A robust Bayesian genome-based median regression model

Abstract

Key message

Current genome-enabled prediction models assumed errors normally distributed, which are sensitive to outliers. We propose a model with errors assumed to follow a Laplace distribution to deal better with outliers.

Abstract

Current genome-enabled prediction models use regressions that fit the expected value (mean) of a response variable with errors assumed normally distributed, which are often sensitive to outliers, either genetic or environmental. For this reason, we propose a robust Bayesian genome median regression (BGMR) model that fits regressions to the medians of a distribution, with errors assumed to follow a Laplace distribution to deal better with outliers. The BGMR model was evaluated under a Bayesian framework with Markov Chain Monte Carlo sampling using a location–scale mixture representation of the Laplace distribution. The BGMR was implemented with two simulated and two real genomic data sets, and we compared its prediction performance with that of a conventional genomic best linear unbiased prediction (GBLUP) model and the Laplace maximum a posteriori (LMAP) method. The prediction accuracies of BGMR were higher than those of the GBLUP and LMAP methods when there were outliers. The BGMR model could be useful to breeders who need to predict and select genotypes based on data with unknown outliers.

Appendices

Appendix 1: Deriving full conditional distributions for the Bayesian Laplace regression model

From Eq. (2), given the random effects and $u = (u_{1} , \ldots ,u_{n} )^{\text{T}} ,$ the joint conditional density of the vector of responses is given by

$$\begin{aligned} f(\varvec{y}|\varvec{b}, \varvec{u},\mu , \sigma_{1}^{2} ,\sigma^{2} ) & \propto \prod\limits_{j = 1}^{n} {\frac{1}{{\sqrt {\sigma^{2} u_{j} } }}\exp \left[ { - \frac{{\left( {y_{j} - \mu - z_{j}^{T} b} \right)^{2} }}{{2\left( 8 \right)\sigma^{2} u_{j} }}} \right]} \\ & \propto \exp \left[ { - \frac{1}{{2\sigma^{2} }}\left( {\varvec{Y} - 1\varvec{\mu}- \varvec{b}} \right)^{T} \varvec{D}_{u}^{ - 1} \left( {\varvec{Y} - 1\varvec{\mu}- \varvec{b}} \right)} \right]. \\ \end{aligned}$$

Fully conditional for $\mu$

$$\begin{aligned} f\left( {\mu | {\text{ELSE}}} \right) & \propto f\left( {\varvec{y}|\varvec{b}, \varvec{u},\mu , \sigma_{1}^{2} ,\sigma^{2} } \right)f\left( {\mu |\sigma_{0}^{2} } \right) \\ & \propto \exp \left[ { - (\varvec{Y} - 1\mu - \varvec{b})^{\text{T}} \varvec{D}_{u}^{ - 1} (\varvec{Y} - 1\mu - \varvec{b}) - \frac{1}{{2\sigma_{0}^{2} }}(\mu - \mu_{0} )^{2} } \right] \\ & \propto \exp \left[ { - \frac{1}{{2\tilde{\sigma }_{0}^{2} }}(\mu - \tilde{\mu }_{0} )^{2} } \right] \\ \end{aligned}$$

(4)

where $\tilde{\sigma }_{0}^{2} = \frac{1}{{\sigma_{0}^{ - 2} + 8^{ - 1} \sum\nolimits_{j = 1}^{n} {u_{j}^{ - 1} \sigma^{ - 2} } }}$. and $\tilde{\mu }_{0} = \tilde{\sigma }_{0}^{2} [\mu_{0} \sigma_{0}^{ - 2} + \sigma^{ - 2} 1^{\text{T}} \varvec{D}_{u}^{ - 1} (\varvec{Y} - \varvec{b})]$. Then $\mu |{\text{ELSE}}\,\sim\,N\left( {\tilde{\mu }_{0} ,\tilde{\sigma }_{0}^{2} } \right)$.

Fully conditional for $\varvec{b}$

Similarly, we have that

$$\begin{aligned} f(\varvec{b} | {\text{ELSE}}) & \propto f(\varvec{y}|\varvec{b}, \varvec{u},\mu , \sigma_{1}^{2} ,\sigma^{2} )f(\varvec{b} |\sigma_{1}^{2} ) \\ & \propto \exp \left[ { - \frac{1}{{2\sigma^{2} }}(\varvec{Y} - 1\mu - \varvec{b})^{\text{T}} \varvec{D}_{u}^{ - 1} (\varvec{Y} - 1\mu - \varvec{b}) - \frac{1}{{2\sigma_{0}^{2} }}(\mu - \mu_{0} )^{2} } \right] \\ & \propto \exp \left[ { - \frac{1}{2}(\varvec{b} - \tilde{\varvec{b}})^{\text{T}} \tilde{\varvec{\varSigma }}_{1}^{ - 1} \left( { \varvec{b} - \tilde{\varvec{b}}} \right)} \right] \\ \end{aligned}$$

