Comparison of a one- and two-stage mixed model analysis of Australia’s National Variety Trial Southern Region wheat data

Abstract

A one-stage analysis of a series of variety trials involves a combined analysis of the individual plot data across trials. Together with prudent modelling of the genetic effects across trials, this is considered to be the gold standard analysis of multi-environment field trial data. An alternative is a two-stage approach in which the variety means from an analysis of the individual trials in stage one are combined into a weighted mixed model analysis in stage two to give the full set of predicted variety by environment effects and an estimate of their associated variance structure. The two-stage analysis will exactly reproduce the one-stage analysis if the full variance-covariance matrix of the means from stage one is known and is utilised in stage two. Typically the full matrix is not stored and a diagonal approximation is used. This introduces a compromise to the full analysis. The impacts of a diagonal approximation are greater in the presence of sophisticated models for the genetic effects. A second compromise is through a loss of information in estimating the non-genetic variance parameters using the two-stage approach. In this paper we draw a direct link between the one and two-stage analysis approaches for crop variety evaluation data in Australia. We now have the computing power to analyse large and complex multi-environment variety trial data sets using the one-stage approach without the need for a two-stage approximation. This should motivate a move away from the two-stage approach in a range of contexts.

Appendix

Derivation of the loss of information for variance parameter estimation

Under the assumptions for model (1) for the one-stage analysis,

$$\begin{aligned} \varvec{y}\sim & {} N\left( \varvec{X}\varvec{\tau }, \; \varvec{H}(\varvec{\sigma }_g, \varvec{\sigma }_p,\varvec{\phi })\right) \end{aligned}$$

where $\varvec{\sigma }_g$, $\varvec{\sigma }_p$ and $\varvec{\phi }$ are vectors of variance parameters for the genetic (VE), non-genetic and residual terms, respectively, and

$$\begin{aligned} \varvec{H}(\varvec{\sigma }_g, \varvec{\sigma }_p,\varvec{\phi })= & {} \varvec{Z}_g\varvec{G}_g(\varvec{\sigma }_g)\varvec{Z}_g^{{\mathsf {T}}} + \varvec{Z}_p\varvec{G}_p(\varvec{\sigma }_p)\varvec{Z}_p^{{\mathsf {T}}} + \varvec{R}(\varvec{\phi }). \end{aligned}$$

(13)

The REML log-likelihood function for estimation of $\varvec{\sigma }_g$, $\varvec{\sigma }_p$ and $\varvec{\phi }$ is

$$\ell _R(\varvec{\sigma }_g, \varvec{\sigma }_p,\varvec{\phi }) = -\frac{1}{2}\left\{ \log | \varvec{H}|+ \log | \varvec{X}^{{\mathsf {T}}}\varvec{H}^{-1}\varvec{X}| + \varvec{y}^{{\mathsf {T}}}\varvec{P}\varvec{y}\right\}$$

where

$$\begin{aligned} \varvec{P}= \varvec{H}^{-1} - \varvec{H}^{-1}\varvec{X}( \varvec{X}^{{\mathsf {T}}} \varvec{H}^{-1} \varvec{X})^{-1} \varvec{X}^{{\mathsf {T}}} \varvec{H}^{-1}. \end{aligned}$$

Now consider stage 1 of a two-stage analysis and note that model (7) can be written equivalently as

$$\begin{aligned} \varvec{y}=\, & {} \varvec{X}_{1}\varvec{\tau }_{1} + \varvec{\epsilon }\end{aligned}$$

where $\varvec{X}_1 = [\varvec{X}_g\;\varvec{X}_p]$ and $\varvec{\tau }_1 = [\varvec{\eta }_{d}^{{\mathsf {T}}}\;\varvec{\tau }_{p}^{{\mathsf {T}}}]^{{\mathsf {T}}}.$ Under the assumptions for this model,

$$\begin{aligned} \varvec{y}\sim & {} N\left( \varvec{X}_{1}\varvec{\tau }_{1}, \; \varvec{V}(\varvec{\sigma }_{p},\varvec{\phi })\right) \end{aligned}$$

