A non-parametric mixture model for genome-enabled prediction of genetic value for a quantitative trait

Abstract

A Bayesian nonparametric form of regression based on Dirichlet process priors is adapted to the analysis of quantitative traits possibly affected by cryptic forms of gene action, and to the context of SNP-assisted genomic selection, where the main objective is to predict a genomic signal on phenotype. The procedure clusters unknown genotypes into groups with distinct genetic values, but in a setting in which the number of clusters is unknown a priori, so that standard methods for finite mixture analysis do not work. The central assumption is that genetic effects follow an unknown distribution with some “baseline” family, which is a normal process in the cases considered here. A Bayesian analysis based on the Gibbs sampler produces estimates of the number of clusters, posterior means of genetic effects, a measure of credibility in the baseline distribution, as well as estimates of parameters of the latter. The procedure is illustrated with a simulation representing two populations. In the first one, there are 3 unknown QTL, with additive, dominance and epistatic effects; in the second, there are 10 QTL with additive, dominance and additive × additive epistatic effects. In the two populations, baseline parameters are inferred correctly. The Dirichlet process model infers the number of unique genetic values correctly in the first population, but it produces an understatement in the second one; here, the true number of clusters is over 900, and the model gives a posterior mean estimate of about 140, probably because more replication of genotypes is needed for correct inference. The impact on inferences of the prior distribution of a key parameter (M), and of the extent of replication, was examined via an analysis of mean body weight in 192 paternal half-sib families of broiler chickens, where each sire was genotyped for nearly 7,000 SNPs. In this small sample, it was found that inference about the number of clusters was affected by the prior distribution of M. For a set of combinations of parameters of a given prior distribution, the effects of the prior dissipated when the number of replicate samples per genotype was increased. Thus, the Dirichlet process model seems to be useful for gauging the number of QTLs affecting the trait: if the number of clusters inferred is small, probably just a few QTLs code for the trait. If the number of clusters inferred is large, this may imply that standard parametric models based on the baseline distribution may suffice. However, priors may be influential, especially if sample size is not large and if only a few genotypic configurations have replicate phenotypes in the sample.

Appendices

Appendix A: Technical details on drawing genomic effects

Computing K _i is critical for implementing the DP methodology. When the baseline distribution is $N( g_{i}|0,\sigma_{g}^{2}) ,$ the integral in (10) represented as K _i is expressible in closed form, because the integrand is the product of two normal densities. Here, $\varvec{\beta }$ and all $g^{\prime }s$ other than g _i enter as fixed parameters, and the latter follows the $N\left( g_{i}|0,\sigma _{g}^{2}\right)$ distribution. Standard integration yields

$$ \begin{aligned} K_{i}&=N\left( {\bf y|X\beta }+\sum\limits_{j\neq i}^{C}{\bf z} _{g_{j}}g_{j},{\bf V}_{i}\right) \\ &=\frac{1}{\left( 2\pi \right)^{\frac{n}{2}}\left\vert {\bf V} _{i}\right\vert ^{\frac{1}{2}}}\exp \left[ -\frac{\left({\bf y-X\beta}-\sum_{j\neq i}^{C}{\bf z}_{g_{j}}g_{j}\right) ^{\prime }{\bf V }_{i}^{-1}\left( {\bf y-X\beta }-\sum_{j\neq i}^{C}{\bf z}_{g_{j}}g_{j}\right)}{2}\right] \end{aligned} $$

(26)

where ${\bf V}_{i}$ is the $n{\times}n$ matrix

$$ {\bf V}_{i}={\bf z}_{g_{i}}{\bf z}_{g_{i}}^{\prime }\sigma _{g}^{2}+ {\bf I}_{n}\sigma _{e}^{2}. $$

It is shown in “Appendix B” that the form of ${\bf V}_{i}$ (after rearrangement of observations, such that the first n _i records are those from individuals with configuration i) is

$$ {\bf V}_{i}=\left[ \begin{array}{cc} {\bf J}_{n_{i}}\sigma _{g}^{2}+{\bf I}_{n_{i}}\sigma _{e}^{2} & {\bf 0} \\ {\bf 0} & {\bf I}_{n-n_{i}}\sigma _{e}^{2} \end{array}\right] , $$

where ${\bf J}_{n_{i}}$ is a matrix of ones of order $n_{i}\times n_{i}$, and n − n _i is the number of individuals with records that have genotypes other than i. Using results in Searle (1971)

