Bayesian joint mapping of quantitative trait loci for Gaussian and categorical characters in line crosses

Abstract

Methodology for joint mapping of quantitative trait loci (QTL) affecting continuous and binary characters in experimental crosses is presented. The procedure consists of a Bayesian Gaussian-threshold model implemented via Markov chain Monte Carlo, which bypasses bottlenecks due to high-dimensional integrals required in maximum likelihood approaches. The method handles multiple binary traits and multiple QTL. Modeling of ordered categorical traits is discussed as well. Features of the method are illustrated using simulated datasets representing a backcross design, and the data are analyzed using mixed-trait and single-trait models. The mixed-trait analysis provides greater detection power of a QTL than a single-trait analysis when the QTL affects two or more traits. The number of QTL inferred in the mixed-trait analysis does not pertain to a specific trait, but the roles of each QTL on specific traits can be assessed from estimates of its effects. The impacts of varying incidence level and sample size on the mixed-trait QTL mapping analysis are investigated as well.

Appendix

Bayesian QTL mapping involving ordered categorical traits

Ordered categorical characters can be handled similarly to binary characters, except that the former involve c > 2 categories and c + 1 thresholds that delimit these categories. In the Bayesian analysis, if the first threshold for each trait is fixed at zero, the remaining thresholds need to be estimated in the analysis. Let ${\varvec{\uptheta}} = \left( {{\varvec{\upkappa}},{\mathbf{l}},{\mathbf{q}},{\varvec{\upbeta}},{\mathbf{Q}},{\mathbf{R}}} \right)$, where ${\varvec{\upkappa}} = \left\{ {\kappa _{{ki}} |i = 1, \ldots ,c-1;k = g + 1, \ldots ,t} \right\}$ contains all unknown thresholds. If a uniform prior is used for the thresholds, the joint posterior distribution of the unknowns is as in (6), because the prior density of κ is a constant.

The fully conditional distribution of a threshold, say κ _ki , can be shown to be the uniform process (Sorensen and Gianola 2002). Alternatively, thresholds can be sampled more efficiently using the Metropolis-Hastings algorithm described by Albert and Chib (1997). In the MCMC, an extra step is needed to sample unknown thresholds, after liabilities are generated.

For the sake of parameter identification, the first threshold is often set to be 0 and the variance to 1 for each ordinal character. In this case, the residual covariance matrix is sampled exactly as in the model with binary trait(s). Alternatively, one may choose to fix the first two thresholds of each ordinary character to 0 and 1, respectively, allowing for the residual covariance matrix to be sampled conveniently from the standard inverse Wishart distribution:

$$ {\mathbf{R}}|{\mathbf{ELSE}}\sim W^{{ - 1}} \left( {\upsilon _{R} + n,{\varvec{\Upsigma}}_{\varvec{R}} + \sum\limits_{j}^{n} {\left[ {\left( {{\mathbf{y}}_{j} - \sum\limits_{{i = 1}}^{m} {{\mathbf{W}}_{{ji}} } {\mathbf{q}}_{i} - {\mathbf{X}}_{j} {\varvec{\upbeta}}} \right)\left( {{\mathbf{y}}_{j} - \sum\limits_{{i = 1}}^{m} {{\mathbf{W}}_{{ji}} } {\mathbf{q}}_{i} - {\mathbf{X}}_{j} {\varvec{\upbeta}}} \right)'} \right]} } \right) $$

(A1)