Bayesian Composite Model Space Approach for Mapping Quantitative Trait Loci in Variance Component Model

Abstract

In this article, we successfully apply the novel model selection method, Bayesian composite model space approach which has been used to map quantitative trait loci (QTL) for allelic substitution model, to map QTL for variance component model. The novel model selection approach has two advantages compared to the reversible jump Markov chain Monte Carlo method. Firstly, it mixes well due to the fixedness of the model dimension; secondly, it can map multiple QTL with higher power especially in genome-wide QTL mapping; finally, in the new method, it is also easy to incorporate our prior information about the variance components, which may bring precise estimate for variance components. A series of simulation experiments were conducted to demonstrate the general characters of the proposed method. The computer program is written in FORTRAN language, which is also built into a software “BayesMapQTL”, and they also can be used for real data analysis and are available for request.

Appendix

Update the covariate effects

We assume the prior distribution of the covariate effects β follow multivariate normal distribution, ${\varvec{\beta}}\sim N({\varvec{\beta}}_{0} ,{\mathbf{V}}_{0} )$ with hyper-parameter β ₀ and V ₀, then the efficient Gibbs sampling approach can be used to update the covariate effects. The posterior distribution of it follows multivariate normal distribution,

$$ {\varvec{\beta}}\left| {{\mathbf{y}},{\varvec{\uptheta}}_{{ - {\varvec{\beta}}}} } \right.,{\mathbf{X}}\sim N(\bar{\varvec{{\beta }}},{\mathbf{V}}_{{\varvec{\beta}}} ) $$

(A1)

where, ${\bar{\varvec{\beta }}} = \left( {{\mathbf{X}}^{T} {\mathbf{V}}^{ - 1} {\mathbf{X}} + {\mathbf{V}}_{{_{0} }}^{ - 1} } \right)^{ - 1} \left( {{\mathbf{X}}^{T} {\mathbf{V}}^{ - 1} {\mathbf{y}} + {\mathbf{V}}_{{_{0} }}^{ - 1} {\varvec{\beta}}_{0} } \right),\,{\mathbf{V}}_{{\varvec{\beta}}} = \left( {{\mathbf{X}}^{T} {\mathbf{V}}^{ - 1} {\mathbf{X}} + {\mathbf{V}}_{{_{0} }}^{ - 1} } \right)^{ - 1} .$

Update the polygenic variance and the residual variance

The approaches of updating the polygenic variance $\sigma_{A}^{2}$ and the residual variance $\sigma_{e}^{2}$ are very similar to that of updating the QTL variance. We also use Browne’s approach to generate the new value and then accept it according to the acceptance probability. The acceptance rate for updating the polygenic variance

$$ r_{A} = \frac{{f({\mathbf{y}}\left| {{\mathbf{\phi }},{\varvec{\uptheta}}_{{ - \sigma_{{^{A} }}^{2} }} ,\sigma_{A}^{2( * )} ,{\varvec{\uplambda}},{\mathbf{\hat{\Uptheta }}}} \right.,{\mathbf{A}},{\mathbf{X}}) \cdot f(\sigma_{A}^{2( * )} \left| {s_{A}^{2} } \right.)}}{{f({\mathbf{y}}\left| {{\mathbf{\phi }},{\varvec{\uptheta}}_{{\sigma_{{^{A} }}^{2} }} ,\sigma_{A}^{2} ,{\varvec{\uplambda}},{\mathbf{\hat{\Uptheta }}}} \right.,{\mathbf{A}},{\mathbf{X}}) \cdot f(\sigma_{A}^{2} \left| {s_{A}^{2} } \right.)}} \cdot hr_{A} $$

(A2)

where, $p(\sigma_{A}^{2} \left| {s_{A}^{2} } \right.)\sim {\text{Inv}} - {{\upchi}}^{2} (\omega_{A} ,s_{A}^{2} ) \propto s_{A}^{{\omega_{A} }} (\sigma_{A}^{2} )^{{ - (\omega_{A} /2 + 1)}} \cdot \exp \left( { - \omega_{A}^{{}} s_{A}^{2} /(2\sigma_{{^{A} }}^{2} )} \right)$, and