$$ \begin{aligned} \left\vert {\bf V}_{i}\right\vert &=\left\vert {\bf J}_{n_{i}}\sigma _{g}^{2}+{\bf I}_{n_{i}}\sigma _{e}^{2}\right\vert \left\vert {\bf I} _{n-n_{i}}\sigma _{e}^{2}\right\vert , \\ \left\vert {\bf J}_{n_{i}}\sigma _{g}^{2}+{\bf I}_{n_{i}}\sigma _{e}^{2}\right\vert &=\sigma _{e}^{2\left( n_{i}-1\right) }\left( \sigma _{e}^{2}+n_{i}\sigma _{g}^{2}\right) , \\ \left\vert {\bf I}_{n-n_{i}}\sigma _{e}^{2}\right\vert &=\sigma _{e}^{2\left( n-n_{i}\right) }, \\ \end{aligned} $$

so that

$$ \left\vert {\bf V}_{i}\right\vert =\sigma _{e}^{2\left( n-1\right) }\left( \sigma _{e}^{2}+n_{i}\sigma _{g}^{2}\right) . $$

(27)

Likewise,

$$ \begin{aligned} {\bf V}_{i}^{-1} &=\left[\begin{array}{cc} {\bf J}_{n_{i}}\sigma _{g}^{2}+{\bf I}_{n_{i}}\sigma _{e}^{2} & {\bf 0}\\ {\bf 0} & {\bf I}_{n-n_{i}}\sigma _{e}^{2} \end{array}\right] ^{-1} \\ &=\left[ \begin{array}{cc}\frac{1}{\sigma _{e}^{2}}{\bf I}_{n_{i}}+\frac{1}{n_{i}}\left( \frac{1}{ \sigma _{e}^{2}+n_{i}\sigma _{g}^{2}}-\frac{1}{\sigma _{e}^{2}}\right) {\bf J}_{n_{i}} & {\bf 0} \\ {\bf 0} & \frac{1}{\sigma _{e}^{2}}{\bf I}_{n-n_{i}} \end{array}\right] . \end{aligned} $$

(28)

Now, with the records arranged such that those of the n _i individuals with configuration i precede data points of the n − n _i individuals having a different genotype, let such rearrangement (indexed by i) lead to

$$ \left( {\bf y-X\beta }-\sum\limits_{j\neq i}^{C}{\bf z} _{g_{j}}g_{j}\right) _{i}=\left[ \begin{array}{c} {\bf z}_{y\in i}\left( {\bf y},\varvec{\beta },{\bf g}\right) \\ {\bf z}_{y\notin i}\left( {\bf y},\varvec{\beta },{\bf g}\right) \end{array}\right] , $$

where ${\bf z}_{y\in i}\left( {\bf y},\varvec{\beta },{\bf g} \right)$ denotes elements of $\left( {\bf y-X\beta }-\sum\limits_{j\neq i}^{C}{\bf z}_{g_{j}}g_{j}\right)$ involving g _i , and ${\bf z} _{y\notin i}\left( {\bf y},\varvec{\beta },{\bf g}\right)$ indicates records in the complement. Using (27) and (28) in (26)

$$ \begin{aligned} K_{i} &=\frac{1}{\left( 2\pi \right)^{\frac{n}{2}}\left[ \sigma _{e}^{2\left( n-1\right) }\left(\sigma _{e}^{2}+n_{i}\sigma _{g}^{2}\right) \right] ^{\frac{1}{2}}} \\ &\times \exp \left[ -\frac{\left( {\bf y-X\beta } -\sum_{j\neq i}^{C}{\bf z}_{g_{j}}g_{j}\right)_{i}^{\prime }{\bf V}_{i}^{-1}\left( {\bf y-X\beta }-\sum_{j\neq i}^{C}{\bf z}_{g_{j}}g_{j}\right) _{i}}{2}\right]\\&=\frac{\exp \left[ -\frac{\left( {\bf y-X\beta }-\sum_{j\neq i}^{C}{\bf z}_{g_{j}}g_{j}\right) ^{\prime }\left( {\bf y-X\beta }- \sum_{j\neq i}^{C}{\bf z}_{g_{j}}g_{j}\right)}{2\sigma _{e}^{2}} \right] }{\left( 2\pi \right)^{\frac{n}{2}}\left[\sigma_{e}^{2\left( n-1\right)}\left( \sigma _{e}^{2}+n_{i}\sigma_{g}^{2}\right) \right] ^{ \frac{1}{2}}}\\ &\times \exp \left[-\frac{\frac{1}{n_{i}}\left( \frac{1}{\sigma _{e}^{2}+n_{i}\sigma_{g}^{2}}-\frac{1}{\sigma _{e}^{2}}\right) \left(\sum\limits_{y\in i,j=1}^{n}z_{y\in i,j}\left( {\bf y},\varvec{\beta },{\bf g}\right) \right)^{2}}{2}\right] ,\end{aligned} $$

