Chapter 13 The Cochran-Satterthwaite Approximation for Linear Combination of Mean Squares

Suppose \(M_1, \ldots, M_k\) are independent mean squares and that \(d_iM_i/E(M_i)\sim \chi_{d_i}^2\) for \(i = 1,\ldots, k\) . That is, \(M_i \sim \frac{E(M_i)}{d_i} \chi_{d_i}^2\). Consider \(M = a_1M_1 + \ldots + a_kM_k.\) The Cochran-Satterthwaite approximation works by assuming that \(M\) is approximately distributed as a scaled \(\chi^2\), \(\frac{dM}{E(M)} \sim \chi_d^2 \Leftrightarrow M \sim \frac{E(M)}{d} \chi_d^2\).

• If \(M \sim \frac{E(M)}{d} \chi_d^2\), then \(Var(M) \approx (\frac{E(M)}{d})^2Var(\chi_d^2) \approx \frac{2M^2}{d}\)
• Note that \(Var(M) = \sum_{i=1}^k 2a_i^2[E(M_i)]^2/d_i \approx 2\sum_{i=1}^k a_i^2M_i^2/d_i\)

Therefore, \[ \begin{aligned}d = \frac{M^2}{\sum_{i=1}^k a_i^2M_i^2/d_i} = \frac{(\sum_{i=1}^k a_iM_i)^2}{\sum_{i=1}^k a_i^2 M_i^2/d_i}\end{aligned} \]