# Multiplicative mixed models for analyzing variety-by-environment data

I got some nice data on multi-environment trials (MET) data for genotype evaluation and would like to use some new developed techniques as discussed in Smith et al. 2005. I'm specifically interested in Factor Analytic (FA) structure. Authors mentioned codes for these methods will be available on request.

There are now several statistical packages (including ASReml, GENSTAT, S-language packages and SAS; Littel et al. 1996) that allow REML estimation of a range of mixed models. The present authors have found the packages ASReml and GENSTAT and the samm functions (through S-language environments) to be the most suitable for the analysis of MET data, both in terms of the generality of models that can be fitted and the ease with which predictions and inference about varietal effects can be made. All models in the current paper are easily fitted and summarized using these software (code is available from the authors on request).

Even after multiple requests to the authors I haven't heard back from the authors. I'm wondering if someone has tried these models and be kind enough to share worked examples, that would be great. I'm looking forward for some positive response.

Smith et al. (2001) used the following mixed model version of multiplicative models

${\small \begin{eqnarray*} (\boldsymbol{I}_{e}\otimes\boldsymbol{I}_{g})\boldsymbol{\eta}_{eg\times1} & = & (\boldsymbol{1}_{e}\otimes\boldsymbol{1}_{g})\mu+(\boldsymbol{I}_{e}\otimes\boldsymbol{1}_{g})\boldsymbol{E}_{e\times1}+(\boldsymbol{1}_{e}\otimes\boldsymbol{I}_{g})\boldsymbol{G}_{g\times1}+\underbrace{(\boldsymbol{I}_{e}\otimes\boldsymbol{I}_{g})(\boldsymbol{GE})_{eg\times1}}\\ & = & (\boldsymbol{1}_{e}\otimes\boldsymbol{1}_{g})\mu+(\boldsymbol{I}_{e}\otimes\boldsymbol{1}_{g})\boldsymbol{E}_{e\times1}+(\boldsymbol{1}_{e}\otimes\boldsymbol{I}_{g})\boldsymbol{G}_{g\times1}\\ & & +\underbrace{(\boldsymbol{\Lambda}_{E_{e\times k}}\otimes\boldsymbol{I}_{g})\:\boldsymbol{f}_{G_{kg\times1}}+(\boldsymbol{I}_{e}\otimes\boldsymbol{I}_{g})\delta_{eg\times1}}\end{eqnarray*} }{\tiny }$

where $\boldsymbol{\Lambda}_{E_{e\times k}}$ is a matrix of environment loadings, $f_{G_{kg\times1}}$ is the associated vector of genotype scores and k is the number of components (multiplicative terms) included in the model.

The authors assumed that the environments are fixed and $\boldsymbol{G}_{g\times1}$, $\boldsymbol{f}_{G_{kg\times1}}$, and $\delta_{eg\times1}$ are random effects with

$\small \left(\begin{array}{l} \boldsymbol{G}_{g\times1}\\ \boldsymbol{f}_{G_{kg\times1}}\\ \boldsymbol{\delta}_{eg\times1}\end{array}\right)\sim\mathcal{N}\left(\begin{array}{ccc} \left[\begin{array}{c} \boldsymbol{0}\\ \boldsymbol{0}\\ \boldsymbol{0}\end{array}\right] & , & \left[\begin{array}{ccc} \sigma_{g}^{2}\,\boldsymbol{I}_{g} & \boldsymbol{0} & \boldsymbol{0}\\ \boldsymbol{0} & \boldsymbol{I}_{k}\otimes\boldsymbol{I}_{g} & \boldsymbol{0}\\ \boldsymbol{0} & \boldsymbol{0} & \boldsymbol{\Psi}_{e}\otimes\boldsymbol{I}_{g}\end{array}\right]\end{array}\right)$