(29)

where $z_{y\in i,j}\left( {\bf y},\varvec{\beta },{\bf g}\right)$ is element j of ${\bf z}_{y\in i}\left( {\bf y},\varvec{\beta },{\bf g}\right) .$ The form of K _i in (29) does not involve matrix inverses or matrix computations.

Appendix B: Matrix manipulations

Form of matrix${\bf V}_{i}.$ To see the pattern, suppose that n = 4, C = 5, and that $\varvec{\beta }={\bf 1}_{4}\mu$ (${\bf 1}_{4}$ is a $4{\times}1$ vector of ones), so that model (2) is

$$ \left[ \begin{array}{c} y_{1} \\ y_{2} \\ y_{3} \\ y_{4} \end{array}\right] =\left[ \begin{array}{c} 1 \\ 1 \\ 1 \\ 1 \end{array}\right] \mu +\left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right] \left[ \begin{array}{c} g_{1} \\ g_{2} \\ g_{3} \\ g_{4} \\ g_{5} \end{array}\right] +\left[ \begin{array}{c} e_{1} \\ e_{2} \\ e_{3} \\ e_{4} \end{array}\right] . $$

Consider effect g₁, so that

$$ \begin{aligned} {\bf z}_{g_{1}}&=\left[ \begin{array}{c} 1 \\ 1 \\ 0 \\ 0 \end{array}\right] ,{\bf z}_{g_{1}}{\bf z}_{g_{1}}^{\prime } =&\left[ \begin{array}{ccccc} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] , \quad {\bf V}_{1}=\left[ \begin{array}{cccc} \sigma _{g}^{2}+\sigma _{e}^{2} & \sigma _{g}^{2} & 0 & 0 \\ \sigma _{g}^{2} & \sigma _{g}^{2}+\sigma _{e}^{2} & 0 & 0 \\ 0 & 0 & \sigma _{e}^{2} & 0 \\ 0 & 0 & 0 & \sigma _{e}^{2} \end{array}\right] , \\ {\bf V}_{1}^{-1} &=\left[ \begin{array}{ccc} \left(\begin{array}{cc}\sigma _{g}^{2}+\sigma _{e}^{2} & \sigma _{g}^{2} \\ \sigma _{g}^{2} & \sigma _{g}^{2}+\sigma _{e}^{2} \end{array}\right)^{-1} & {\bf 0} & {\bf 0} \\ {\bf 0} & \frac{1}{\sigma _{e}^{2}} & 0 \\ {\bf 0} & 0 & \frac{1}{\sigma _{e}^{2}} \end{array}\right] . \end{aligned} $$

For g₂

$$ \begin{aligned} {\bf z}_{g_{2}}&=\left[ \begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array}\right] ,{\bf z}_{g_{2}}{\bf z}_{g_{2}}^{\prime } =\left[ \begin{array}{ccccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] , \quad {\bf V}_{2}=\left[ \begin{array}{cccc} \sigma _{e}^{2} & 0 & 0 & 0 \\ 0 & \sigma _{e}^{2} & 0 & 0 \\ 0 & 0 & \sigma _{g}^{2}+\sigma _{e}^{2} & 0 \\ 0 & 0 & 0 & \sigma _{e}^{2} \end{array}\right] , \\ {\bf V}_{2}^{-1} &=\left[ \begin{array}{ccccc} \frac{1}{\sigma _{e}^{2}} & 0 & 0 & 0 \\ 0 & \frac{1}{\sigma _{e}^{2}} & 0 & 0 \\ 0 & 0 & \frac{1}{\sigma _{g}^{2}+\sigma _{e}^{2}} & 0 \\ 0 & 0 & 0 & \frac{1}{\sigma _{e}^{2}} \end{array}\right] . \end{aligned} $$