where $\varvec{\phi }= [\varvec{\phi }_1^{{\mathsf {T}}},\ldots ,\varvec{\phi }_{t}^{{\mathsf {T}}}]^{{\mathsf {T}}}$ is the full set of variance parameters associated with the residual term, $\varvec{\sigma }_p = [\varvec{\sigma }_{p_1}^{{\mathsf {T}}},\ldots ,\varvec{\sigma }_{p_t}^{{\mathsf {T}}}]^{{\mathsf {T}}}$ contains the remaining non-genetic variance parameters, and $\varvec{V}(\varvec{\sigma }_{p},\varvec{\phi }) = \varvec{Z}_{p}\varvec{G}_{p}(\varvec{\sigma }_{p})\varvec{Z}_{p}^{{\mathsf {T}}} + \varvec{R}(\varvec{\phi })$. The REML log-likelihood function for estimation of $\varvec{\sigma }_{p}$ and $\varvec{\phi }$ is

$$\ell _{1}(\varvec{\sigma }_{p}, \varvec{\phi }) = -\frac{1}{2}\left\{ \log | \varvec{V}|+ \log | \varvec{X}_{1}^{{\mathsf {T}}}\varvec{V}^{-1}\varvec{X}_{1}| + \varvec{y}^{{\mathsf {T}}}\varvec{P}_{1} \varvec{y}\right\}$$

where $\varvec{P}_{1} = \varvec{V}^{-1} - \varvec{V}^{-1}\varvec{X}_{1} ( \varvec{X}_{1}^{{\mathsf {T}}} \varvec{V}^{-1} \varvec{X}_{1})^{-1} \varvec{X}_{1}^{{\mathsf {T}}} \varvec{V}^{-1}.$ Finally, under the assumptions for model (9) for stage 2 of the two-stage analysis

$$\hat{\varvec{\eta }}_d\sim N\left( \varvec{D}_{e}\varvec{\tau }_{e}, \; \varvec{V}_{2}(\varvec{\sigma }_{g})\right)$$

where $\varvec{\sigma }_{g}$ is the vector of variance parameters for the VE effects and

$$\varvec{V}_{2}(\varvec{\sigma }_{g})=\, \varvec{D}\varvec{G}_{g}(\varvec{\sigma }_{g})\varvec{D}^{{\mathsf {T}}} + \mathbf {\Pi }(\varvec{\pi }),$$

(14)

where $\varvec{\pi }_j$ contains the individual weights $\pi _{jk}$ for trial j and $\varvec{\pi }= [\varvec{\pi }_1^{{\mathsf {T}}},\ldots ,\varvec{\pi }_{t}^{{\mathsf {T}}}]^{{\mathsf {T}}}.$ The REML log-likelihood function for estimation of $\varvec{\sigma }_{g}$ is

$$\ell _{2}(\varvec{\sigma }_{g}) = -\frac{1}{2}\left\{ \log | \varvec{V}_{2} |+ \log | \varvec{D}_e^{{\mathsf {T}}}\varvec{V}_{2}^{-1}\varvec{D}_e| + \hat{\varvec{\eta }}_d^{{\mathsf {T}}}\varvec{P}_2 \hat{\varvec{\eta }}_d\right\}$$

where

$$\varvec{P}_2 = \varvec{V}_{2}^{-1} - \varvec{V}_{2}^{-1}\varvec{D}_e ( \varvec{D}_e^{{\mathsf {T}}} \varvec{V}_{2}^{-1} \varvec{D}_e)^{-1} \varvec{D}_e^{{\mathsf {T}}} \varvec{V}_{2}^{-1}.$$

Derivation of (11)

