3a.5 - Summary

By coding treatment or factor levels in to numerical terms, we can use regression methods to perform the ANOVA.

 See Textbook: Section 16.8

To state the ANOVA model

\(Y_{ij}=\mu_. + \tau_i + \epsilon_{ij}\)

as a regression model, we need to represent the parameters \(\mu_., \tau_i , ... , \tau_r\) in the model. However, constraint 16.64 (from text) for the case of equal weightings:

\(\sum_{i=1}^{r} \tau_i =0\)

implies that one of the r parameters \(\tau_i\) is not needed since it can be expressed in terms of the other r - 1 parameters. We shall drop the parameter \(\tau_r\), which according to constraint 16.64 (from text) can be expressed in terms of the other r - 1 parameters \(\tau_i\) as follows:

\(\tau_r\ = -\tau_1 - \tau_2 - \cdots \tau_{r-1}\)

Thus, we shall use only the parameters \(\mu_., \tau_1 , ... , \tau_{r-1}\) for the linear model.

To illustrate how a linear model is developed with this approach, consider a single-factor study with r = 3 factor levels when \(n_1=n_2=n_3=2\). The Y, X, \(\boldsymbol{\beta}\) and \(\boldsymbol{\epsilon}\) matrices for this case are as follows:

\(\mathbf{Y} = \begin{bmatrix}Y_{11}\\ Y_{12}\\ Y_{21}\\ Y_{22}\\ Y_{31}\\ Y_{32}\end{bmatrix} \mathbf{X} = \begin{bmatrix}1 & 1 & 0\\ 1 & 1 & 0\\ 1 & 0 & 1\\ 1 & 0 & 1\\ 1 & -1 & -1\\ 1 & -1 & -1\end{bmatrix} \boldsymbol{\beta} = \begin{bmatrix}\beta_{0}\\ \beta_{1}\\ \beta_{2}\end{bmatrix} \boldsymbol{\epsilon} = \begin{bmatrix}\epsilon_{11}\\ \epsilon_{12}\\ \epsilon_{21}\\ \epsilon_{22}\\ \epsilon_{31}\\ \epsilon_{32}\end{bmatrix}\)

Note that the vector of expected values \(\mathbf{E}\{\mathbf{Y}\} =\mathbf{X}\boldsymbol{\beta}\), yields the following:

\(\mathbf{E}\{\mathbf{Y}\} = \begin{bmatrix}E\{Y_{11}\}\\ E\{Y_{12}\}\\ E\{Y_{21}\}\\ E\{Y_{22}\}\\E\{Y_{31}\}\\E\{Y_{32}\}\end{bmatrix} =\mathbf{X}\boldsymbol{\beta}= \begin{bmatrix}1 & 1 & 0\\ 1 & 1 & 0\\ 1 & 0 & 1\\ 1 & 0 & 1\\ 1 & -1 & -1\\ 1 & -1 & -1\end{bmatrix}\begin{bmatrix}\beta_{0}\\ \beta_{1}\\ \beta_{2}\end{bmatrix}=\begin{bmatrix}\mu_{.}+\tau_{1}\\ \mu_{.}+\tau_{1}\\ \mu_{.}+\tau_{2}\\ \mu_{.}+\tau_{2}\\ \mu_{.}-\tau_{1}-\tau_{2}\\ \mu_{.}-\tau_{1}-\tau_{2}\end{bmatrix}\)

Since \(\tau_3=-\tau_{1}-\tau_{2}\), see above, we see that \(E\{Y_{31}\} =E\{Y_{32}\}=\mu_{.}+\tau_{3}\). Thus, the above X matrix and \(\beta\) vector representation provides in all cases the appropriate expected values:

\(E\{Y_{ij}\} =\mu_{.}+\tau_{i}\)