On a mixed model analysis of multi-environment variety trials: a reconsideration of the one-stage and the two-stage models and analyses

Abstract

Of interest is the analysis of results of a series of experiments conducted at several environments with the same set of plant varieties (called multi-environment variety trials). The most common practice is first to analyze individual trials and then to perform a kind of “synthesis” of the results obtained. This is considered as a two-stage approach to the analysis of the trial data. More recently a combined analysis of the raw plot data from all trials taken simultaneously has been advocated, as a one-stage approach to the analysis. The purpose of this article is to reconsider these two approaches with regard to the underlying models and the analyses based on them. The indicated differences between them are illustrated by a thorough analysis of a set of data from a series of trials with rye varieties. The required computations have been accomplished with the use of R.

Appendices

Appendix 1

When solving, for an individual trial (jth experiment), the Eq. (3.8), $\alpha = 1,\,2,$ it may be helpful to utilize the general balance (GB) property of the considered design (see Nelder 1968; also Caliński and Kageyama 2000, Sects. 3.6, 5.4).

Due to the GB property, the matrices ${{\varvec{C}}}_{1(j)} = {{\varvec{X}}}_{j}^{\prime }{{\varvec{\phi }}}_{1(j)}{{\varvec{X}}}_{j}$ and ${{\varvec{C}}}_{2(j)} = {{\varvec{X}}}_{j}^{\prime }{{\varvec{\phi }}}_{2(j)}{{\varvec{X}}}_{j}$ can be presented in the following spectral decomposition forms (though not formally specified, the realization of these and the following formulae depends on the design of the analyzed trial):

$$\begin{aligned} {{\varvec{C}}}_{1(j)} = a^{2}\sum \limits _{\beta =0}^{f-1}\varepsilon _{1 \beta }{{\varvec{S}}}_{\beta }{{\varvec{S}}}_{\beta }^{\prime } = a\sum \limits _{\beta =0}^{f-1}\varepsilon _{1 \beta }{{\varvec{L}}}_{\beta }, \quad {{\varvec{C}}}_{2(j)} = a^{2}\sum \limits _{\beta =1}^{f-1}\varepsilon _{2 \beta }{{\varvec{S}}}_{\beta }{{\varvec{S}}}_{\beta }^{\prime } = a\sum \limits _{\beta =0}^{f-1}\varepsilon _{2 \beta }{{\varvec{L}}}_{\beta }, \end{aligned}$$

where ${{\varvec{S}}}_{\beta }{{\varvec{S}}}_{\beta }^{\prime } = \sum \nolimits _{\ell =1}^{\rho _{\beta }}{{\varvec{s}}}_{\beta \ell }{{\varvec{s}}}_{\beta \ell }^{\prime }$ and ${{\varvec{L}}}_{\beta } = a{{\varvec{S}}}_{\beta }{{\varvec{S}}}_{\beta }^{\prime },$ chosen so that ${{\varvec{C}}}_{\alpha (j)}{{\varvec{S}}}_{\beta } = a\varepsilon _{\alpha \beta }{{\varvec{S}}}_{\beta },$ with ${{\varvec{S}}}_{\beta } = [{{\varvec{s}}}_{\beta 1}{:} {{\varvec{s}}}_{\beta 2}{:} \cdots {:} {{\varvec{s}}}_{\beta \rho _{\beta }}],$ for $\alpha = 1,\,2$ and $\beta = 0,\,1,\ldots , f - 1,$ i.e., ${{\varvec{C}}}_{\alpha (j)}{{\varvec{s}}}_{\beta \ell } = a\varepsilon _{\alpha \beta }{{\varvec{s}}}_{\beta \ell }$ for $\ell = 1,\,2,\ldots , \rho _{\beta }$ and $\alpha = 1,\,2,$ with $\varepsilon _{10} = 1$ and, hence, $\varepsilon _{20} = 0.$ Furthermore, the eigenvectors ${{\varvec{s}}}_{\beta 1},\,{{\varvec{s}}}_{\beta 2},\ldots , {{\varvec{s}}}_{\beta \rho _{\beta }}$ are orthonormalized in such a way that $a{{\varvec{S}}}_{\beta }^{\prime }{{\varvec{S}}}_{\beta } = {{\varvec{I}}}_{\rho _{\beta }}$ for any $\beta$ and $a{{\varvec{S}}}_{\beta }^{\prime }{{\varvec{S}}}_{\beta ^{\prime }} = \mathbf{0}$ for $\beta \ne \beta ^{\prime }.$ With this notation, the following equalities can be presented:

