Bayesian discrete lognormal regression model for genomic prediction

Abstract

Key message

Genomic prediction models for quantitative traits assume continuous and normally distributed phenotypes. In this research, we proposed a novel Bayesian discrete lognormal regression model.

Abstract

Genomic selection is a powerful tool in modern breeding programs that uses genomic information to predict the performance of individuals and select those with desirable traits. It has revolutionized animal and plant breeding, as it allows breeders to identify the best candidates without labor-intensive and time-consuming phenotypic evaluations. While several statistical models have been developed, most of them have been for quantitative continuous traits and only a few for count responses. In this paper, we propose a discrete lognormal regression model in the Bayesian context, that with a Gibbs sampler to explore the corresponding posterior distribution and make the predictions. Two datasets of resistance disease is used in the wheat crop and are then evaluated against the traditional Gaussian model and a lognormal model. The results indicate the proposed model is a competitive and natural model for predicting count genomic traits.

Appendices

Appendix 1

The observed response variable in model (1) is the result of applying the floor function to a continuous Lognormal regression model, that is, $Y_{i} = \left\lfloor {L_{i}^{*} } \right\rfloor$, where given ${{\varvec{x}}}_{i}$ the latent variable ${L}_{i}^{*}$ follows a Lognormal distribution with parameters ${\mu }_{i}={\beta }_{0}+\sum_{j=1}^{p}{x}_{ij}{\beta }_{j}$ and ${\sigma }_{i}^{2}={\sigma }^{2}$, $i=1,\dots ,n.$ Then, by expressing ${L}_{i}^{*}={\text{exp}}({L}_{i})$ where ${L}_{i}|{{\varvec{x}}}_{i}\sim N\left({\mu }_{i},{\sigma }^{2}\right)$, $i=1,\dots ,n$, and by augmenting the posterior distribution of the parameters of model (1), ${\beta }_{0},{\varvec{\beta}},{\sigma }_{\beta }^{2},{\sigma }^{2}$, with latent variables ${L}_{i}$ (${\varvec{L}}={\left({L}_{1},..,{L}_{n}\right)}^{T}$), the joint posterior of ${\beta }_{0},{\varvec{\beta}},{\sigma }_{\beta }^{2},{\sigma }^{2}$ and ${\varvec{L}}$ is given by

$${f}_{{\beta }_{0},{\varvec{\beta}},{\sigma }_{\beta }^{2},{\sigma }^{2},{\varvec{L}}|{\varvec{Y}}}\left({\beta }_{0},{\varvec{\beta}},{\sigma }_{\beta }^{2},{\sigma }^{2},{\varvec{l}}|{\varvec{y}}\right)\propto \left\{\prod_{i=1}^{n}\frac{1}{\sqrt{{\sigma }^{2}}}{\text{exp}}\left[-\frac{1}{2{\sigma }^{2}}{\left({l}_{i}-{\beta }_{0}-\sum_{j=1}^{p}{x}_{ij}{\beta }_{j}\right)}^{2}\right]{I}_{\left\{{\text{log}}\left({y}_{i}\right)\le {l}_{i}\le \mathit{log}\left({y}_{i}+1\right)\right\}}\right\}\times {f}_{{\varvec{\beta}}|{\sigma }_{\beta }^{2}}\left({\varvec{\beta}}|{\sigma }_{\beta }^{2}\right){f}_{{\sigma }_{\beta }^{2}}\left({\sigma }_{\beta }^{2}\right){f}_{{\sigma }^{2}}\left({\sigma }^{2}\right)$$

(A1)

where ${\varvec{l}}={\left({l}_{1},..,{l}_{n}\right)}^{T}$. From here and doing simple algebraic manipulations, the full conditional posterior for ${\beta }_{0}$ is a normal distribution with mean $\frac{1}{n}\sum_{i=1}^{n}{l}_{i}^{\left(0\right)}$ and variance $\frac{{\sigma }^{2}}{n}$ where ${l}_{i}^{(0)}={l}_{i}-\sum_{\begin{array}{c}j=1\end{array}}^{p}{x}_{ij}{\beta }_{j}$.