Let $\varvec{T}= \left[ \varvec{T}_{_1}\;\varvec{T}_{_2}\right]$ be an $(n \times n)$ non-singular transformation matrix such that ${ \varvec{T}}_{_1}$ and ${ \varvec{T}}_{_2},$ of dimension $(n \times t)$ and $(n \times (n-t)),$ satisfy

$$\varvec{T}_{_1}^{{\mathsf {T}}}{\varvec{X}} = {\varvec{I}}_{t} \varvec{T}_{_2}^{{\mathsf {T}}}{\varvec{X}} = \mathbf {0} \Longleftrightarrow \mathcal{R}(\varvec{T}_{_2}) \perp { \mathcal{R} (\varvec{X}).}$$

Likewise, let $\varvec{Q}= \left[ \varvec{Q}_{_1}\;\varvec{Q}_{_2} \right]$ be an $((n-t) \times (n-t))$ non-singular transformation matrix such that ${ \varvec{Q}}_{_1}$ and ${ \varvec{Q}}_{_2},$ of dimension $((n-t) \times d)$ and $((n-t) \times (n-t-d)),$ satisfy

$$\varvec{Q}_{_1}^{{\mathsf {T}}}{ \varvec{T}_{_2}^{{\mathsf {T}}} \varvec{X}_{g}} = {\varvec{I}}_{d} \varvec{Q}_{_2}^{{\mathsf {T}}}\varvec{T}_{_2}^{{\mathsf {T}}} \varvec{X}_{g} = \mathbf {0} \Longleftrightarrow {\mathcal{R}(\varvec{Q}}_{_2}) \perp { \mathcal{R} (\varvec{T}_{_2} ^{{\mathsf {T}}}\varvec{X}_g ).}$$

(15)

Using the results of Verbyla (1990), $\ell _R(\varvec{\sigma }_g, \varvec{\sigma }_p, \varvec{\phi })$ is the REML log-likelihood function for the marginal distribution of $\varvec{y}_{_2} = \varvec{T}_{_2} ^{{\mathsf {T}}}\varvec{y}$, where

$$\varvec{y}_{_2}\sim N(\mathbf {0},\; \varvec{T}_{_2}^{{\mathsf {T}}}\varvec{H}\varvec{T}_{_2}).$$

Now consider the transformation $\varvec{Q}^{{\mathsf {T}}}\varvec{y}_{_2}.$ Since $\varvec{Q}$ is a non-singular transformation matrix, then for estimation

$$\ell _{R}(\varvec{\sigma }_g, \varvec{\sigma }_p, \varvec{\phi })\equiv \ell _Q(\varvec{\sigma }_g, \varvec{\sigma }_p, \varvec{\phi })$$

where $\ell _Q(\varvec{\sigma }_g, \varvec{\sigma }_p, \varvec{\phi })$ is the log-likelihood function for

$$\varvec{Q}^{{\mathsf {T}}}\varvec{y}_{_2}\sim N(\mathbf {0},\; \varvec{Q}^{{\mathsf {T}}}\varvec{T}_{_2}^{{\mathsf {T}}}\varvec{H}\varvec{T}_{_2}\varvec{Q}).$$

(16)

If $\varvec{Q}^{{\mathsf {T}}}\varvec{y}_{_2} = \varvec{Q}^{{\mathsf {T}}}\varvec{T}_{_2}^{{\mathsf {T}}}\varvec{y}= \left[ \begin{array}{c} \varvec{Q}_{_1}^{{\mathsf {T}}}\varvec{T}_{_2}^{{\mathsf {T}}}\varvec{y}\\ \varvec{Q}_{_2}^{{\mathsf {T}}}\varvec{T}_{_2}^{{\mathsf {T}}}\varvec{y}\end{array}\right] = \left[ \begin{array}{c}\varvec{q}_{_1}\\ \varvec{q}_{_2}\end{array}\right] ,$ then

$$\ell _{R}(\varvec{\sigma }_g, \varvec{\sigma }_p, \varvec{\phi })\equiv\, \ell _{q_{_1}|q_{_2}} + \ell _{q_{_2}}$$

