9.1 The Model and Its Notation

To establish notation conforming to the classical linear model.
Let \(\begin {bmatrix} Z_{j0}, & Z_{j1}, \ldots& Z_{jn}\\ \end {bmatrix} \) denote values for the predictor variables for the \(j^{\text {th}}\) trial.
Let \(\underline {Y}_j= \begin {pmatrix} Y_{j1} & Y_{j2} & \cdots& Y_{jn}\\ \end {pmatrix}'\) be the responses and let \(\underline {\varepsilon }_j = \begin {pmatrix} \varepsilon _{j1} & \varepsilon _{j2} & \cdots& \varepsilon _{jn}\\ \end {pmatrix}'\) be the errors.
In matrix notation the design matrix \[ Z_{n \times (r+1)} = \begin {pmatrix} Z_{10} & Z_{11} & \cdots& Z_{1r}\\ Z_{20} & Z_{21} & \cdots& Z_{2r}\\ \vdots & \vdots & & \vdots \\ Z_{n0} & Z_{n1} & \cdots& Z_{nr}\\ \end {pmatrix} \]

this is same as that for a single response regression model. The other matrix multivariate counter parts

\[Y_{n\times m} = \begin {pmatrix} Y_{11} & Y_{12} & \cdots& Y_{1m}\\ Y_{21} & Y_{22} & \cdots& Y_{2m}\\ \vdots & \vdots & & \vdots \\ Y_{n1} & Y_{n2} & \cdots& Y_{nm}\\ \end {pmatrix} = \begin {bmatrix} \underline {Y}_{(1)} & \vdots & \underline {Y}_{(2)} & \vdots & \cdots& \vdots & \underline {Y}_{(m)}\\ \end {bmatrix} \]

\[B_{(r+1)\times m} = \begin {pmatrix} \beta _{01} & \beta _{02} & \cdots& \beta _{0m}\\ \beta _{11} & \beta _{12} & \cdots& \beta _{1m}\\ \vdots & \vdots & & \vdots \\ \beta _{r1} & \beta _{r2} & \cdots& \beta _{rm}\\ \end {pmatrix} = \begin {bmatrix} \underline {B}_{(1)} & \vdots & \underline {B}_{(2)} & \vdots & \cdots& \vdots & \underline {B}_{(m)}\\ \end {bmatrix} \]

and

\[ \varepsilon _{n\times m} = \begin {pmatrix} \varepsilon _{11} & \varepsilon _{12} & \cdots& \varepsilon _{1m}\\ \varepsilon _{21} & \varepsilon _{22} & \cdots& \varepsilon _{2m}\\ \vdots & \vdots & &\vdots \\ \varepsilon _{n1} & \varepsilon _{n2} &\cdots& \varepsilon _{nm}\\ \end {pmatrix} = \begin {bmatrix} \underline {\varepsilon }_{(1)} & \vdots & \underline {\varepsilon }_{(2)} & \vdots & \cdots& \vdots & \underline {\varepsilon }_{(m)}\\ \end {bmatrix} = \begin {bmatrix} \underline {\varepsilon }_{(1)}\\ \underline {\varepsilon }_{(2)}\\ \vdots \\ \vdots \\ \underline {\varepsilon }_{(n)}\\ \end {bmatrix} \]

The multivariate regression model is \begin {equation} \tag {2} Y_{n\times m} = Z_{n\times (r+1)}B_{(r+1)\times m} + \varepsilon _{n\times m} \end {equation} with \(E(\varepsilon _{ij})=\underline {0}\) and \(cov(\varepsilon _{(i)},\varepsilon _{(j)})=\sigma _{ij}I\quad i,j= 1,2,\ldots , m\).

The \(m\) observations on the \(j^{\text {th}}\) trial have covariance matrix \(\Sigma =(\sigma _{ij})\) but observations from different trials are uncorrelated. Here \(B\) and \(\varepsilon _{ij}\) are unknown parameters. The design matrix \(Z\) has \(j^{\text {th}}\) row \[ \begin {bmatrix} Z_{j0}, & Z_{j1}, & \cdots , Z_{jr}\\ \end {bmatrix} \] Simply stated the \(i^{\text {th}}\) response \begin {equation} \tag {3} \underbrace {\underline {Y}_{(i)}}_{n\times 1}= ZB_{(i)} + \varepsilon _{(i)}\qquad i=1,2,\ldots , m \end {equation} with \(cov(\varepsilon _{(i)} = \sigma _{ii}I\).

However, the errors for different responses for the same trial can be correlated.

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