Likewise for $g_{3}\left( g_{4}\right)$

$$ \begin{aligned} {\bf z}_{g_{3}} &=\left[ \begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}\right] ,{\bf z}_{g_{3}}{\bf z}_{g_{3}}^{\prime }=\left[ \begin{array}{ccccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] , \quad {\bf V}_{3}=\left[ \begin{array}{cccc} \sigma _{e}^{2} & 0 & 0 & 0 \\ 0 & \sigma _{e}^{2} & 0 & 0 \\ 0 & 0 & \sigma _{e}^{2} & 0 \\ 0 & 0 & 0 & \sigma _{e}^{2} \end{array}\right] , \\ {\bf V}_{3}^{-1} &=\left[ \begin{array}{cccc} \frac{1}{\sigma _{e}^{2}}& 0 & 0 & 0 \\ 0 & \frac{1}{\sigma _{e}^{2}} & 0 & 0 \\ 0 & 0 & \frac{1}{\sigma _{e}^{2}} & 0 \\ 0 & 0 & 0 & \frac{1}{\sigma _{e}^{2}} \end{array}\right]. \end{aligned} $$

Finally,

$$ \begin{aligned} {\bf z}_{g_{5}} &=\left[ \begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array}\right] ,{\bf z}_{g_{5}}{\bf z}_{g_{5}}^{\prime }=\left[ \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] , \quad {\bf V}_{5}=\left[ \begin{array}{cccc} \sigma _{e}^{2} & 0 & 0 & 0 \\ 0 & \sigma _{e}^{2} & 0 & 0 \\ 0 & 0 & \sigma _{e}^{2} & 0 \\ 0 & 0 & 0 & \sigma _{g}^{2}+\sigma _{e}^{2} \end{array}\right] , \\ {\bf V}_{5}^{-1} &=\left[\begin{array}{cccc} \frac{1}{\sigma _{e}^{2}} & 0 & 0 & 0 \\ 0 & \frac{1}{\sigma _{e}^{2}} & 0 & 0 \\ 0 & 0 & \frac{1}{\sigma _{e}^{2}} & 0 \\ 0 & 0 & 0 & \frac{1}{\sigma _{g}^{2}+\sigma _{e}^{2}} \end{array}\right]. \end{aligned} $$

The general form of ${\bf V}_{i}$ (after rearrangement of observations, such that the first n _i records are those from individuals with configuration i) is

$$ {\bf V}_{i}=\left[ \begin{array}{cc} {\bf J}_{n_{i}}\sigma _{g}^{2}+{\bf I}_{n_{i}}\sigma _{e}^{2} & {\bf 0} \\ {\bf 0} & {\bf I}_{n-n_{i}}\sigma _{e}^{2} \end{array}\right] , $$

where ${\bf J}_{n_{i}}$ is an $n_{i}{\times}n_{i}$ matrix of ones, and ${\bf I}_{n_{i}}$ and ${\bf I}_{n-n_{i}}$ are identity matrices of order n _i and n − n _i , respectively.

Appendix C: Conditional posterior distribution of M

The normalized density (22) is

$$ \begin{aligned} &p\left( M|x,t,ELSE\right) \\ &\quad =\frac{M^{t+a-1}\exp \left[ -M\left( b-\log x\right) \right] +CM^{t+a-2}\exp \left[ -M\left( b-\log x\right) \right] } { \int_{0}^{\infty }\left\{ M^{t+a-1}\exp \left[ -M\left( b-\log x\right) \right] +CM^{t+a-2}\exp \left[ -M\left( b-\log x\right) \right] \right\} dM}. \end{aligned} $$

The integrals in the denominator yield

$$ \begin{aligned} \int\limits_{0}^{\infty }M^{t+a-1}\exp \left[ -M\left( b-\log x\right) \right] dM &=\frac{\Upgamma \left( t+a\right) } {\left( b-\log x\right) ^{t+a}} \\ \int\limits_{0}^{\infty }CM^{t+a-2}\exp \left[ -M\left( b-\log x\right) \right] dM &= \frac{C\Upgamma \left( t+a-1\right) } {\left( b-\log x\right)^{t+a-1}}. \end{aligned} $$

