Discussion
Under one way fixed effects ANOVA which of the following parametric functions are estimable, where the model is :
\(y_{ij} = \mu + \alpha_{i} + \epsilon_{ij}\) \(i = 1 \ldots p,\ j = 1 \ldots n_{i},\)
\(\epsilon_{ij}\) are independent identically distributed with zero mean and finite variance.
Answer: D
The estimable function is \(\alpha_{1} + \alpha_{p} - 2\alpha_{3}\).
In the one-way model, \(E(y_{ij}) = \mu + \alpha_{i}\), and a linear function \(c_{0}\mu + \sum c_{i}\alpha_{i}\) is estimable exactly when it is a combination of these expectations, that is, when \(c_{0} = \sum c_{i}\). With no \(\mu\) term, this means \(\sum c_{i} = 0\): a treatment contrast.
\(\alpha_{1} + \alpha_{p} - 2\alpha_{3}\) has coefficients summing to \(1 + 1 - 2 = 0\), so it is a contrast and estimable. \(\alpha_{1} + \alpha_{3}\) and \(2\alpha_{1} - \alpha_{2}\) have sums 2 and 1, and \(\mu + \alpha_{1} - \alpha_{2}\) has \(c_{0} = 1\) but \(\sum c_{i} = 0\), so none of these is estimable.
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