$$\begin{aligned} \left\| {{\varvec{\phi }}}_{1(j)}\left( {{\varvec{I}}}_{am} - {{\varvec{P}}}_{{{\varvec{}}X}_{j}({{\varvec{}}\tilde{V}}_{(j)}^{-1})}\right) {{\varvec{y}}}_{j}\right\| ^{2}= & {} {{\varvec{y}}}_{j}^{\prime }{{\varvec{\psi }}}_{1(j)}{{\varvec{y}}}_{j} + \sum \limits _{\beta =1}^{f-1}\varepsilon _{1 \beta }w_{2 \beta }^{2}\left( \varepsilon _{1 \beta }^{-1}{{\varvec{Q}}}_{1(j)}^{\prime }\right. \\&\left. - \varepsilon _{2 \beta }^{-1}{{\varvec{Q}}}_{2(j)}^{\prime }\right) {{\varvec{S}}}_{\beta }{{\varvec{S}}}_{\beta }^{\prime }\left( \varepsilon _{1 \beta }^{-1}{{\varvec{Q}}}_{1(j)} - \varepsilon _{2 \beta }^{-1}{{\varvec{Q}}}_{2(j)}\right) , \\ \left\| {{\varvec{\phi }}}_{2(j)}\left( {{\varvec{I}}}_{am} - {{\varvec{P}}}_{{{\varvec{}}X}_{j}({{\varvec{}}\tilde{V}}_{(j)}^{-1})}\right) {{\varvec{y}}}_{j}\right\| ^{2}= & {} {{\varvec{y}}}_{j}^{\prime }{{\varvec{\psi }}}_{2(j)}{{\varvec{y}}}_{j} + \sum \limits _{\beta =1}^{f-1}\varepsilon _{2 \beta }w_{1 \beta }^{2}\left( \varepsilon _{1 \beta }^{-1}{{\varvec{Q}}}_{1(j)}^{\prime }\right. \\&\left. - \varepsilon _{2 \beta }^{-1}{{\varvec{Q}}}_{2(j)}^{\prime }\right) {{\varvec{S}}}_{\beta }{{\varvec{S}}}_{\beta }^{\prime }\left( \varepsilon _{1 \beta }^{-1}{{\varvec{Q}}}_{1(j)} - \varepsilon _{2 \beta }^{-1}{{\varvec{Q}}}_{2(j)}\right) , \end{aligned}$$

where ${{\varvec{\psi }}}_{\alpha (j)} = {{\varvec{\phi }}}_{\alpha (j)} - {{\varvec{\phi }}}_{\alpha (j)}{{\varvec{X}}}_{j}{{\varvec{C}}}_{\alpha (j)}^{-}{{\varvec{X}}}_{j}^{\prime }{{\varvec{\phi }}}_{\alpha (j)} = {{\varvec{\phi }}}_{\alpha (j)}({{\varvec{I}}}_{am} - {{\varvec{X}}}_{j}{{\varvec{C}}}_{\alpha (j)}^{-}{{\varvec{X}}}_{j}^{\prime }){{\varvec{\phi }}}_{\alpha (j)}$ for $\alpha = 1,\,2,\,{{\varvec{Q}}}_{1(j)} = {{\varvec{X}}}_{j}^{\prime }{{\varvec{\phi }}}_{1(j)}{{\varvec{y}}}_{j}$ and ${{\varvec{Q}}}_{2(j)} = {{\varvec{X}}}_{j}^{\prime }{{\varvec{\phi }}}_{2(j)}{{\varvec{y}}}_{j},$ and where the weights $w_{1 \beta },\,w_{2 \beta }$ are defined as

$$\begin{aligned} w_{1 \beta } = \frac{\varepsilon _{1 \beta }\sigma _{2(j)}^{2}}{\varepsilon _{1 \beta }\sigma _{2(j)}^{2} + \varepsilon _{2 \beta }\sigma _{1(j)}^{2}}, \quad w_{2 \beta } = \frac{\varepsilon _{2 \beta }\sigma _{1(j)}^{2}}{\varepsilon _{1 \beta }\sigma _{2(j)}^{2} + \varepsilon _{2 \beta }\sigma _{1(j)}^{2}}. \end{aligned}$$