$\small E((\boldsymbol{I}_{e}\otimes\boldsymbol{I}_{g})\boldsymbol{\eta}_{eg\times1})=(\boldsymbol{1}_{e}\otimes\boldsymbol{1}_{g})\mu+(\boldsymbol{I}_{e}\otimes\boldsymbol{1}_{g})\boldsymbol{E}_{e\times1}$

${\small \begin{eqnarray*} \mathrm{var}((\boldsymbol{I}_{e}\otimes\boldsymbol{I}_{g})\boldsymbol{\eta}_{eg\times1}) & = & \sigma_{g}^{2}\,(\boldsymbol{J}_{e}\otimes\boldsymbol{I}_{g})+(\boldsymbol{\Lambda}_{E_{e\times k}}\otimes\boldsymbol{I}_{g})(\boldsymbol{I}_{k}\otimes\boldsymbol{I}_{g})(\boldsymbol{\Lambda}_{E_{e\times k}}\otimes\boldsymbol{I}_{g})^{T}+\boldsymbol{\Psi}_{e}\otimes\boldsymbol{I}_{g}\\ & = & \sigma_{g}^{2}\,(\boldsymbol{J}_{e}\otimes\boldsymbol{I}_{g})+(\boldsymbol{\Lambda}_{E_{e\times k}}\boldsymbol{\Lambda}_{E_{e\times k}}^{T}\otimes\boldsymbol{I}_{g})+\boldsymbol{\Psi}_{e}\otimes\boldsymbol{I}_{g}\\ & = & \sigma_{g}^{2}\,(\boldsymbol{J}_{e}\otimes\boldsymbol{I}_{g})+(\boldsymbol{\Lambda}_{E_{e\times k}}\boldsymbol{\Lambda}_{E_{e\times k}}^{T}+\boldsymbol{\Psi}_{e})\otimes\boldsymbol{I}_{g}\\ & = & (\underbrace{\sigma_{g}^{2}\,\boldsymbol{J}_{e}+\boldsymbol{\Lambda}_{E_{e\times k}}\boldsymbol{\Lambda}_{E_{e\times k}}^{T}}+\boldsymbol{\Psi}_{e})\otimes\boldsymbol{I}_{g}\\ & = & (\underbrace{\boldsymbol{\Lambda}_{E}^{*}\boldsymbol{\Lambda}_{E}^{*T}}+\boldsymbol{\Psi}_{e})\otimes\boldsymbol{I}_{g}\end{eqnarray*} }$

where $\boldsymbol{\Psi}_{e}$ is a diagonal $e\times e$ matrix with elements commonly referred to as specific variances.

• Could you describeyour data and/or the model you want to fit in a little more detail? Your reference seems to be behind a paywall. From your quote it seems you're after mixed effect models; if so then there are a multitude of R packages which do so as there are many models which fall under this umbrella. – JMS Apr 21 '11 at 23:34
• @JMS: Thanks for your comment. In multienvironment trials, we have a matrix genotype-by-environment interaction effects and implement singular value decomposition (SVD), equivalent to PCA, on this matrix assuming both genotypes and environments fixed. Most often one or both of genotypes and environments are random results in the use of Factor Analytic (FA) structure. Please have a look on the eqs (12) and (13) on page 453 in the given reference. I'd highly appreciate if you help me on this. Thanks – MYaseen208 Apr 22 '11 at 8:03
• Like I said, your reference is behind a paywall. If you reproduce the equations here maybe we can help; again, packages exist for doing factor analysis and PCA (eg, factanal). – JMS Apr 22 '11 at 14:46
• Hmm, it looks to me like you ought to be able to fit this with one of the mixed effects packages in R (for example) but I'm afraid I haven't done anything similar. Are there constraints on the $\Lambda$'s? (you mentioned PCA & the SVD so I'm wondering if they're supposed to be orthogonal) – JMS Apr 30 '11 at 15:47
• @JMS: If you like I can send the said article for your consideration. Thanks – MYaseen208 Apr 30 '11 at 16:40