Table 5.

6 shows data from an agricultural experiment where crop yield was measured for two levels of pesticide and three levels of fertilizer. There are three responses for each combination.

(a) Organize the data in standard form, where each row corresponds to a single measurement and the columns correspond to the response variable and the two factor variables.

(b) Let \(Y_{i j k}\) be the response for the \(k\)-th replication at level \(i\) for factor 1 and level \(j\) for factor 2.

To assess which factors best explain the response variable, we use the ANOVA model

\[ \begin{equation*} Y_{i j k}=\mu+\alpha_{i}+\beta_{j}+\gamma_{i j}+\varepsilon_{i j k} \tag{5.43} \end{equation*} \]

where \(\Sigma_{i} \alpha_{i}=\Sigma_{j} \beta_{j}=\Sigma_{i} \gamma_{i j}=\Sigma_{j} \gamma_{i j}=0\). Define \(\boldsymbol{\beta}=\left[\mu, \alpha_{1}, \alpha_{2}, \beta_{1}, \beta_{2}, \beta_{3}, \gamma_{11}, \gamma_{12}, \gamma_{13}, \gamma_{21}\right.\), \(\left.\gamma_{22}, \gamma_{23}\right]^{\top}\). Give the corresponding \(18 \times 12\) model matrix.

(c) Note that the parameters are linearly dependent in this case. For example, \(\alpha_{2}=-\alpha_{1}\) and \(\gamma_{13}=+\gamma_{12}\) ). To retain only 6 linearly independent variables consider the 6 -dimensional parameter vector \(\widetilde{\boldsymbol{\beta}}=\left[\mu, \alpha_{1}, \beta_{1}, \beta_{2}, \gamma_{11}, \gamma_{12}\right]^{\top}\). Find the matrix \(\mathbf{M}\) such that \(\mathbf{M} \widetilde{\boldsymbol{\beta}}=\boldsymbol{\beta}\).

(d) Give the model matrix corresponding to \(\widetilde{\boldsymbol{\beta}}\).

Transcribed Image Text:

Table 5.6: Crop yields for pesticide and fertilizer combinations. Pesticide No Low 3.23, 3.20, 3.16 Fertilizer Medium 2.99, 2.85, 2.77 High 5.72, 5.77, 5.62 Yes 6.78, 6.73, 6.79 9.07, 9.09, 8.86 8.12, 8.04, 8.31