On the other hand, it can be shown, for the considered GL designs, that

$$\begin{aligned} {tr}\left\{ {{\varvec{\phi }}}_{1(j)}\left( {{\varvec{I}}}_{am} - {{\varvec{P}}}_{{{\varvec{}}X}_{j}({{\varvec{}}\tilde{V}}_{(j)}^{-1})}\right) \right\}= & {} am - b - m + 1 + \sum \limits _{\beta =0}^{f-1}\left( 1 - w_{1 \beta }\right) \rho _{\beta }, \\ {tr}\left\{ {{\varvec{\phi }}}_{2(j)}\left( {{\varvec{I}}}_{am} - {{\varvec{P}}}_{{{\varvec{}}X}_{j}({{\varvec{}}\tilde{V}}_{(j)}^{-1})}\right) \right\}= & {} b - a - m + 1 + \rho _{0} + \sum \limits _{\beta =1}^{f-1}\left( 1 - w_{2 \beta }\right) \rho _{\beta }. \end{aligned}$$

Now, it is interesting to note that in the special case of an affine resolvable design the Eq. (3.8), used for estimating the variances $\sigma _{1(j)}^{2}$ and $\sigma _{2(j)}^{2},$ reduce to the following forms:

$$\begin{aligned}&{{\varvec{y}}}_{j}^{\prime }{{\varvec{\psi }}}_{1(j)}{{\varvec{y}}}_{j} + a^{-1}\varepsilon _{11}w_{21}^{2}\left( \varepsilon _{11}^{-1}{{\varvec{Q}}}_{1(j)}^{\prime } - \varepsilon _{21}^{-1}{{\varvec{Q}}}_{2(j)}^{\prime }\right) {{\varvec{L}}}_{1}\left( \varepsilon _{11}^{-1}{{\varvec{Q}}}_{1(j)} - \varepsilon _{21}^{-1}{{\varvec{Q}}}_{2(j)}\right) \\&\quad =\sigma _{1(j)}^{2}\left( d_{1} + \ w_{21}\rho _{1}\right) , \\&a^{-1}\varepsilon _{21}w_{11}^{2}\left( \varepsilon _{11}^{-1}{{\varvec{Q}}}_{1(j)}^{\prime } - \varepsilon _{21}^{-1}{{\varvec{Q}}}_{2(j)}^{\prime }\right) {{\varvec{L}}}_{1}\left( \varepsilon _{11}^{-1}{{\varvec{Q}}}_{1(j)} - \varepsilon _{21}^{-1}{{\varvec{Q}}}_{2(j)}\right) \\&\quad = \sigma _{2(j)}^{2}w_{11}\rho _{1}, \end{aligned}$$

where $\varepsilon _{11} = (a - 1)/a,\,\varepsilon _{21} = 1/a,\, {{\varvec{L}}}_{1} = a{{\varvec{S}}}_{1}{{\varvec{S}}}_{1}^{\prime } = {{\varvec{C}}}_{2(j)},$ of the rank $\rho _{1} = b - a,$ and $d_{1} = am - b - m + 1.$ Note that in this case ${{\varvec{\psi }}}_{2(j)} = {{\varvec{0}}}.$ A desirable advantage of these equations is that they need not to be solved by an iterative procedure, as it is required in the general case. In fact, their solutions are

