# Table 5. 6 shows data from an agricultural experiment where crop yield was measured for two levels

## Question:

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 \(\widetilde{\boldsymbol{\beta}}\).

## Step by Step Answer:

**Related Book For**

## Data Science And Machine Learning Mathematical And Statistical Methods

**ISBN:** 9781118710852

1st Edition

**Authors:** Dirk P. Kroese, Thomas Taimre, Radislav Vaisman, Zdravko Botev