A simple linear regression method for quantitative trait loci linkage analysis with censored observations

Standard quantitative trait loci (QTL) mapping techniques commonly assume that the trait is both fully observed and normally distributed. When considering survival or age-at-onset traits these assumptions are often incorrect. Methods have been developed to map QTL for survival traits; however, they are both computationally intensive and not available in standard genome analysis software packages. We propose a grouped linear regression method for the analysis of continuous survival data. Using simulation we compare this method to both the Cox and Weibull proportional hazards models and a standard linear regression method that ignores censoring. The grouped linear regression method is of equivalent power to both the Cox and Weibull proportional hazards methods and is significantly better than the standard linear regression method when censored observations are present. The method is also robust to the proportion of censored individuals and the underlying distribution of the trait. On the basis of linear regression methodology, the grouped linear regression model is computationally simple and fast and can be implemented readily in freely available statistical software.     

A pseudolikelihood method for analyzing interval censored data

We introduce a method based on a pseudolikelihood ratio forestimating the distribution function of the survival time ina mixed-case interval censoring model. In a mixed-case model,an individual is observed a random number of times, and at eachtime it is recorded whether an event has happened or not. Oneseeks to estimate the distribution of time to event. We usea Poisson process as the basis of a likelihood function to constructa pseudolikelihood ratio statistic for testing the value ofthe distribution function at a fixed point, and show that thisconverges under the null hypothesis to a known limit distribution,that can be expressed as a functional of different convex minorantsof a two-sided Brownian motion process with parabolic drift.Construction of confidence sets then proceeds by standard inversion.The computation of the confidence sets is simple, requiringthe use of the pool-adjacent-violators algorithm or a standardisotonic regression algorithm. We also illustrate the superiorityof the proposed method over competitors based on resamplingtechniques or on the limit distribution of the maximum pseudolikelihoodestimator, through simulation studies, and illustrate the differentmethods on a dataset involving time to HIV seroconversion ina group of haemophiliacs.     

A method for analyzing censored survival phenotype with gene expression data

Background  

Survival time is an important clinical trait for many disease studies. Previous works have shown certain relationship between patients' gene expression profiles and survival time. However, due to the censoring effects of survival time and the high dimensionality of gene expression data, effective and unbiased selection of a gene expression signature to predict survival probabilities requires further study.     

Quantitative trait linkage analysis using Gaussian copulas

Mapping and identifying variants that influence quantitative traits is an important problem for genetic studies. Traditional QTL mapping relies on a variance-components (VC) approach with the key assumption that the trait values in a family follow a multivariate normal distribution. Violation of this assumption can lead to inflated type I error, reduced power, and biased parameter estimates. To accommodate nonnormally distributed data, we developed and implemented a modified VC method, which we call the "copula VC method," that directly models the nonnormal distribution using Gaussian copulas. The copula VC method allows the analysis of continuous, discrete, and censored trait data, and the standard VC method is a special case when the data are distributed as multivariate normal. Through the use of link functions, the copula VC method can easily incorporate covariates. We use computer simulations to show that the proposed method yields unbiased parameter estimates, correct type I error rates, and improved power for testing linkage with a variety of nonnormal traits as compared with the standard VC and the regression-based methods.     

Boosting method for nonlinear transformation models with censored survival data

We propose a general class of nonlinear transformation models for analyzing censored survival data, of which the nonlinear proportional hazards and proportional odds models are special cases. A cubic smoothing spline-based component-wise boosting algorithm is derived to estimate covariate effects nonparametrically using the gradient of the marginal likelihood, that is computed using importance sampling. The proposed method can be applied to survival data with high-dimensional covariates, including the case when the sample size is smaller than the number of predictors. Empirical performance of the proposed method is evaluated via simulations and analysis of a microarray survival data.     

A nonparametric test for association with censored data

A nonparametric procedure is proposed for the problem of testing association between two continuous variables when one is subject to arbitrary censoring. The motivation for the procedure derives from our finding that Cox's likelihood procedure may not adequately control the size of the test. The proposed procedure allows the censoring mechanism to depend on the independent variable, is simple computationally, and provides accurate control over the size of the test even for quite small samples. Asymptotic results suggest that it may provide a sensitive alternative to Cox's procedure. An example dealing with survival following operation for myasthenia gravis is provided, wherein a method for testing after adjustment for covariate information is described.     

Segregation and linkage analysis for longitudinal measurements of a quantitative trait

We present a method for using slopes and intercepts from a linear regression of a quantitative trait as outcomes in segregation and linkage analyses. We apply the method to the analysis of longitudinal systolic blood pressure (SBP) data from the Framingham Heart Study. A first-stage linear model was fit to each subject's SBP measurements to estimate both their slope over time and an intercept, the latter scaled to represent the mean SBP at the average observed age (53.7 years). The subject-specific intercepts and slopes were then analyzed using segregation and linkage analysis. We describe a method for using the standard errors of the first-stage intercepts and slopes as weights in the genetic analyses. For the intercepts, we found significant evidence of a Mendelian gene in segregation analysis and suggestive linkage results (with LOD scores >or= 1.5) for specific markers on chromosomes 1, 3, 5, 9, 10, and 17. For the slopes, however, the data did not support a Mendelian model, and thus no formal linkage analyses were conducted.     

