Motivation: A simultaneous search is necessary for maximizing the power to detect epistatic quantitative trait loci (QTL). The computational complexity demands that the traditional exhaustive search be replaced by a more efficient global optimization algorithm. Results: We have adapted DIRECT, an algorithm presented in (Jones et al., 1993), to the problem of simultaneous mapping of multiple QTL. We have compared DIRECT with standard exhaustive search and a genetic algorithm previously used for QTL mapping in two dimensions. In all two- and three-QTL test cases, DIRECT accurately finds the global optimum two to four orders of magnitude faster than when using an exhaustive search, and one order of magnitude faster than when using the genetic algorithm. Thus randomization testing for determining empirical significance thresholds for at least three QTL is made feasible by the use of DIRECT. Availability: The code of the prototype implementation is available at

A tutorial derivation of the reversible jump Markov chain Monte Carlo (MCMC) algorithm is given. Various examples illustrate how reversible jump MCMC is a general framework for Metropolis-Hastings algorithms where the proposal and the target distribution may have densities on spaces of varying dimension. It is nally discussed how reversible jump MCMC can be applied in genetics to compute the posterior distribution of the number, locations, eects, and genotypes of putative quantitative trait loci.

The integrated likelihood (also called the marginal likelihood or the normalizing constant) is a central quantity in Bayesian model selection and model averaging. It is defined as the integral over the parameter space of the likelihood times the prior density. The Bayes factor for model comparison and Bayesian testing is a ratio of integrated likelihoods, and the model weights in Bayesian model averaging are proportional to the integrated likelihoods. We consider the estimation of the integrated likelihood from posterior simulation output, aiming at a generic method that uses only the likelihoods from the posterior simulation iterations. The key is the harmonic mean identity, which says that the reciprocal of the integrated likelihood is equal to the posterior harmonic mean of the likelihood. The simplest estimator based on the identity is thus the harmonic mean of the likelihoods. While this is an unbiased and simulation-consistent estimator, its reciprocal can have infinite variance and so it is unstable in general. We describe two methods for stabilizing the harmonic mean estimator. In the first one, the parameter space is reduced in such a way that the modified estimator involves a harmonic mean of heavier-tailed densities, thus resulting in a finite variance estimator. The resulting

Identifying quantitative trait loci in experimental crosses by Karl William Broman Doctor of Philosophy in Statistics University of California, Berkeley Professor Terence P. Speed, Chair Identifying the genetic loci responsible for variation in traits which are quantitative in nature (such as the yield from an agricultural crop or the number of abdominal bristles on a fruit fly) is a problem of great importance to biologists. The number and effects of such loci help us to understand the biochemical basis of these traits, and of their evolution in populations over time. Moreover, knowledge of these loci may aid in designing selection experiments to improve the traits. We focus on data from a large experimental cross. The usual methods for analyzing such data use multiple tests of hypotheses. We feel the problem is best viewed as one of model selection. After a brief review of the major methods in this area, we discuss the use of model selection to identify quantitative trait loci. Forwa...

this paper we use a Bayesian model determination approach via MCMC to estimate the number of loci by including the number of QTL as an unknown parameter following Green (1995).

Identification of quantitative trait loci (QTLs) in experimental animals is critical for understanding the biochemical bases of complex traits, and thus for the identification of drug targets. The author reviews the basic statistical methods for mapping QTLs in experimental crosses and comments on a number of the statistical issues to consider in the application of these methods. Traditional genetic studies have concentrated on dichotomous traits such as the presence or absence of a disease. Such traits are often the result of a mutation at a single gene. However, many interesting traits, like blood pressure or survival time after an infection, are quantitative in nature, and are affected by many genes and by environmental factors. There are several reviews of the statistical methods for mapping quantitative trait loci (QTLs, the genes responsible for variation in quantitative traits) in experimental crosses 1-3. Here, the attempt is to describe these methods to the reader with little detailed knowledge of statistics. This paper will sidestep

Multipoint linkage analyses ofgenetic data on extended pedigrees can involve exact computations which are infeasible. Markov chain Monte Carlo methods represent an attractive alternative, greatly extending the range of models and data sets for which analysis is practical. In this paper, several advances in Markov chain Monte Carlo theory, namely joint updates of latent variables across loci and meioses, integrated proposals, Metropolis-Hastings restarts via sequential imputation and Rao Blackwellized estimators, are incorporated into a sampling strategy which mixes well and produces accurate results in real time. The methodology is demonstrated through its application to several data sets originating from a study of early-onset Alzheimer's disease in families of Volga-German ethnic origin.

The interval mapping method has been shown to be a powerful tool for mapping QTL. However, it is still a challenge to perform a simultaneous analysis of several linked QTLs, and to isolate multiple linked QTLs. To circumvent these problems, multiple regression analysis has been suggested for experimental species. In this paper, the multiple regression approach is extended to human sib-pair data through multiple regression of the squared difference in trait values between two sibs on the proportions of alleles shared identical by descent by sib pairs at marker loci. We conduct an asymptotic analysis of the partial regression coefficients, which provide a basis for the estimation of the additive genetic variance and of locations of the QTLs. We demonstrate how the magnitude of the regression coefficients can be used to separate multiple linked QTLs. Further, we shall show that the multiple regression model using sib pairs is identifiable, and our proposed procedure for locating QTLs is robust in the sense that it can detect the number of QTLs and their locations in the presence of several linked (QTLs) in an interval, unlike a simple regression model which may find a “ghost ” QTL with no effect on the trait in the interval with several linked QTLs. Moreover, we give procedures for computing the threshold values for prespecified significance levels and for computing the power for detecting (QTLs). Finally, we investigate the consistency of the estimator for QTL locations. Using the concept of epi-convergence and variation analysis theory, we shall prove the consistency of the estimator of map location in the framework of the multiple regression approach. Since the true IBD status is not always known, the multiple regression of the squared sib difference on the estimated IBD sharing is also considered. 1. Introduction. Mapping

Abstract – The advent of molecular markers has created opportunities for a better understanding of quantitative inheritance and for developing novel strategies for genetic improvement of agricultural species, using information on quantitative trait loci (QTL). A QTL analysis relies on accurate genetic marker maps. At present, most statistical methods used for map construction ignore the fact that molecular data may be read with error. Often, however, there is ambiguity about some marker genotypes. A Bayesian MCMC approach for inferences about a genetic marker map when random miscoding of genotypes occurs is presented, and simulated and real data sets are analyzed. The results suggest that unless there is strong reason to believe that genotypes are ascertained without error, the proposed approach provides more reliable inference on the genetic map. genetic map construction / miscoded genotypes / Bayesian inference 1.

Since Lander and Botstein proposed the interval mapping method for QTL mapping data analysis in 1989, tremendous progress has been made in the last many years to advance new and powerful statistical methods for QTL analysis. Recent research progress has been focused on statistical methods and issues for mapping multiple QTL together. In this article, we review this progress. We focus the discussion on the statistical methods for mapping multiple QTL by maximum likelihood and Bayesian methods and also on determining appropriate thresholds for the analysis. Copyright © 2008 W. Zou and Z.-B. Zeng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.