4.7: Try It!
- Page ID
- 33495
Below is a design matrix for a data set of a recent study.
\[\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 1 & -1 & -1 & -1 \\ 1 & -1 & -1 & -1 \\ 1 & -1 & -1 & -1 \end{bmatrix} \nonumber\]
a) Identify the number of treatment levels and replicates.
- Solution
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4 treatment levels and 3 replicates
b) Name the model and write its equation.
- Solution
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This design matrix corresponds to the effects model, and the model equation is \(Y_{ij} = \mu + \tau_{i} + \epsilon_{ij}\), where \(i=1,2,3,4\), \(j=1,2,3\), and \(\sum_{i=1}^{4} \tau_{i} = 0\).
c) Write the equation and the design matrix that corresponds to the cell means model.
- Solution
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The equation for the cell means model is: \(Y_{ij} = \mu + \epsilon_{ij}\), where \(i=1,2,3,4\) and \(j=1,2,3\). The design matrix corresponding to the cell means model is: \[\begin{bmatrix} 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \end{bmatrix} \nonumber\]
d) Write the equation and the design matrix that corresponds to the dummy variable regressions model.
- Solution
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The equation for the 'dummy variable regression' model is: \(Y_{ij} = \mu + \mu_{i} + \epsilon_{ij}\) for \(i=1,2,3\) and \(j=1,2,3\). \(Y_{4j} = \mu + \epsilon_{4j}\)
The design matrix is given below. Note that the last 3 rows correspond to the 4th treatment level which is the reference category and its effect is estimated by the model intercept.
\[\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{bmatrix} \nonumber\]