A Monte Carlo method for Bayesian analysis of linkage between single markers and quantitative trait loci. II. A simulation study

A Bayesian approach to the statistical mapping of Quantitative Trait Loci (QTLs) using single markers was implemented via Markov Chain Monte Carlo (MCMC) algorithms for parameter estimation and hypothesis testing. Parameters were estimated by marginal posterior means computed with a Gibbs sampler with data augmentation. Variables sampled included the augmented data (marker-QTL genotypes, polygenic effects), the event of linkage or nonlinkage, and the parameters (allele frequencies, QTL substitution effect, recombination rate, polygenic and residual variances). The analysis was evaluated empirically via application to simulated granddaughter designs consisting of 2000 sons, 20 related sires and their ancestors. Results obtained in this study and preliminary work on multiple linked markers and multiple QTLs support the usefulness of the Bayesian method for the statistical mapping of QTLs.     

High-resolution joint linkage disequilibrium and linkage mapping of quantitative trait loci based on sibship data

This paper proposes variance component models for high resolution joint linkage disequilibrium (LD) and linkage mapping of quantitative trait loci (QTL) based on sibship data; this can include population data if independent individuals are treated as single sibships. One application of these models is late onset complex disease gene mapping, when parental data are not available. The models simultaneously incorporate both LD and linkage information. The LD information is contained in mean coefficients of sibship data. The linkage information is contained in the variance-covariance matrices of trait values for sibships with at least two siblings. We derive formulas for calculating the probability of sharing two trait alleles identical by descent (IBD) for sibpairs in interval mapping of QTL; this is the coefficient of dominant variance of the trait covariance of sibpairs on major QTL. To investigate the performance of the formulas, we calculate the numerical values via the formulas and get satisfactory approximations. We compare the power and sample sizes for both LD and linkage mapping. By simulation and theoretical analysis, we compare the results with those of Fulker and Abecasis "AbAw" approach. It is well known that the resolution of linkage analysis can be low for complex disease gene mapping. LD mapping, on the other hand, can increase mapping precision and is useful in high resolution mapping. Linkage analysis is less sensitive to population subdivisions and admixtures. The level of LD is sensitive to population stratification which may easily lead to spurious association. Performing a joint analysis of LD and linkage mapping can help to overcome the limits of both approaches. Moreover, the advantages of the two complementary strategies can be utilized maximally. In practice, linkage analysis may be performed using pedigree data to identify suggestive linkage between markers and trait loci based on a sparse marker map. In the presence of linkage, joint LD and linkage mapping can be carried out to do fine gene mapping based on a dense genetic map using both pedigree and population data. Population and pedigree data of any type can be combined to perform a joint analysis of high resolution LD and linkage mapping of QTL by generalizing the 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.     

A powerful likelihood method for the analysis of linkage disequilibrium between trait loci and one or more polymorphic marker loci.

Historically, most methods for detecting linkage disequilibrium were designed for use with diallelic marker loci, for which the analysis is straightforward. With the advent of polymorphic markers with many alleles, the normal approach to their analysis has been either to extend the methodology for two-allele systems (leading to an increase in df and to a corresponding loss of power) or to select the allele believed to be associated and then collapse the other alleles, reducing, in a biased way, the locus to a diallelic system. I propose a likelihood-based approach to testing for linkage disequilibrium, an approach that becomes more conservative as the number of alleles increases, and as the number of markers considered jointly increases in a multipoint test for linkage disequilibrium, while maintaining high power. Properties of this method for detecting associations and fine mapping the location of disease traits are investigated. It is found to be, in general, more powerful than conventional methods, and it provides a tractable framework for the fine mapping of new disease loci. Application to the cystic fibrosis data of Kerem et al, is included to illustrate the method.     

A simple nonparametric test for linkage with sib-pair censored event time observations

Consider the case that individual phenotype and genotype observations were collected from a large or moderate number of pedigrees. Some of the pedigrees have multi-generation nuclear families. For each nuclear family, the phenotype trait value of each sibling is the time to onset for a specific event (e.g., disease). Often, this event time may be right censored, that is, an individual is event-free at the study examination time point. In this article, we propose a purely nonparametric test for testing if the distribution of a Haseman-Elston distance measure between two siblings' event times is independent of their mean genetic sharing identical by descent at a genetic marker based on such incomplete observations from all the nuclear families. The new test can be implemented easily and is illustrated with a data set from the Genetic Analysis Workshop 12. The validity of the new test is examined via a simulation study.