(17)

where $\ell _{q_{_1} | q_{_2}}$ and $\ell _{q_{_2}}$ are the conditional and marginal log-likelihood functions for $\varvec{q}_{_1} | \varvec{q}_{_2}$ and $\varvec{q}_{_2},$ respectively. We show that

$$\ell _{q_{_2}}= \ell _{_1}(\varvec{\sigma }_p, \varvec{\phi })$$

so that

$$\ell _{R}(\varvec{\sigma }_g, \varvec{\sigma }_p, \varvec{\phi })\equiv\, \ell _{q_{_1}|q_{_2}} + \ell _{_1}(\varvec{\sigma }_p, \varvec{\phi }).$$

This shows that in a one-stage analysis estimation of the variance parameters associated with the genetic (VE) effects is restricted to $\ell _{q_{_1}|q_{_2}}$ while the full likelihood $\ell _R$ is used for estimation of the non-genetic variance parameters.

Consider $\ell _{q_{_2}}.$ Using (16)

$$\begin{aligned} \left[ \begin{array}{l} { \varvec{q}}_{_1} \\ { \varvec{q}}_{_2} \\ \end{array} \right]\sim & {} { N}\left( \; \left[ \begin{array}{l} \mathbf{0} \\ \mathbf{0} \\ \end{array} \right] , \; \left[ \begin{array}{cc} {\varvec{Q}_{_1}}^{{\mathsf {T}}}\varvec{T}_{_2} ^{{\mathsf {T}}}\varvec{H}\varvec{T}_{_2}\, \varvec{Q}_{_1} &{} {\varvec{Q}_{_1}}^{{\mathsf {T}}}\varvec{T}_{_2} ^{{\mathsf {T}}}\varvec{H}\varvec{T}_{_2}\, \varvec{Q}_{_2} \\ {\varvec{Q}_{_2}}^{{\mathsf {T}}}\varvec{T}_{_2} ^{{\mathsf {T}}} \varvec{H}\varvec{T}_{_2}\, \varvec{Q}_{_1} &{} {\varvec{Q}_{_2}}^{{\mathsf {T}}}\varvec{T}_{_2} ^{{\mathsf {T}}}\varvec{H}\varvec{T}_{_2}\, \varvec{Q}_{_2} \\ \end{array} \right] \;\right) \\ \end{aligned}$$

$$\begin{aligned} \Longrightarrow \;\;\;\;\varvec{q}_{_2} \sim N(\mathbf{0}, \;{\varvec{Q}_{_2}}^{{\mathsf {T}}} \varvec{T}_{_2} ^{{\mathsf {T}}}\varvec{H}\varvec{T}_{_2}{\varvec{Q}_{_2}}). \end{aligned}$$

However, using the form of $\varvec{H}$ in (13) and the definition of $\varvec{V},$

$$\begin{aligned} \begin{array}{lcl} \varvec{H}(\varvec{\sigma }_g, \varvec{\sigma }_p,\varvec{\phi }) &{} = &{} \varvec{Z}_g\varvec{G}_g(\varvec{\sigma }_g)\varvec{Z}_g^{{\mathsf {T}}} + \varvec{Z}_p\varvec{G}_p(\varvec{\sigma }_p)\varvec{Z}_p^{{\mathsf {T}}} + \varvec{R}(\varvec{\phi })\\ &{} = &{} \varvec{Z}_g\varvec{G}_g(\varvec{\sigma }_g)\varvec{Z}_g^{{\mathsf {T}}} + \varvec{V}(\varvec{\sigma }_p, \varvec{\phi }).\end{array} \end{aligned}$$