$$\begin{aligned} \hat{\sigma }_{1(j)}^{2}= & {} d_{1}^{-1}{{\varvec{y}}}_{j}^{\prime }{{\varvec{\psi }}}_{1(j)}{{\varvec{y}}}_{j}, \\ \hat{\sigma }_{2(j)}^{2}= & {} \frac{1}{b - a}\biggl [\frac{\varepsilon _{21}}{a}\left( \varepsilon _{11}^{-1}{{\varvec{Q}}}_{1(j)}^{\prime }- \varepsilon _{21}^{-1}{{\varvec{Q}}}_{2(j)}^{\prime }\right) {{\varvec{L}}}_{1}\left( \varepsilon _{11}^{-1}{{\varvec{Q}}}_{1(j)} - \varepsilon _{21}^{-1}{{\varvec{Q}}}_{2(j)}\right) \\&-\,\frac{\varepsilon _{21}}{\varepsilon _{11}} \ \frac{b - a}{d_{1}}{{\varvec{y}}}_{j}^{\prime }{{\varvec{\psi }}}_{1(j)}{{\varvec{y}}}_{j}\biggl ]. \end{aligned}$$

For details of this solution see Caliński and Kageyama (2008, Sect. 4).

Appendix 2

To find ${{\varvec{\zeta }}}_{E}$ as a solution of Eq. (3.10), it may be helpful to represent the $m \times m$ matrix ${{\varvec{\zeta }}}_{E}$ by an $m^{2} \times 1$ vector, $\mathrm{vec}({{\varvec{\zeta }}}_{E}),$ writing vertically all the elements of ${{\varvec{\zeta }}}_{E}$ starting with the first element and proceeding in the lexicographical order row by row. Then, the equation can be written in a more manageable form, as

$$\begin{aligned}&\left[ \sum \limits _{j=1}^{p}\sum \limits _{j^{\prime }=1}^{p}\left\{ \left( {{\varvec{U}}}_{j}^{\prime }{{\varvec{R}}}{{\varvec{U}}}_{j^{\prime }}\right) \otimes \left( {{\varvec{U}}}_{j}^{\prime }{{\varvec{R}}}{{\varvec{U}}}_{j^{\prime }}\right) \right\} \right] \mathrm{vec}\left( {{\varvec{\zeta }}}_{E}\right) \\&\quad = \mathrm{vec}\left( \sum \limits _{j=1}^{p}{{\varvec{U}}}_{j}^{\prime }{{\varvec{R}}}{{\varvec{y}}}{{\varvec{y}}}^{\prime }{{\varvec{R}}}{{\varvec{U}}}_{j}- \sum \limits _{j=1}^{p}{{\varvec{U}}}_{j}^{\prime }{{\varvec{R}}}\tilde{{{\varvec{V}}}}{{\varvec{R}}}{{\varvec{U}}}_{j}\right) \end{aligned}$$

(see Rao and Kleffe 1988, p. 10). To simplify it further, note that, when adopting the notation ${{\varvec{R}}} = [{{\varvec{R}}}_{jj^{\prime }}],$ one can use the equalities

$$\begin{aligned} {{\varvec{U}}}_{j}^{\prime }{{\varvec{R}}}{{\varvec{U}}}_{j^{\prime }}= & {} \left( \mathbf{1}_{a}^{\prime } \otimes {{\varvec{I}}}_{m}\right) {{\varvec{R}}}_{jj^{\prime }}\left( \mathbf{1}_{a} \otimes {{\varvec{I}}}_{m}\right) , \\ {{\varvec{U}}}_{j}^{\prime }{{\varvec{R}}}= & {} \left[ \left( \mathbf{1}_{a}^{\prime } \otimes {{\varvec{I}}}_{m}\right) {{\varvec{R}}}_{j1}{:} \left( \mathbf{1}_{a}^{\prime } \otimes {{\varvec{I}}}_{m}\right) {{\varvec{R}}}_{j2}{:} \cdots {:}\left( \mathbf{1}_{a}^{\prime } \otimes {{\varvec{I}}}_{m}\right) {{\varvec{R}}}_{jp}\right] . \end{aligned}$$