(5)

where $\tilde{\varvec{\varSigma }}_{1} = (\varvec{G}^{ - 1} \sigma_{1}^{ - 2} + \sigma^{ - 2} \varvec{D}_{u}^{ - 1} )^{ - 1}$ and $\tilde{\varvec{b}} = \sigma^{ - 2} \tilde{\varvec{\varSigma }}_{{b_{1} }} \varvec{D}_{u}^{ - 1} (\varvec{Y} - 1\mu )$. So $\varvec{b}|{\text{ELSE}}\,\sim\,N(\tilde{\varvec{b}},\tilde{\varvec{\varSigma }}_{1} )$.

Fully conditional for $\sigma_{1}^{2}$

$$\begin{aligned} f\left( {\sigma_{1}^{2} | {\text{ELSE}}} \right) & \propto f(\varvec{b} |\sigma_{1}^{2} )f(\sigma_{1}^{2} ) \\ & \propto (\sigma_{1}^{2} )^{{ - \frac{{\nu_{1} + J}}{2} - 1}} \exp \left\{ { - \left( {\frac{{\varvec{b}^{\text{T}} \varvec{G}^{ - 1} \varvec{b} + S_{1} }}{{2\sigma_{1}^{2} }}} \right)} \right\} \\ & \propto \chi^{ - 2} \left( {\nu_{1} + J,\varvec{b}^{\text{T}} \varvec{G}^{ - 1} \varvec{b} + S_{1} } \right) \\ \end{aligned}$$

(6)

Fully conditional for $\sigma^{2}$

$$\begin{aligned} f(\sigma^{2} | {\text{ELSE}}) & \propto f\left( {\varvec{y}|\varvec{b}, \varvec{u},\mu , \sigma_{1}^{2} ,\sigma^{2} } \right)f(\sigma^{2} ) \\ & \propto (\sigma^{2} )^{{ - \frac{df + n}{2} - 1}} \exp \left[ { - \frac{{\left( {\varvec{y} - 1\mu - \varvec{b}} \right)^{\text{T}} \varvec{D}_{u}^{ - 1} \left( {\varvec{y} - 1\mu - \varvec{b}} \right) + S}}{{2\sigma^{2} }}} \right] \\ & \propto \chi^{ - 2} \left( {df + n,\left( {\varvec{y} - 1\mu - \varvec{b}} \right)^{\text{T}} \varvec{D}_{u}^{ - 1} \left( {\varvec{y} - 1\mu - \varvec{b}} \right) + S} \right) \\ \end{aligned}$$

(7)

Fully conditional for $\varvec{u}$

$$\begin{aligned} P(\varvec{u}\left| {\text{ELSE}} \right.) & \propto f\left( {\varvec{y}|\varvec{b}, \varvec{u},\mu , \sigma_{1}^{2} ,\sigma^{2} } \right)\prod\limits_{j = 1}^{n} {f(u_{j} )} \\ & \propto \mathop \prod \limits_{j = 1}^{n} \frac{1}{{\sqrt {u_{j} } }}{ \exp }\left( { - \frac{{\left( {y_{j} - \mu - b_{j} } \right)^{2} }}{{2\left( 8 \right)\sigma^{2} u_{j} }}} \right){\text{exp(}} - u_{j} )\\ & \propto \mathop \prod \limits_{j = 1}^{n} u_{j}^{{\frac{1}{2} - 1}} { \exp }\left( { - \frac{1}{2}\left[ {\frac{{\left( {y_{j} - \mu - b_{j} } \right)^{2} }}{{8\sigma^{2} }}u_{j}^{ - 1} + 2u_{j} } \right]} \right) \\ & \propto \mathop \prod \limits_{j = 1}^{n} {\text{GIG}}\left( {\frac{1}{2},\frac{{\left( {y_{j} - \mu - b_{j} } \right)^{2} }}{{8\sigma^{2} }},2} \right) \\ \end{aligned}$$

(8)

where ${\text{GIG(}}v,a,b )$ denotes the generalized inverse Gaussian distribution with parameters $v$, $a$ and $b$ (Kozumi and Kobayashi 2011).