Since $\varvec{Q}_{_2}^{\mathsf {T}}\varvec{T}_{_2}^{\mathsf {T}}\varvec{X}_g = \mathbf{0}$ it follows that $\varvec{Q}_{_2}^{\mathsf {T}}\varvec{T}_{_2}^{\mathsf {T}}\varvec{X}_g\varvec{D}= \varvec{Q}_{_2}^{\mathsf {T}}\varvec{T}_{_2}^{\mathsf {T}}\varvec{Z}_g = \mathbf{0},$ in which case

$$\varvec{Q}_{_2}^{{\mathsf {T}}}\varvec{T}_{_2}^{{\mathsf {T}}}\varvec{H}\varvec{T}_{_2}\varvec{Q}_{_2} = \varvec{Q}_{_2}^{{\mathsf {T}}}\varvec{T}_{_2} ^{{\mathsf {T}}}\varvec{Z}_g \varvec{G}_g(\varvec{\sigma }_g) \varvec{Z}_g ^{{\mathsf {T}}}\varvec{T}_{_2}\varvec{Q}_{_2} + \varvec{Q}_{_2}^{{\mathsf {T}}}\varvec{T}_{_2}^{{\mathsf {T}}} \varvec{V}\varvec{T}_{_2}\varvec{Q}_{_2} = \varvec{Q}_{_2}\varvec{T}_{_2}^{{\mathsf {T}}}\varvec{V}\varvec{T}_{_2}\varvec{Q}_{_2}$$

$$\begin{aligned} \Longrightarrow \;\;\;\;\varvec{q}_{_2} \sim N(\mathbf{0}, \; {\varvec{Q}_{_2}}^{{\mathsf {T}}} \varvec{T}_{_2} ^{{\mathsf {T}}}\varvec{V}\varvec{T}_{_2}{\varvec{Q}_{_2}}) \end{aligned}$$

and the corresponding REML log-likelihood function for estimation of the variance parameters in $\varvec{\sigma }_p$ and $\varvec{\phi }$ is

$$\begin{aligned} \ell _{q_{_2}}= & {} -\frac{1}{2}\left\{ \log |\varvec{Q}_{_2} ^{{\mathsf {T}}}\varvec{T}_{_2} ^{{\mathsf {T}}}\varvec{V}\varvec{T}_{_2}\varvec{Q}_{_2} | + \varvec{q}_{_2} ^{{\mathsf {T}}}(\varvec{Q}_{_2} ^{{\mathsf {T}}}\varvec{T}_{_2} ^{{\mathsf {T}}}\varvec{V}\varvec{T}_{_2}\varvec{Q}_{_2})^{-1}\varvec{q}_{_2} \right\} .\nonumber \\ \end{aligned}$$

Now, $\varvec{T}_{_2} ^{{\mathsf {T}}} \varvec{X}= \mathbf{0}$ $\Longrightarrow \; \varvec{T}_{_2} ^{{\mathsf {T}}} \varvec{X}_{p} = \mathbf{0}$ and $\varvec{Q}_{_2}^{{\mathsf {T}}}\varvec{T}_{_2} ^{{\mathsf {T}}} \varvec{X}_{p} = \mathbf{0}.$ Also, by the definition of $\varvec{Q}_{_2}$, $\varvec{Q}_{_2} ^{{\mathsf {T}}}\varvec{T}_{_2} ^{{\mathsf {T}}}\varvec{X}_{g} = \mathbf{0}.$ We then have $Q_{_2} ^{{\mathsf {T}}}\varvec{T}_{_2} ^{{\mathsf {T}}}\;[\varvec{X}_{g}\;\varvec{X}_{p}] = Q_{_2} ^{{\mathsf {T}}}\varvec{T}_{_2} ^{{\mathsf {T}}}\;\varvec{X}_{1} = \mathbf{0}.$ If $\varvec{X}_1$ is of full column rank it follows that

