified at 158 cM (LOD = 3.8) in the full sample and at 108 in the non-alcoholic subgroup (LOD = 3.1). Further, we detected QTL-specific genotype x alcoholism interaction at these loci. This work demonstrates the importance of considering the complexity of common complex traits in our search for genes that predispose to alcoholism.     

Methods for linkage analysis of quantitative trait loci in humans

This paper reviews linkage analysis methods for detecting loci associated with quantitative traits in humans. All such methods are based on the underlying principle that family members who have similar trait values should have higher than expected levels of sharing of genetic material (identity by descent) near the genes that influence those traits. A number of different statistical methods for testing that association between shared trait values and shared identity by descent have been developed over the past 30 or more years. These different types of tests are reviewed here, with emphasis on their theory and derivations. Robustness and power are also discussed.     

Quantitative trait locus analysis of longitudinal quantitative trait data in complex pedigrees

There is currently considerable interest in genetic analysis of quantitative traits such as blood pressure and body mass index. Despite the fact that these traits change throughout life they are commonly analyzed only at a single time point. The genetic basis of such traits can be better understood by collecting and effectively analyzing longitudinal data. Analyses of these data are complicated by the need to incorporate information from complex pedigree structures and genetic markers. We propose conducting longitudinal quantitative trait locus (QTL) analyses on such data sets by using a flexible random regression estimation technique. The relationship between genetic effects at different ages is efficiently modeled using covariance functions (CFs). Using simulated data we show that the change in genetic effects over time can be well characterized using CFs and that including parameters to model the change in effect with age can provide substantial increases in power to detect QTL compared with repeated measure or univariate techniques. The asymptotic distributions of the methods used are investigated and methods for overcoming the practical difficulties in fitting CFs are discussed. The CF-based techniques should allow efficient multivariate analyses of many data sets in human and natural population genetics.     

A pathway for multivariate analysis of ecological communities using copulas

We describe a new pathway for multivariate analysis of data consisting of counts of species abundances that includes two key components: copulas, to provide a flexible joint model of individual species, and dissimilarity‐based methods, to integrate information across species and provide a holistic view of the community. Individual species are characterized using suitable (marginal) statistical distributions, with the mean, the degree of over‐dispersion, and/or zero‐inflation being allowed to vary among a priori groups of sampling units. Associations among species are then modeled using copulas, which allow any pair of disparate types of variables to be coupled through their cumulative distribution function, while maintaining entirely the separate individual marginal distributions appropriate for each species. A Gaussian copula smoothly captures changes in an index of association that excludes joint absences in the space of the original species variables. A permutation‐based filter with exact family‐wise error can optionally be used a priori to reduce the dimensionality of the copula estimation problem. We describe in detail a Monte Carlo expectation maximization algorithm for efficient estimation of the copula correlation matrix with discrete marginal distributions (counts). The resulting fully parameterized copula models can be used to simulate realistic ecological community data under fully specified null or alternative hypotheses. Distributions of community centroids derived from simulated data can then be visualized in ordinations of ecologically meaningful dissimilarity spaces. Multinomial mixtures of data drawn from copula models also yield smooth power curves in dissimilarity‐based settings. Our proposed analysis pathway provides new opportunities to combine model‐based approaches with dissimilarity‐based methods to enhance understanding of ecological systems. We demonstrate implementation of the pathway through an ecological example, where associations among fish species were found to increase after the establishment of a marine reserve.     

Quantitative trait loci analysis of potato tuber greening

Molecular Biology Reports - A conversion of amyloplasts into chloroplasts in the potato tuber after light exposure is known as tuber greening and is one of the major causes of tuber loss. We report...     

The efficiency of designs for fine-mapping of quantitative trait loci using combined linkage disequilibrium and linkage

In a simulation study, different designs were compared for efficiency of fine-mapping of QTL. The variance component method for fine-mapping of QTL was used to estimate QTL position and variance components. The design of many families with small size gave a higher mapping resolution than a design with few families of large size. However, the difference is small in half sib designs. The proportion of replicates with the QTL positioned within 3 cM of the true position is 0.71 in the best design, and 0.68 in the worst design applied to 128 animals with a phenotypic record and a QTL explaining 25% of the phenotypic variance. The design of two half sib families each of size 64 was further investigated for a hypothetical population with effective size of 1000 simulated for 6000 generations with a marker density of 0.25 cM and with marker mutation rate 4 × 10^-4 per generation. In mapping using bi-allelic markers, 42~55% of replicated simulations could position QTL within 0.75 cM of the true position whereas this was higher for multi allelic markers (48~76%). The accuracy was lowest (48%) when mutation age was 100 generations and increased to 68% and 76% for mutation ages of 200 and 500 generations, respectively, after which it was about 70% for mutation ages of 1000 generations and older. When effective size was linearly decreasing in the last 50 generations, the accuracy was decreased (56 to 70%). We show that half sib designs that have often been used for linkage mapping can have sufficient information for fine-mapping of QTL. It is suggested that the same design with the same animals for linkage mapping should be used for fine-mapping so gene mapping can be cost effective in livestock populations.     