Letting $a^{{\ast}}=(t+a)$ and $b^{\ast }=\left( b-\log x\right)$ the conditional posterior distribution becomes

$$ \begin{aligned} &p\left( M|x,t,ELSE\right) \\ &\quad =\frac{M^{a^{\ast }-1}\exp \left( -Mb^{\ast }\right) +CM^{a^{\ast }-1-1}\exp \left( -Mb^{\ast }\right) }{\frac{\Upgamma \left( a^{\ast }\right) } {b^{\ast a^{\ast }}}+\frac{C\Upgamma \left( a^{\ast }-1\right) } {b^{\ast a^{\ast }-1}}} \\ &\quad =\frac{\frac{\Upgamma \left( a^{\ast }\right) } {b^{\ast a^{\ast }}}\frac{ b^{\ast a^{\ast }}}{\Upgamma \left( a^{\ast }\right) }M^{a^{\ast }-1}\exp \left( -Mb^{\ast }\right) +C\frac{\Upgamma \left( a^{\ast }-1\right) } {b^{\ast a^{\ast }-1}}\frac{b^{\ast a^{\ast }-1}}{\Upgamma \left( a^{\ast }-1\right) } M^{a^{\ast }-1-1}\exp \left( -Mb^{\ast }\right) } {\frac{\Upgamma \left( a^{\ast }\right) } {b^{\ast a^{\ast }}}+\frac{C\Upgamma \left( a^{\ast }-1\right) } {b^{\ast a^{\ast }-1}}} \\ &\quad =\pi _{x}Gamma\left( a^{\ast },b^{\ast }\right) +\left( 1-\pi _{x}\right) Gamma\left( a^{\ast }-1,b^{\ast }\right) . \end{aligned} $$

(30)

This is a mixture of Gamma distributions indicated, with mixing probabilities ${\pi}_{x}$ and $1-{\pi}_{x}.$ Note that

$$ \pi _{x}=\frac{\frac{\Upgamma \left( a^{\ast }\right) }{b^{\ast a^{\ast }}}} { \frac{\Upgamma \left( a^{\ast }\right) } {b^{\ast a^{\ast }}}+\frac{C\Upgamma \left( a^{\ast }-1\right) } {b^{\ast a^{\ast }-1}}}=\frac{\Upgamma \left( a^{\ast }\right) } {\Upgamma \left( a^{\ast }\right) +Cb^{\ast }\Upgamma \left( a^{\ast }-1\right) }. $$

Since $\Upgamma \left( a^{\ast }\right) =\left( a^{\ast }-1\right) \Upgamma \left( a^{\ast }-1\right) ,$

$$ \pi _{x}=\frac{a^{\ast }-1} {a^{\ast }-1+Cb^{\ast }} $$

$$ 1-\pi _{x}=\frac{Cb^{\ast }}{a^{\ast }-1+Cb^{\ast }}. $$

At the end of MCMC sampling there will be S, say, samples of the number of clusters t and of the auxiliary variables x. The density of the marginal posterior distribution of M can be estimated (West 1992) using the Rao-Blackwell estimator

$$ \begin{aligned} & p\left( M|ELSE\right)\\ &\quad =\frac{1} {S}\sum\limits_{s=1}^{S}p\left( M|x^{\left( s\right) },t^{\left( s\right) },ELSE\right) \\ &\quad =\frac{1} {S}\sum\limits_{s=1}^{S}\left[ \pi _{x^{\left( s\right) }}Gamma\left( a^{\ast \left( s\right) },b^{\ast \left( s\right) }\right) +\left( 1-\pi _{x^{\left( s\right) }}\right) Gamma\left( a^{\ast \left( s\right) }-1,b^{\ast \left( s\right) }\right) \right]. \end{aligned} $$

(31)

where $a^{\ast \left( s\right) }=t^{\left( s\right) }+a,$ and $b^{\ast \left( s\right) }=b-\log x^{\left( s\right) }$. If a = 0 and b = 0, the Gamma prior degenerates to $p(M) \varpropto M^{-1}$ or, equivalently, to $p\left[ \log \left( M\right) \right] \varpropto$ constant. If such improper prior is adopted, the Rao-Blackwell estimator reduces to

$$ \begin{aligned} p&\left( M|ELSE\right) \\ &\quad =\frac{1}{S}\sum\limits_{s=1}^{S}\left[ \pi _{x^{(s) }}Gamma\left( t^{(s)},-\log x^{(s)}\right) +\left( 1-\pi _{x^{(s)}}\right) Gamma\left( t^{\left( s\right) }-1,-\log x^{\left( s\right) }\right) \right] , \end{aligned} $$

with a* = t and b* =  −logx used in the calculation of $\pi _{x^{\left( s\right) }}.$ The conditional posterior distribution of M is well defined for a = b = 0, but this does not guarantee that its marginal posterior distribution will be always proper. Hence, the uniform prior on ${\log}(M)$ should be used cautiously.