Also note that because ${{\varvec{\zeta }}}_{E},$ as a part of the covariance matrix ${{\varvec{V}}}$ defined in (2.10), appears also in the matrix ${{\varvec{R}}} = {{\varvec{V}}}^{-1} - {{\varvec{V}}}^{-1}{{\varvec{X}}}({{\varvec{X}}}^{\prime }{{\varvec{V}}}^{-1}{{\varvec{X}}})^{-1}{{\varvec{X}}}^{\prime }{{\varvec{V}}}^{-1},$ the above equation is to be solved by an iterative procedure. To start it, one may use, instead of the matrix ${{\varvec{\zeta }}}_{E}$ appearing in ${{\varvec{R}}},$ a matrix $\zeta _{E}{{\varvec{J}}}_{m},$ taking some initial value for the scalar $\zeta _{E}.$ A suitable choice will be $\zeta _{E} = (am)^{-1}(\sigma _{4}^{2} - \sigma _{3}^{2}),$ where $\sigma _{3}^{2}$ and $\sigma _{4}^{2}$ are obtainable as solutions of the equations ${{\varvec{y}}}^{\prime }{{\varvec{\phi }}}_{3}{{\varvec{y}}} =\sigma _{3}^{2}p(a - 1)$ and ${{\varvec{y}}}^{\prime }{{\varvec{\phi }}}_{4}^{*}{{\varvec{y}}} =\sigma _{4}^{2}(p - 1),$ with ${{\varvec{\phi }}}_{3} = {{\varvec{I}}}_{p} \otimes ({{\varvec{I}}}_{a} - a^{-1}{{\varvec{J}}}_{a}) \otimes m^{-1}{{\varvec{J}}}_{m}$ and ${{\varvec{\phi }}}_{4}^{*} = ({{\varvec{I}}}_{p} - p^{-1}{{\varvec{J}}}_{p}) \otimes a^{-1}{{\varvec{J}}}_{a} \otimes m^{-1}{{\varvec{J}}}_{m}.$

This would mean to start the iterations under the assumption of complete additivity. Continuing then the iterative procedure one has to secure, at any iteration cycle, that the obtained provisional estimate of ${{\varvec{\zeta }}}_{E},$ denoted by ${{\varvec{\zeta }}}_{E,0},$ is n.n.d. For that, it is advisable to take its spectral decomposition, ${{\varvec{\zeta }}}_{E,0} = \sum \nolimits _{i=1}^{m}\lambda _{i}{{\varvec{p}}}_{i}{{\varvec{p}}}_{i}^{\prime },$ and use

$$\begin{aligned} {{\varvec{\zeta }}}_{E,0}^{(+)} = \sum \limits _{i=1}^{m}\lambda _{i}^{*}{{\varvec{p}}}_{i}{{\varvec{p}}}_{i}^{\prime }, \quad \mathrm{with} \quad \lambda _{i}^{*} = \lambda _{i}, \quad \mathrm{if} \quad \lambda _{i} > 0, \quad \mathrm{or} \quad \lambda _{i}^{*} = 0, \,\mathrm{otherwise}. \end{aligned}$$

Appendix 3

To show the equivalence of the two forms appearing in formula (4.10), first write

$$\begin{aligned} {{\varvec{u}}}_{V(E,j)} = \left( {{\varvec{I}}}_{m}- m^{-1}{{\varvec{J}}}_{m}\right) \left\{ \left( \mathbf{1}_{a}^{\prime }\otimes {{\varvec{I}}}_{m}\right) {{\varvec{V}}}_{*(j)}^{-1}\left( \mathbf{1}_{a} \otimes {{\varvec{I}}}_{m}\right) \right\} ^{-1}\left( \mathbf{1}_{a}^{\prime } \otimes {{\varvec{I}}}_{m}\right) {{\varvec{V}}}_{*(j)}^{-1}{{\varvec{y}}}_{*j}, \end{aligned}$$

