I got some nice data on multi-environmet trials (MET) data for genotype evaluation and would like to use some new developed techniques as discussed in Smith et. al 2005 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. (2001b) 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.
Thanks
