Bayesian Mapping of Genomewide Interacting Quantitative Trait Loci for Ordinal Traits
Yi, Nengjun, Banerjee, Samprit, Pomp, Daniel, Yandell, Brian S., Genetics
Development of statistical methods and software for mapping interacting QTL has been the focus of much recent research. We previously developed a Bayesian model selection framework, based on the composite model space approach, for mapping multiple epistatic QTL affecting continuous traits. In this study we extend the composite model space approach to complex ordinal traits in experimental crosses. We jointly model main and epistatic effects of QTL and environmental factors on the basis of the ordinal probit model (also called threshold model) that assumes a latent continuous trait underlies the generation of the ordinal phenotypes through a set of unknown thresholds. A data augmentation approach is developed to jointly generate the latent data and the thresholds. The proposed ordinal probit model, combined with the composite model space framework for continuous traits, offers a convenient way for genomewide interacting QTL analysis of ordinal traits. We illustrate the proposed method by detecting new QTL and epistatic effects for an ordinal trait, dead fetuses, in a F^sub 2^ intercross of mice. Utility and flexibility of the method are also demonstrated using a simulated data set. Our method has been implemented in the freely available package R/qtlbim, which greatly facilitates the general usage of the Bayesian methodology for genomewide interacting QTL analysis for continuous, binary, and ordinal traits in experimental crosses.
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MOST complex traits are influenced by interacting networks of multiple genetic (QTL) and environmental factors. Recently several statistical methods and software have been developed to map multiple interacting QTL for continuous traits (Kao et al. 1999; Carlborg et al. 2000; Reifsnyder et al. 2000; Bogdan et al. 2004; Yi et al. 2005; Baierl et al. 2006). However, many complex traits in humans and other organisms aremeasured in an ordinalmanner. For example, many diseases are scored in several ordered categories on the basis of the magnitude of the disease symptom. Although the phenotypes of these characters are discrete, their inheritance is determined by many factors, including multiple genes and environmental components (Lynch and Walsh 1998). Theoretically, the statistical methods for continuous traits are not optimal for ordinal traits because the normality assumption is violated (Johnson and Albert 1999; Gelman et al. 2003). Therefore,mapping QTL for ordinal traits requires new methods.
The probit model is commonly used to analyze discrete binary and ordinal data (Albert and Chib 1993; Johnson and Albert 1999). An important way for the statistical inference and interpretation of the probit model is to postulate the existence of a latent (unobserved) continuous variable associated with each response through a series of unknown thresholds (Albert and Chib 1993; Johnson and Albert 1999). In quantitative genetics, the latent presentation of the probit model is called the threshold model, which has been widely used to analyze the genetic architecture of binary and ordinal traits (Wright 1934; Lynch and Walsh 1998). Under the threshold model, one can treat the latent variable as an unobservable quantitative trait, and genes controlling ordinal traits can be treated as quantitative trait loci and handled using a QTL mapping approach.
A number of statisticalmethods have been developed to identify QTL for binary or ordinal traits in experimental crosses based on the threshold model of single QTL (Hackett and Weller 1995; Xu and Atchley 1996; Rao and Xu 1998; Xu et al. 2003, 2005). Recently, several methods have been proposed to simultaneously identify multiple QTL for ordinal traits (Coffman et al. 2005; Li et al. 2006). The method of Li et al. (2006) is based on multiple-interval mapping (MIM) of Kao et al. (1999) that fits a multiple-QTL model including epistasis and simultaneously searches for the number, positions, and interaction of QTL using a non-Bayesian model selection procedure and criterion. …