and then use the following notation: ${{\varvec{A}}}_{(j)} = \tilde{{{\varvec{V}}}}_{*(j)} = \sigma _{1(j)}^{2}{{\varvec{\phi }}}_{1(j)} + \sigma _{2(j)}^{2}({{\varvec{I}}}_{am} - {{\varvec{\phi }}}_{1(j)}),\,{{\varvec{\Sigma }}}_{VE} = {{\varvec{R}}}^{VE}({{\varvec{R}}}^{VE})^{\prime },\,{{\varvec{B}}} = \mathbf{1}_{a} \otimes {{\varvec{R}}}^{VE},\,{{\varvec{D}}} = {{\varvec{I}}}_{t},$ where $t = \mathrm{rank}({{\varvec{\Sigma }}}_{VE}) = \mathrm{rank}({{\varvec{R}}}^{VE}).$ With it, one can write for ${{\varvec{V}}}_{*(j)},$ defined in (4.6), ${{\varvec{V}}}_{*(j)} = {{\varvec{A}}}_{(j)} + {{\varvec{B}}}{{\varvec{D}}}{{\varvec{B}}}^{\prime }$ and ${{\varvec{V}}}_{*(j)}^{-1} = {{\varvec{A}}}_{(j)}^{-1} - {{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}({{\varvec{B}}}^{\prime }{{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}} + {{\varvec{D}}})^{-1}{{\varvec{B}}}^{\prime }{{\varvec{A}}}_{(j)}^{-1}.$ Furthermore, note that ${{\varvec{B}}} = (\mathbf{1}_{a} \otimes {{\varvec{I}}}_{m}){{\varvec{R}}}^{VE} = {{\varvec{B}}}_{1}{{\varvec{B}}}_{2},$ say. So, one can also write ${{\varvec{V}}}_{*(j)}^{-1} = {{\varvec{A}}}_{(j)}^{-1} - {{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}_{1}{{\varvec{B}}}_{2}({{\varvec{B}}}_{2}^{\prime }{{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}_{1}{{\varvec{B}}}_{2} + {{\varvec{D}}})^{-1}{{\varvec{B}}}_{2}^{\prime }{{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1}.$ From this, $(\mathbf{1}_{a}^{\prime } \otimes {{\varvec{I}}}_{m}){{\varvec{V}}}_{*(j)}^{-1}(\mathbf{1}_{a} \otimes {{\varvec{I}}}_{m}) = {{\varvec{B}}}_{1}^{\prime }{{\varvec{V}}}_{*(j)}^{-1}{{\varvec{B}}}_{1} = {{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}_{1} - {{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}_{1}{{\varvec{B}}}_{2}({{\varvec{B}}}_{2}^{\prime }{{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}_{1}{{\varvec{B}}}_{2} + {{\varvec{D}}})^{-1}{{\varvec{B}}}_{2}^{\prime }{{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}_{1},$ where $\mathrm{rank}({{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}_{1}) =\mathrm{rank}({{\varvec{B}}}_{1}).$ This implies that $\{(\mathbf{1}_{a}^{\prime } \otimes {{\varvec{I}}}_{m}){{\varvec{V}}}_{*(j)}^{-1}(\mathbf{1}_{a} \otimes {{\varvec{I}}}_{m})\}^{-1} = ({{\varvec{B}}}_{1}^{\prime }{{\varvec{V}}}_{*(j)}^{-1}{{\varvec{B}}}_{1})^{-1} = ({{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}_{1})^{-1} + {{\varvec{B}}}_{2}{{\varvec{D}}}{{\varvec{B}}}_{2}^{\prime }.$ Hence,

$$\begin{aligned}&\left\{ \left( \mathbf{1}_{a}^{\prime } \otimes {{\varvec{I}}}_{m}\right) {{\varvec{V}}}_{*(j)}^{-1}\left( \mathbf{1}_{a} \otimes {{\varvec{I}}}_{m}\right) \right\} ^{-1}\left( \mathbf{1}_{a}^{\prime } \otimes {{\varvec{I}}}_{m}\right) {{\varvec{V}}}_{*(j)}^{-1} = \left\{ \left( {{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}_{1}\right) ^{-1}\right. \\&\left. \quad +\,{{\varvec{B}}}_{2}{{\varvec{D}}}{{\varvec{B}}}_{2}^{\prime }\right\} \left\{ {{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1} - {{\varvec{B}}}_1^{\prime }{{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}_{1}{{\varvec{B}}}_{2}\left( {{\varvec{B}}}_2^{\prime }{{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}_{1}{{\varvec{B}}}_{2} + {{\varvec{D}}}\right) ^{-1}{{\varvec{B}}}_{2}^{\prime }{{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1}\right\} \\&= \left( {{\varvec{B}}}_{1}{{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}_{1}\right) ^{-1}{{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1} + {{\varvec{B}}}_{2}\left\{ {{\varvec{D}}} - \left( {{\varvec{B}}}_{2}^{\prime }{{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}_{1}{{\varvec{B}}}_{2} + {{\varvec{D}}}\right) ^{-1}\right. \\&\left. \quad -\, {{\varvec{D}}}{{\varvec{B}}}_{2}^{\prime }{{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}_{1}{{\varvec{B}}}_{2}\left( {{\varvec{B}}}_2^{\prime }{{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}_{1}{{\varvec{B}}}_{2} + {{\varvec{D}}}\right) ^{-1}\right\} {{\varvec{B}}}_{2}^{\prime }{{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1}. \end{aligned}$$