Fully conditional for missing values

$$\begin{aligned} f\left( {\varvec{y}_{\text{miss}} | {\text{ELSE}}} \right) & \propto f\left( {\varvec{y}_{\text{miss}} |\varvec{b}, \varvec{u},\mu , \sigma_{1}^{2} ,\sigma^{2} } \right) \\ & \propto N\left( {\varvec{\eta}^{*} ,\sigma^{2} \varvec{D}_{u}^{*} } \right) \\ \end{aligned}$$

(9)

where $\varvec{\eta}^{*} = 1^{\varvec{*}} \mu + \varvec{g}^{\varvec{*}}$ is the corresponding linear predictor of the missing values in the model in Eq. (2) and $\varvec{D}_{u}^{*}$ is a diagonal matrix that retains the elements in $\varvec{D}_{u}$ corresponding to the missing values.

Appendix 2: Setting hyperparameters for the prior distributions of the BGMR model

The prior mean ($\mu_{0}$) for the general mean ($\mu$) was settled as the mean response sample in the training data, while the rest of the hyperparameters for the BGMR model were set similarly to those used in the BGLR software (Pérez-Rodríguez and de los Campos 2014). These rules provide proper, but weakly informative prior distributions. We partitioned the total variance–covariance of the phenotypes into two components: (1) the error and (2) the linear predictor. First, the variance of the phenotypes $y_{i}$ under the model is given by

$${\text{Var}}(y_{j} ) = {\text{Var}}(b_{j} ) + 8\sigma^{2}$$

Therefore, the average of the variance of the individuals, called total variance, is equal to

$$\frac{1}{n}\mathop \sum \limits_{j = 1}^{n} {\text{Var}}(y_{j} ) = \frac{1}{n}\mathop \sum \limits_{j = 1}^{n} {\text{Var(}}b_{j} )+ 8\sigma^{2} = \frac{1}{n}{\text{tr}}(\varvec{G})\sigma_{1}^{2} + 8\sigma^{2} = V_{1} + V_{\epsilon } .$$

Then, by setting $R_{1}^{2}$ as the proportion of the total variance (${\mathbf{V}}_{y}$) that is explained by lines a priori, $V_{g} = R_{1}^{2} {\mathbf{V}}_{y}$, and replacing $\sigma_{1}^{2}$ in $V_{1}$ by its prior mode, $\frac{{S_{1} }}{{df_{1} + 2}}$. Once we have set a value for $df_{1}$, the scale parameter is given by

$$S_{1} = \frac{{R_{1}^{2} {\mathbf{V}}_{y} }}{{\frac{1}{n}{\text{tr(}}\varvec{G} )}}\left( {df_{1} + 2} \right).$$

For the shape parameter by default, we set $df_{1} = 5$ and $R_{1}^{2} = 0.5$.

Similarly, once there is a value for the shape parameter of the prior distribution of $\sigma^{2}$, $df$, the value of the scale parameter is given by

$$S = \frac{{\left( {1 - R_{1}^{2} } \right){\mathbf{V}}_{y} }}{8}\left( {df + 2} \right)$$

where $1 - R_{1}^{2}$ is the proportion of the total variance (${\mathbf{V}}_{y}$) that is explained by the error a priori. By default, we set $df = 5.$

The pdf of the scaled inverse Chi-square distribution with $v$ degrees of freedom and scale parameter $S$, $\chi^{ - 2} \left( {v,S} \right)$, is given by

$$f(x;\,df,S) = \frac{{\left( {\frac{S}{2}} \right)^{df/2} }}{{\Gamma (df/2)}}x^{ - 1 - df/2} \exp \left( { - \frac{S}{2x}} \right), x > 0$$

and the mean, mode, and variance of this distribution are given by $\frac{S}{df - 2}$, $\frac{S}{df + 2}$, and $\frac{{2S^{2} }}{{\left( {df - 2} \right)^{2} \left( {df - 4} \right)}}$, respectively. Specifically, the prior mean, mode, and variance for the variance components are:

$$\begin{aligned} & E\left( {\sigma_{1}^{2} } \right) = \frac{{S_{1} }}{{df_{1} - 2}}, \quad {\text{Mode}}\left( {\sigma_{1}^{2} } \right) = \frac{{S_{1} }}{{df_{1} + 2}}\;{\text{and}}\;{\text{Var}}\left( {\sigma_{1}^{2} } \right) = \frac{{2S_{1}^{2} }}{{\left( {df_{1} - 2} \right)^{2} \left( {df_{1} - 4} \right)}} \\ & E\left( {\sigma^{2} } \right) = \frac{S}{df - 2},\quad {\text{Mode}}\left( {\sigma^{2} } \right) = \frac{S}{df + 2} \;{\text{and}}\;{\text{Var}}\left( {\sigma^{2} } \right) = \frac{{2S^{2} }}{{\left( {df - 2} \right)^{2} \left( {df - 4} \right)}}. \\ \end{aligned}$$

Appendix 3