3.8 Power and sample size
By Theorem 3.3.3, when \(H_0\) is false \(F_0\) has the non-central \(F_{t-1,\,N-t}(\lambda )\) distribution, and in a balanced design with \(r\) replicates and \(\sum \tau _i = 0\), \[ \lambda = \frac {r\sum _{i=1}^t\tau _i^2}{\sigma ^2} . \] The power \(P\bigl (F_{t-1,\,N-t}(\lambda ) > F_{\alpha ,\,t-1,\,N-t}\bigr )\) increases with \(\lambda \), and so with \(r\) and with the size of the effects relative to \(\sigma \). To choose \(r\) the experimenter must say what effects matter. A common specification is the smallest difference \(D\) between any two treatment means that it is important to detect.
Proposition 3.8.1. If two treatment means differ by \(D\), then \(\sum _i(\tau _i - \bar \tau )^2 \ge D^2/2\), with equality when those two effects are \(\pm D/2\) about the mean and all others equal it. Hence \(\lambda \ge rD^2/(2\sigma ^2)\).
Proof. Let \(\tau _a - \tau _b = D\). Dropping the other terms, \(\sum _i(\tau _i - \bar \tau )^2 \ge (\tau _a - \bar \tau )^2 + (\tau _b - \bar \tau )^2\), and for fixed \(\tau _a - \tau _b\) the right side is smallest when \(\bar \tau \) is midway between them, where it equals \(2(D/2)^2 = D^2/2\). □
Using \(\lambda = rD^2/(2\sigma ^2)\) therefore guarantees at least the stated power for every configuration of means with a difference of \(D\); it is the least favourable case.
Example 3.8.2. An experiment with \(t = 4\) treatments is to detect a difference of \(D = 2\sigma \) between any two means with power at least \(0.90\) at \(\alpha = 0.05\). Then \(\lambda = r(2\sigma )^2/(2\sigma ^2) = 2r\), and the power for successive \(r\), from the non-central \(F_{3,\,4(r-1)}\) distribution, is
| \(r\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) |
| \(\lambda \) | \(8\) | \(10\) | \(12\) | \(14\) | \(16\) | \(18\) |
| power | \(0.504\) | \(0.644\) | \(0.755\) | \(0.836\) | \(0.894\) | \(0.933\) |
so \(r = 9\) replicates per treatment, \(36\) runs in all, are needed.
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.