Now, after some algebraical derivations, one can show that ${{\varvec{D}}} - ({{\varvec{B}}}_{2}^{\prime }{{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}_{1}{{\varvec{B}}}_{2} + {{\varvec{D}}})^{-1} - {{\varvec{D}}}{{\varvec{B}}}_{2}^{\prime }{{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}_{1}{{\varvec{B}}}_{2}({{\varvec{B}}}_{2}^{\prime }{{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}_{1}{{\varvec{B}}}_{2} + {{\varvec{D}}})^{-1} = \mathbf{0},$ which finally gives the equality $\{(\mathbf{1}_{a}^{\prime } \otimes {{\varvec{I}}}_{m}){{\varvec{V}}}_{*(j)}^{-1}(\mathbf{1}_{a} \otimes {{\varvec{I}}}_{m})\}^{-1}(\mathbf{1}_{a}^{\prime } \otimes {{\varvec{I}}}_{m}){{\varvec{V}}}_{*(j)}^{-1} = ({{\varvec{B}}}_{1}{{\varvec{A}}}_{(j)}^{-1}{{\varvec{B}}}_{1})^{-1}{{\varvec{B}}}_{1}^{\prime }{{\varvec{A}}}_{(j)}^{-1} = \{(\mathbf{1}_{a}^{\prime } \otimes {{\varvec{I}}}_{m})\tilde{{{\varvec{V}}}}_{*(j)}^{-1}(\mathbf{1}_{a} \otimes {{\varvec{I}}}_{m})\}^{-1}(\mathbf{1}_{a}^{\prime }\otimes {{\varvec{I}}}_{m})\tilde{{{\varvec{V}}}}_{*(j)}^{-1}.$ This further implies that

$$\begin{aligned}&{{\varvec{u}}}_{V(E,j)} \\&\quad =\left( {{\varvec{I}}}_{m} - m^{-1}{{\varvec{J}}}_{m}\right) \left\{ \left( \mathbf{1}_{a}^{\prime } \otimes {{\varvec{I}}}_{m}\right) \tilde{{{\varvec{V}}}}_{*(j)}^{-1}\left( \mathbf{1}_{a} \otimes {{\varvec{I}}}_{m}\right) \right\} ^{-1}\left( \mathbf{1}_{a}^{\prime } \otimes {{\varvec{I}}}_{m}\right) \tilde{{{\varvec{V}}}}_{*(j)}^{-1}{{\varvec{y}}}_{*j} \quad \mathrm{and} \\&\mathrm{Cov}\left( {{\varvec{u}}}_{V(E,j)}\right) \\&\quad =\left( {{\varvec{I}}}_{m} - m^{-1}{{\varvec{J}}}_{m}\right) \left\{ \left( \mathbf{1}_{a}^{\prime } \otimes {{\varvec{I}}}_{m}\right) \tilde{{{\varvec{V}}}}_{*(j)}^{-1}\left( \mathbf{1}_{a} \otimes {{\varvec{I}}}_{m}\right) \right\} ^{-1}\left( {{\varvec{I}}}_{m} - m^{-1}{{\varvec{J}}}_{m}\right) + {{\varvec{\Sigma }}}_{VE}. \end{aligned}$$

The consequences of these results for presenting (4.12) and its covariance matrix are obvious. Note that the results presented here do not depend on the variances $\sigma _{1(j)}^{2}$ and $\sigma _{2(j)}^{2},$ whether they differ among js (the trials) or not.