Quantitative trait locus analysis for hemostasis and thrombosis

Susceptibility to thrombosis varies in human populations as well as many in inbred mouse strains. The objective of this study was to characterize the genetic control of thrombotic risk on three chromosomes. Previously, utilizing a tail-bleeding/rebleeding assay as a surrogate of hemostasis and thrombosis function, three mouse chromosome substitution strains (CSS) (B6-Chr5^A/J, Chr11^A/J _, Chr17^A/J) were identified (Hmtb1, Hmtb2, Hmtb3). The tail-bleeding/rebleeding assay is widely used and distinguishes mice with genetic defects in blood clot formation or dissolution. In the present study, quantitative trait locus (QTL) analysis revealed a significant locus for rebleeding (clot stability) time (time between cessation of initial bleeding and start of the second bleeding) on chromosome 5, suggestive loci for bleeding time (time between start of bleeding and cessation of bleeding) also on chromosomes 5, and two suggestive loci for clot stability on chromosome 17 and one on chromosome 11. The three CSS and the parent A/J had elevated clot stability time. There was no interaction of genes on chromosome 11 with genes on chromosome 5 or chromosome 17. On chromosome 17, twenty-three candidate genes were identified in synteny with previously identified loci for thrombotic risk on human chromosome 18. Thus, we have identified new QTLs and candidate genes not previously known to influence thrombotic risk. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Quantitative trait associated microarray gene expression data analysis

Selection on phenotypes may cause genetic change. To understand the relationship between phenotype and gene expression from an evolutionary viewpoint, it is important to study the concordance between gene expression and profiles of phenotypes. In this study, we use a novel method of clustering to identify genes whose expression profiles are related to a quantitative phenotype. Cluster analysis of gene expression data aims at classifying genes into several different groups based on the similarity of their expression profiles across multiple conditions. The hope is that genes that are classified into the same clusters may share underlying regulatory elements or may be a part of the same metabolic pathways. Current methods for examining the association between phenotype and gene expression are limited to linear association measured by the correlation between individual gene expression values and phenotype. Genes may be associated with the phenotype in a nonlinear fashion. In addition, groups of genes that share a particular pattern in their relationship to phenotype may be of evolutionary interest. In this study, we develop a method to group genes based on orthogonal polynomials under a multivariate Gaussian mixture model. The effect of each expressed gene on the phenotype is partitioned into a cluster mean and a random deviation from the mean. Genes can also be clustered based on a time series. Parameters are estimated using the expectation-maximization algorithm and implemented in SAS. The method is verified with simulated data and demonstrated with experimental data from 2 studies, one clusters with respect to severity of disease in Alzheimer's patients and another clusters data for a rat fracture healing study over time. We find significant evidence of nonlinear associations in both studies and successfully describe these patterns with our method. We give detailed instructions and provide a working program that allows others to directly implement this method in their own analyses.     

A tobit variance-component method for linkage analysis of censored trait data

Variance-component (VC) methods are flexible and powerful procedures for the mapping of genes that influence quantitative traits. However, traditional VC methods make the critical assumption that the quantitative-trait data within a family either follow or can be transformed to follow a multivariate normal distribution. Violation of the multivariate normality assumption can occur if trait data are censored at some threshold value. Trait censoring can arise in a variety of ways, including assay limitation or confounding due to medication. Valid linkage analyses of censored data require the development of a modified VC method that directly models the censoring event. Here, we present such a model, which we call the "tobit VC method." Using simulation studies, we compare and contrast the performance of the traditional and tobit VC methods for linkage analysis of censored trait data. For the simulation settings that we considered, our results suggest that (1) analyses of censored data by using the traditional VC method lead to severe bias in parameter estimates and a modest increase in false-positive linkage findings, (2) analyses with the tobit VC method lead to unbiased parameter estimates and type I error rates that reflect nominal levels, and (3) the tobit VC method has a modest increase in linkage power as compared with the traditional VC method. We also apply the tobit VC method to censored data from the Finland-United States Investigation of Non-Insulin-Dependent Diabetes Mellitus Genetics study and provide two examples in which the tobit VC method yields noticeably different results as compared with the traditional method.     

Optimal sibship selection for genotyping in quantitative trait locus linkage analysis

In this paper we present a novel method for selecting optimally informative sibships of any size for quantitative trait locus (QTL) linkage analysis. The method allocates a quantitative index of potential informativeness to each sibship on the basis of observed trait scores and an assumed true QTL model. Any sample of phenotypically screened sibships can therefore be easily rank-ordered for selective genotyping. The quantitative index is the sibship's expected contribution to the non-centrality parameter. This expectation represents the weighted sum of chi(2) test statistics that would be obtained given the observed trait values over all possible sibship genotypic configurations; each configuration is weighted by the likelihood of it occurring given the assumed true genetic model. The properties of this procedure are explored in relation to the accuracy of the assumed true genetic model and sibship size. In comparison to previous methods of selecting phenotypically extreme sibships for genotyping, the proposed method is considerably more efficient and is robust with regard to the specification of the genetic model.