Similarly, the full conditional posterior for each ${\beta }_{k},$ $k=1,..,p,$ is a normal distribution with variance $\frac{1}{{\sigma }_{\beta }^{-2}+{\sigma }^{-2}\sum_{i=1}^{n}{x}_{ik}^{2}}$ and mean $\frac{{\sigma }^{-2}}{{\sigma }_{\beta }^{-2}+{\sigma }^{-2}\sum_{i=1}^{n}{x}_{ik}^{2}}\sum_{i=1}^{n}{l}_{i}^{(k)}{x}_{ik}$ where ${l}_{i}^{(k)}={l}_{i}-{\beta }_{0}-\sum_{\begin{array}{c}j=1\\ j\ne k\end{array}}^{p}{x}_{ij}{\beta }_{j}$.

The full conditional posterior for ${\sigma }_{\beta }^{2}$ is

$${f}_{{\sigma }_{\beta }^{2}}\left({\sigma }_{\beta }^{2}|{\varvec{y}},-\right)\propto \frac{1}{{{\sigma }_{\beta }^{2}}^{p/2}}{\text{exp}}\left[-\frac{1}{2{\sigma }_{\beta }^{2}}\sum_{j=1}^{p}{\beta }_{j}^{2}\right]\frac{1}{{{\sigma }_{\beta }^{2}}^{1+{v}_{\beta }/2}}{\text{exp}}\left(-\frac{{s}_{\beta }}{2{\sigma }_{\beta }^{2}}\right)\propto \frac{1}{{{\sigma }_{\beta }^{2}}^{1+({v}_{\beta }+p)/2}}{\text{exp}}\left[-\frac{\left({s}_{\beta }+\sum_{j=1}^{p}{\beta }_{j}^{2}\right)}{2{\sigma }_{\beta }^{2}}\right]$$

which corresponds to the density of a scaled inverse chi-squared distribution (${\chi }^{-2}$) and so ${\sigma }_{\beta }^{2}|{\varvec{y}},-\sim {\chi }^{-2}\left({\widetilde{v}}_{\beta },{\widetilde{s}}_{\beta }\right)$, ${\widetilde{v}}_{\beta }={v}_{\beta }+p$ and${\widetilde{s}}_{\beta }={s}_{\beta }+\sum_{j=1}^{p}{\beta }_{j}^{2}$. Likewise, the full conditional posterior for ${\sigma }^{2}$ is ${\sigma }^{2}|{\varvec{y}},-\sim {\chi }^{-2}\left(\widetilde{v},\widetilde{s}\right)$, $\widetilde{v}=v+n$ and$\widetilde{s}=s+\sum_{i=1}^{n}{\left({l}_{i}-{\beta }_{0}-\sum_{j=1}^{p}{x}_{ij}{\beta }_{j}\right)}^{2}$. Here, we denote the rest of the parameters other than the parameter for which the conditional distribution is specified.

Now, from equation (A1) the full conditional posterior for ${\varvec{L}}$ is given by

$${f}_{{\varvec{L}}|{\varvec{Y}}}\left({\varvec{l}}|{\varvec{y}},-\right)\propto \prod_{i=1}^{n}\frac{1}{\sqrt{2\pi {\sigma }^{2}}}{\text{exp}}\left[-\frac{1}{2{\sigma }^{2}}{\left({l}_{i}-{\beta }_{0}-\sum_{j=1}^{p}{x}_{ij}{\beta }_{j}\right)}^{2}\right]{I}_{\left\{{\text{log}}\left({y}_{i}\right)\le {l}_{i}\le {\text{log}}({y}_{i}+1)\right\}}$$

and from here conditioned to ${\varvec{Y}}$ and the parameters of model (1), ${L}_{1},..,{L}_{n}$ are independent random variables, each one with truncated normal distribution on ($\mathit{log}\left({y}_{i}\right),log({y}_{i}+1$) with parameters ${\beta }_{0}+\sum_{j=1}^{p}{x}_{ij}{\beta }_{j}$ and ${\sigma }^{2}$, $i=1,\dots ,n$, respectively.

Appendix 2

See Figs. 3 and 4.

Fig. 3

Trace plots for two chains of the log-posterior density of model (1) with predictor (2)

Fig. 4

Trace plots for two chains of the fixed (environment) effects and variance components of model (1) with predictor (2)