Bayesian mapping of genomewide interacting quantitative trait loci for ordinal traits

Abstract

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(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.

Figure 1.—

Boxplots for week 10 weight by number of dead fetuses per replicate in the F₂ mice.

Figure 2.—

Real F₂ data analysis with the ordinal probit model: one-dimensional profiles of Bayes factors (rescaled as 2 log_eBF and negative values are truncated as zero). (Top) For all combined effects (additive, dominance, and epistatic effects); (middle) for main effects on the selected chromosomes (solid and dashed lines represent additive and dominance effects, respectively); (bottom) for epistatic interactions on the selected chromosomes (solid lines represent additive–additive interactions and other epistatic effects were not detected). On the x-axis, outer tick marks represent chromosomes and inner tick marks represent markers.

Figure 3.—

Real F₂ data analysis with the ordinal probit model: two-dimensional profiles of Bayes factors (rescaled as 2 log_eBF and negative values are truncated as zero). Top triangle shows Bayes factor of epistasis only; bottom triangle shows Bayes factor comparing full model with epistasis to no QTL.

Figure 4.—

Real F₂ data analysis by treating the ordinal trait DF as a continuous trait: one-dimensional profiles of Bayes factors (rescaled as 2 log_eBF and negative values are truncated as zero). (Top) For all combined effects (additive, dominance, and epistatic effects); (middle) for main effects on the selected chromosomes (solid and dashed lines represent additive and dominance effects, respectively); (bottom) for epistatic interactions on the selected chromosomes (solid lines represent additive–additive interactions and other epistatic effects were not detected). On the x-axis, outer tick marks represent chromosomes and inner tick marks represent markers.

Figure 5.—

Simulated F₂ data analyses: one-dimensional profiles of Bayes factors (rescaled as 2 log_eBF and negative values are truncated as zero). (Top) For all combined effects (additive, dominance, and epistatic effects) for all three analyses: solid, dashed, and dotted lines represent analyses with the ordinal probit model, the continuous trait, and the model treating the ordinal phenotype as a continuous trait, respectively. Vertical shaded dashed lines show true location of QTL. (Middle) For main effects on the selected chromosomes (solid and dashed lines represent additive and dominance effects, respectively). (Bottom) For epistatic interactions on the selected chromosomes (solid, dotted, and dashed lines represent additive–additive, additive–dominance, and dominance–additive interactions, respectively). On the x-axis, outer tick marks represent chromosomes and inner tick marks represent markers.

Figure 6.—

Simulated F₂ data analysis with the ordinal probit model: two-dimensional profiles of Bayes factors (rescaled as 2 log_eBF and negative values are truncated as zero) on selected chromosomes. Bayes factor of epistasis only is shown above the diagonal; Bayes factor comparing full model with epistasis to no QTL is shown below the diagonal.