$$\begin{aligned} \begin{array}{lll} \varvec{q}_{_2} ^{{\mathsf {T}}}(\varvec{Q}_{_2} ^{{\mathsf {T}}}\varvec{T}_{_2} ^{{\mathsf {T}}}\varvec{V}\varvec{T}_{_2}\varvec{Q}_{_2})^{-1}\varvec{q}_{_2} \nonumber \\ &{} = &{} \varvec{y}^{{\mathsf {T}}}\varvec{T}_{_2}\varvec{Q}_{_2}(\varvec{Q}_{_2} ^{{\mathsf {T}}}\varvec{T}_{_2} ^{{\mathsf {T}}}\varvec{V}\varvec{T}_{_2}\varvec{Q}_{_2})^{-1}\varvec{Q}_{_2} ^{{\mathsf {T}}} \varvec{T}_{_2} ^{{\mathsf {T}}} \varvec{y}\nonumber \\ &{} = &{} \varvec{y}^{{\mathsf {T}}}\left( \varvec{V}^{-1} - \varvec{V}^{-1} \varvec{X}_1(\varvec{X}_1^{{\mathsf {T}}} \varvec{V}^{-1} \varvec{X}_1)^{-1} \varvec{X}_1^{{\mathsf {T}}}\varvec{V}^{-1}\right) \varvec{y}\\ &{} = &{} \varvec{y}^{{\mathsf {T}}} P_{1} \varvec{y}, \end{array} \end{aligned}$$

see (Searle et al. 1992). Now consider $\log |\varvec{Q}_{_2} ^{{\mathsf {T}}}\varvec{T}_{_2} ^{{\mathsf {T}}}\varvec{V}\varvec{T}_{_2}\varvec{Q}_{_2} |.$ Except for a constant

$$\begin{aligned}\begin{array}{lcl} \log |\varvec{Q}_{_2} ^{{\mathsf {T}}}\varvec{T}_{_2} ^{{\mathsf {T}}}\varvec{V}\varvec{T}_{_2}\varvec{Q}_{_2} | &{} = &{} \log |\varvec{V}| + \log |\varvec{X}_1^{{\mathsf {T}}} \varvec{V}^{-1} \varvec{X}_1|. \\ \end{array} \end{aligned}$$

We then have

$$\begin{aligned} \ell _{q_{_2}} = & -\frac{1}{2}\left\{ \log |\varvec{Q}_{_2} ^{{\mathsf {T}}}\varvec{T}_{_2} ^{{\mathsf {T}}}\varvec{V}\varvec{T}_{_2}\varvec{Q}_{_2} | + \varvec{q}_{_2} ^{{\mathsf {T}}}(\varvec{Q}_{_2} ^{{\mathsf {T}}}\varvec{T}_{_2} ^{{\mathsf {T}}}\varvec{V}\varvec{T}_{_2}\varvec{Q}_{_2})^{-1}\varvec{q}_{_2}\right\} \\ =& -\frac{1}{2}\left\{ \log |\varvec{V}| + \log |\varvec{X}_1^{{\mathsf {T}}}\varvec{V}^{-1} \varvec{X}_1| + \varvec{y}^{{\mathsf {T}}}P_{1} \varvec{y}\right\} \\ = &\,\ell _{_1}(\varvec{\sigma }_p, \varvec{\phi }) \end{aligned}$$

so that

$$\ell _{R}(\varvec{\sigma }_g, \varvec{\sigma }_p, \varvec{\phi })\equiv\, \ell _{q_{_1}|q_{_2}} + \ell _{_1}(\varvec{\sigma }_p, \varvec{\phi }).$$

This implies

$$\begin{aligned} \ell _g(\varvec{\sigma }_g, \varvec{\sigma }_p, \varvec{\phi })\equiv\, & {} \ell _R(\varvec{\sigma }_g, \varvec{\sigma }_p, \varvec{\phi }) - \ell _{_1}(\varvec{\sigma }_p, \varvec{\phi }). \end{aligned}$$

where $\ell _g(\varvec{\sigma }_g, \varvec{\sigma }_p, \varvec{\phi })$ is a new notation for $\ell _{q_{_1}|q_{_2}}$ to indicate that estimation of the variance parameters in $\varvec{\sigma }_g$ is restricted to this part of the likelihood. Finally then,

$$\begin{aligned} \ell _{R}(\varvec{\sigma }_g, \varvec{\sigma }_p, \varvec{\phi })\equiv\, & {} \ell _g(\varvec{\sigma }_g, \varvec{\sigma }_p, \varvec{\phi }) + \ell _{_1}(\varvec{\sigma }_p, \varvec{\phi }) \end{aligned}$$

as required.