FACTORS

HYPOTHESIS

UNBIASED ESTIMATES OF THE MEAN SQUARES

APPROXIMATE

F-STATISTIC

A

$H_{0A}:A_1=\cdots =A_n=0$

$\begin{array}{c}M{S}_{\varphi }={\varphi }_{1}M{S}_{AC}\\ +{\varphi }_{2}M{S}_{ABC}+\left(1-{\varphi }_1-{\varphi }_2\right)M{S}_E\end{array}$

$\frac{M{S}_A}{M{S}_{\varphi }}$

B

$H_{0B}:B_1=\cdots =B_n=0$

$\begin{array}{c}M{S}_{\lambda }={\lambda }_{1}M{S}_{BC}\\ +{\lambda }_{2}M{S}_{ABC}+\left(1-{\lambda }_1-{\lambda }_2\right)M{S}_E\end{array}$

$\frac{M{S}_B}{M{S}_{\lambda }}$

C

$H_{0C}:{\sigma }_C^2=0$

$\begin{array}{l}M{S}_{\gamma }={\gamma }_{1}M{S}_{AC}+{\gamma }_{2}M{S}_{BC}+{\gamma }_{3}M{S}_{ABC}\\ +\left(1-{\gamma }_1-{\gamma }_2-{\gamma }_3\right)M{S}_E\end{array}$

$\frac{M{S}_B}{M{S}_{\gamma }}$

AB

$H_{0AB}:{\left(AB\right)}_1=\cdots ={\left(AB\right)}_n=0$

$M{S}_{\pi }={\pi }_{1}M{S}_{ABC}+\left(1-{\pi }_1\right)M{S}_E$

$\frac{M{S}_{AB}}{M{S}_{\pi }}$

AC

$H_{0AC}:{\sigma }_{AC}^2=0$

$\begin{array}{c}M{S}_{\rho }={\rho }_{1}M{S}_{AC}\\ +{\rho }_{2}M{S}_{ABC}+\left(1-{\rho }_1-{\rho }_2\right)M{S}_E\end{array}$

$\frac{M{S}_{AC}}{M{S}_{\rho }}$

BC

$H_{0BC}:{\sigma }_{BC}^2=0$

$\begin{array}{c}M{S}_{\tau }={\tau }_{1}M{S}_{BC}\\ +{\tau }_{2}M{S}_{ABC}+\left(1-{\tau }_1-{\tau }_2\right)M{S}_E\end{array}$

$\frac{M{S}_{BC}}{M{S}_{\tau }}$

ABC

$H_{0ABC}:{\sigma }_{ABC}^2=0$

$\frac{M{S}_{ABC}}{M{S}_E}$