Discussion
Consider a two-factor factorial, fixed effect model with main effects A and B used at \(a\) and \(b\) levels respectively. Then the estimate of operating characteristic curve parameter for A is :
Answer: B
The parameter for factor A is \(\Phi^{2} = \dfrac{nb\sum_{i=1}^{a}\tau_{i}^{2}}{a\sigma^{2}}\).
In the two-factor fixed effects model with \(n\) replicates, each level of A is observed \(nb\) times. The operating characteristic curves for testing A use
\[ \Phi^{2} = \frac{nb\sum_{i=1}^{a}\tau_{i}^{2}}{a\sigma^{2}} \]
with \(a - 1\) and \(ab(n - 1)\) degrees of freedom (Montgomery). The multiplier is the number of observations per level of A, and the denominator is the number of levels of A.
Option A swaps \(a\) and \(b\). Options C and D sum the \(\beta_{j}^{2}\), which belong to factor B; C is in fact the corresponding parameter for B.
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