Genome-based prediction of testcross values in maize

Abstract

This is the first large-scale experimental study on genome-based prediction of testcross values in an advanced cycle breeding population of maize. The study comprised testcross progenies of 1,380 doubled haploid lines of maize derived from 36 crosses and phenotyped for grain yield and grain dry matter content in seven locations. The lines were genotyped with 1,152 single nucleotide polymorphism markers. Pedigree data were available for three generations. We used best linear unbiased prediction and stratified cross-validation to evaluate the performance of prediction models differing in the modeling of relatedness between inbred lines and in the calculation of genome-based coefficients of similarity. The choice of similarity coefficient did not affect prediction accuracies. Models including genomic information yielded significantly higher prediction accuracies than the model based on pedigree information alone. Average prediction accuracies based on genomic data were high even for a complex trait like grain yield (0.72–0.74) when the cross-validation scheme allowed for a high degree of relatedness between the estimation and the test set. When predictions were performed across distantly related families, prediction accuracies decreased significantly (0.47–0.48). Prediction accuracies decreased with decreasing sample size but were still high when the population size was halved (0.67–0.69). The results from this study are encouraging with respect to genome-based prediction of the genetic value of untested lines in advanced cycle breeding populations and the implementation of genomic selection in the breeding process.

Appendix

In this Appendix we show the functional dependency of the genomic variance components pertaining to Model 2, Model SM and the random regression model suggested by Meuwissen et al. (2001) denoted Model RR.

All three models can be seen as special cases of a more general model

$$ {\mathbf{y = X{\varvec{\upbeta}} + (W - Q)m + e}} $$

where ${\varvec{\upbeta}}$ is a vector of fixed effects and X is a design matrix assigning fixed effects to the phenotypes, W is an N × M design matrix assigning M SNP marker genotypes coded 0 or 2 to N phenotypes. Random marker effects in vector m are assumed to follow a normal distribution with ${\mathbf {m}} \sim N(0,{\mathbf {I}}\sigma_{m}^{2} )$, where I is an identity matrix and $\sigma_{m}^{2}$ denotes the proportion of the testcross variance contributed by each individual SNP marker. Q is an N × M matrix composed of M uniform column vectors ${\mathbf{Q}} = \left\{ {q_{1} {\bf c},q_{2} {\bf c}, \cdots ,q_{M} {\bf c}} \right\}$ where $q_{m}$ is a scalar correction term for marker m and ${\mathbf {c}}$ is a column vector of length N containing only 1s.

The estimated SNP effects and the corresponding variance component $\sigma_{m}^{2}$ in the general model are unaffected by the choice of Q, since subtracting the correction terms shifts the intercept, but does not change the slope of the regression of the phenotype on each SNP (Habier et al. 2007; Piepho 2009). For Model RR, Q is the null matrix, but the same variance components as estimated from Model RR will be received with any other Q.

For Model 2 the kinship between lines i and j is modeled by the matrix U calculated as

$$ {\mathbf{U}} = {\frac{{({\mathbf{W}} - {\mathbf{P}})({\mathbf{W}} - {\mathbf{P}})'}}{{8\sum\nolimits_{m = 1}^{M} {p_{m} (1 - p_{m} )} }}} $$

and the variance–covariance matrix of the phenotype vector y can be written as

$$ {\mathbf{V}}_{\text{Model2}} = {\frac{{({\mathbf{W}} - {\mathbf{P}})({\mathbf{W}} - {\mathbf{P}})'}}{{8\sum\nolimits_{m = 1}^{M} {p_{m} (1 - p_{m} )} }}}\sigma_{u}^{2} + {\mathbf{I}}\sigma_{{}}^{2} $$

Using ${\mathbf{Q}} = {\mathbf{P}}$ in the generalized model, the variance–covariance matrix of the phenotype vector y is

$$ {\mathbf{V}}_{{\rm P}} = ({\mathbf{W}} - {\mathbf{P}})({\mathbf{W}} - {\mathbf{P}})'\sigma_{m}^{2} + {\mathbf{I}}\sigma_{{}}^{2} $$

Hence, ${\mathbf{V}}_{\text{Model2}} = {\mathbf{V}}_{P}$ if

$$ \sigma_{u}^{2} = \sigma_{m}^{2} \times 8\sum\nolimits_{m = 1}^{M} {p_{m} (1 - p_{m} )} $$

In Model SM, the matrix S can be written as

$$ {\mathbf{S}} = {\frac{{({\mathbf{W}} - {\mathbf{J}}_{N \times M} )({\mathbf{W}} - {\mathbf{J}}_{N \times M} )'}}{{2M(1 - s_{\min } )}}} + {\frac{{M - 2Ms_{\min } }}{{2M - 2Ms_{\min } }}}{\mathbf{J}}_{N \times N} $$

The second term reflects a constant, which is added to all elements of the matrix. This is equivalent to a constant random block effect and thus fully confounded with the fixed intercept (Piepho et al. 2008; Williams et al. 2006). Hence, ignoring the second term will not affect the estimated variance components.

The numerator of the first term of S is a special case of the numerator of the generalized model with ${\mathbf{Q}} = {\mathbf{J}}_{N \times M}$, i.e. assuming $q_{m} = 1$ for all loci, which is equivalent to the assumption of allele frequency $p_{m} = 0.5$ for all SNPs in Model 2. Using the same argument as above,

$$ {\mathbf{V}}_{\text{SM}} = {\frac{{({\mathbf{W}} - {\mathbf{J}}_{N \times M} )({\mathbf{W}} - {\mathbf{J}}_{N \times M} )'}}{{2M(1 - s_{\min } )}}}\sigma_{s}^{2} + {\mathbf{I}}\sigma^{2} $$

For the special case ${\mathbf{Q}} = {\mathbf{J}}_{N \times M}$ in the general model, the variance–covariance matrix of the phenotype vector y is

$$ {\mathbf{V}}_{J} = ({\mathbf{W}} - {\mathbf{J}}_{N \times M} )({\mathbf{W}} - {\mathbf{J}}_{N \times M} )'\sigma_{m}^{2} + {\mathbf{I}}\sigma^{2} $$

and ${\mathbf{V}}_{\text{SM}} = {\mathbf{V}}_{J}$ if

$$ \sigma_{s}^{2} = \sigma_{m}^{2} \times 2M(1 - s_{\min } ). $$