Multivariate whole genome average interval mapping: QTL analysis for multiple traits and/or environments

Abstract

A major aim in some plant-based studies is the determination of quantitative trait loci (QTL) for multiple traits or across multiple environments. Understanding these QTL by trait or QTL by environment interactions can be of great value to the plant breeder. A whole genome approach for the analysis of QTL is presented for such multivariate applications. The approach is an extension of whole genome average interval mapping in which all intervals on a linkage map are included in the analysis simultaneously. A random effects working model is proposed for the multivariate (trait or environment) QTL effects for each interval, with a variance–covariance matrix linking the variates in a particular interval. The significance of the variance–covariance matrix for the QTL effects is tested and if significant, an outlier detection technique is used to select a putative QTL. This QTL by variate interaction is transferred to the fixed effects. The process is repeated until the variance–covariance matrix for QTL random effects is not significant; at this point all putative QTL have been selected. Unlinked markers can also be included in the analysis. A simulation study was conducted to examine the performance of the approach and demonstrated the multivariate approach results in increased power for detecting QTL in comparison to univariate methods. The approach is illustrated for data arising from experiments involving two doubled haploid populations. The first involves analysis of two wheat traits, α-amylase activity and height, while the second is concerned with a multi-environment trial for extensibility of flour dough. The method provides an approach for multi-trait and multi-environment QTL analysis in the presence of non-genetic sources of variation.

Appendix: The score statistic under the alternative outlier model

The full (marginal) model for ${\mathbf y}$ under the AOM (9) can be written as

$${\mathbf y} \sim N({\mathbf X} {\mathbf \tau}, \;{\mathbf H})$$

(18)

where if ${\mathbf M}_{E,k} = {\mathbf M}_E {\mathbf D}_k,$ the selected columns of ${\mathbf M}_E$ corresponding to the kth chromosome, the variance matrix ${\mathbf H}$ is given by

$${\mathbf H} = {\mathbf R} + {\mathbf Z}_0 {\mathbf G}_0 {\mathbf Z}_0^T + {\mathbf Z} \left\lbrace ({\mathbf L}_a {\mathbf L}_a^T) \otimes {\mathbf M}_E {\mathbf M}_E^T \right\rbrace {\mathbf Z}^T + {\mathbf Z} \left\lbrace (\sigma_{ak}^2 {\mathbf L}_a {\mathbf L}_a^T) \otimes {\mathbf M}_{E,k} {\mathbf M}_{E,k}^T \right\rbrace {\mathbf Z}^T$$

As in Verbyla et al. (2007), an outlier statistic is developed using the score for σ ² _ak under the null hypothesis $H_0:\; \sigma_{ak}^2 = 0.$ If ${\mathbf P} = {\mathbf H}^{-1} - {\mathbf H}^{-1}{\mathbf X} ({\mathbf X}^T{\mathbf H}^{-1}{\mathbf X})^{-1}{\mathbf X}^T{\mathbf H}^{-1},$ the REML score for σ ² _ak evaluated at zero is

$$\begin{aligned} U_{k}(0) =& -\frac{1}{2} \left\lbrace {\rm tr}\left({{\mathbf P}{\mathbf Z} ({\mathbf L}_a {\mathbf L}_a^T \otimes {\mathbf M}_{E,k}{\mathbf M}_{E,k}^T){\mathbf Z}^T}\right) - {\mathbf y}^T {\mathbf P}{\mathbf Z} ({\mathbf L}_a {\mathbf L}_a^T \otimes {\mathbf M}_{E,k}{\mathbf M}_{E,k}^T){\mathbf Z}^T{\mathbf P}{\mathbf y} \right\rbrace \\ =& -\frac{1}{2} \left\lbrace {\rm tr}\left({{\mathbf P}{\mathbf Z} ({\mathbf G}_a \otimes {\mathbf M}_{E,k}{\mathbf M}_{E,k}^T){\mathbf Z}^T}\right) - {\mathbf y}^T {\mathbf P}{\mathbf Z} ({\mathbf G}_a \otimes {\mathbf M}_{E,k}{\mathbf M}_{E,k}^T){\mathbf Z}^T{\mathbf P}{\mathbf y} \right\rbrace \end{aligned}$$

(19)

Let ${\mathbf a}_k$ be the vector of sizes for all variates for all intervals on chromosome k, with individual intervals having sizes given by ${\mathbf a}_{kl}.$ Then the BLUP for ${\mathbf a}_k$ is

$$\tilde{{\mathbf a}}_k = ({\mathbf G}_a \otimes {\mathbf M}_{E,k})^T{\mathbf Z}^T{\mathbf P}{\mathbf y}$$

(20)

with variance

$$\hbox{var}\left({\tilde{{\mathbf a}}_k}\right) = ({\mathbf G}_a \otimes {\mathbf M}_{E,k})^T{\mathbf Z}^T{\mathbf P} {\mathbf Z}({\mathbf G}_a \otimes {\mathbf M}_{E,k})$$

(21)

It may be that ${\mathbf G}_a$ is non-negative definite, rather than positive definite. Thus ${\mathbf G}_a$ may be singular. If ${\mathbf G}_a^-$ is a generalized inverse of ${\mathbf G}_a,$ ${\mathbf G}_a$ can be replaced in (19) by ${\mathbf G}_a{\mathbf G}_a^-{\mathbf G}_a,$ and hence using (20), (21) and properties of the trace, (19) can be written as

$$\begin{aligned} U_k(0) =& -\frac{1}{2} \left[ {\rm tr}\left({({\mathbf G}_a^{-} \otimes {\mathbf I}_{r_k-1}) \hbox{var}\left({\tilde{{\mathbf a}}_k}\right)}\right) - \tilde{{\mathbf a}}_k^T ({\mathbf G}_a^{-} \otimes {\mathbf I}_{r_k-1}) \tilde{{\mathbf a}}_k \right] \\ =& -\frac{1}{2} \sum_{l=1}^{r_k-1} \left[ {\rm tr}\left({{\mathbf G}_a^{-} \hbox{var}\left({\tilde{{\mathbf a}}_{kl}}\right)}\right) - \tilde{{\mathbf a}}_{kl}^T {\mathbf G}_a^{-} \tilde{{\mathbf a}}_{kl}\right]\\ =& \frac{1}{2} \left( \sum_{l=1}^{r_k-1} {\rm tr}\left({{\mathbf G}_a^{-} \hbox{var}\left({\tilde{{\mathbf a}}_{kl}}\right)}\right) \right) (t_k^2 - 1) \end{aligned}$$

where t ² _k is given by (10). Thus t ² _k indicates the departure from U _k (0) = 0. This statistic therefore provides evidence that σ ² _ak departs from zero for chromosome k. The chromosome most likely to contain a QTL is the one with largest t ² _k .

The statistic t ² _k is made up of components that relate to the intervals on chromosome k. Hence using a similar argument, the outlier statistics for individual intervals is given by t ² _kl in (11).

A fully parameterized ${\mathbf G}_a$ requires many parameters for larger multivariate problems and an approximation becomes both sensible and necessary. It is possible to use a factor analytic approximation for the full covariance model ${\mathbf G}_a$ which mirrors the use of FA models in the analysis of multi-environment trials.