$$ \begin{gathered} hr_{A} = \frac{{p(\sigma_{A}^{2(t)} \left| {\sigma_{A}^{2( * )} } \right.)}}{{p(\sigma_{A}^{2( * )} \left| {\sigma_{A}^{2(t)} } \right.)}} \hfill \\ \quad = \frac{{{\text{Inv}} - {{\upchi}}^{2} \left( {\nu_{A} ,(\nu_{A} - 2)\sigma_{A}^{2( * )} /\nu_{A} } \right)}}{{{\text{Inv}} - {{\upchi}}^{2} \left( {\nu_{A} ,(\nu_{A} - 2)\sigma_{A}^{2(t)} /\nu_{A} } \right)}} \hfill \\ \quad \hfill \\ \quad = \left( {\frac{{\sigma_{A}^{2( * )} }}{{\sigma_{A}^{2(t)} }}} \right)^{v + 1} \cdot \exp \left\{ {\frac{{(\nu_{A} - 2)}}{2}\left( {\frac{{\sigma_{A}^{2(t)} }}{{\sigma_{A}^{2( * )} }} - \frac{{\sigma_{A}^{2( * )} }}{{\sigma_{A}^{2(t)} }}} \right)} \right\}. \hfill \\ \end{gathered} $$

(A3)

The acceptance rate for updating the residual variance

$$ r_{e} = \frac{{f({\mathbf{y}}\left| {{\mathbf{\phi }},{\varvec{\uptheta}}_{{ - \sigma_{{^{e} }}^{2} }} ,\sigma_{e}^{2( * )} ,{\varvec{\uplambda}},{\mathbf{\hat{\Uptheta }}}} \right.,{\mathbf{A}},{\mathbf{X}}) \cdot f(\sigma_{e}^{2( * )} \left| {s_{e}^{2} } \right.)}}{{f({\mathbf{y}}\left| {{\mathbf{\phi }},{\varvec{\uptheta}}_{{\sigma_{{^{e} }}^{2} }} ,\sigma_{e}^{2} ,{\varvec{\uplambda}},{\mathbf{\hat{\Uptheta }}}} \right.,{\mathbf{A}},{\mathbf{X}}) \cdot f(\sigma_{e}^{2} \left| {s_{e}^{2} } \right.)}} \cdot hr_{e} , $$

(A4)

where, $p(\sigma_{e}^{2} \left| {s_{e}^{2} } \right.)\sim {\text{Inv}} - {{\upchi}}^{2} (\omega_{e} ,s_{e}^{2} ) \propto s_{e}^{{\omega_{e} }} (\sigma_{e}^{2} )^{{ - (\omega_{e} /2 + 1)}} \cdot \exp \left( { - \omega_{e}^{{}} s_{e}^{2} /(2\sigma_{{^{e} }}^{2} )} \right)$, and

$$ \begin{gathered} hr_{e} = \frac{{p(\sigma_{e}^{2(t)} \left| {\sigma_{e}^{2( * )} } \right.)}}{{p(\sigma_{e}^{2( * )} \left| {\sigma_{e}^{2(t)} } \right.)}} \hfill \\ \quad = \frac{{{\text{Inv}} - {{\upchi}}^{2} \left( {\nu_{e} ,(\nu_{e} - 2)\sigma_{e}^{2( * )} /\nu_{e} } \right)}}{{{\text{Inv}} - {{\upchi}}^{2} \left( {\nu_{e} ,(\nu_{e} - 2)\sigma_{e}^{2(t)} /\nu_{e} } \right)}} \hfill \\ \quad \hfill \\ \quad = \left( {\frac{{\sigma_{e}^{2( * )} }}{{\sigma_{e}^{2(t)} }}} \right)^{v + 1} \cdot \exp \left\{ {\frac{{(\nu_{e} - 2)}}{2}\left( {\frac{{\sigma_{e}^{2(t)} }}{{\sigma_{e}^{2( * )} }} - \frac{{\sigma_{e}^{2( * )} }}{{\sigma_{e}^{2(t)} }}} \right)} \right\}. \hfill \\ \end{gathered} $$

(A5)

Update QTL position

The proposal position is moved around the old one, $\lambda_{j}^{*} = \lambda_{j}^{{}} + d$, where d is a random number which is sampled from uniform distribution with bound −kcM and kcM, where k is a predetermined tuning parameter, equal to 1 for chromosome segment analysis and 20 for genome-wide scan in our study. The new position is accepted with probability equal to min (1, r), and r is shown in Table 1.