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