91影视

Denote with

\(\begin{aligned}{l}{x_i} = \sum\limits_{j = 1}^J {{x_{i\,j}}} \,;\\{x_{ \cdot \cdot }} = \sum\limits_{i = 1}^I {\,\sum\limits_{j = 1}^J {\,{x_{i\,j}}} \,;\,} \end{aligned}\)

The mean square is

\(MSTr = \frac{1}{{I - 1}}.\,SSTr\)

Where the treatment sum of squares is

\(SSTr = {\sum\limits_{i = 1}^I {\,\sum\limits_{j = 1}^J \, \left( {{{\mathop x\limits^\_ }_{i\,.}} - \mathop x\limits^\_ ..} \right)} ^2} = \sum\limits_{i = 1}^I \, \frac{1}{{{J_i}}}x_i^2 - \frac{1}{n}{x^2}..\)

Let's find expected value of the treatment sum of squares using the definition

\(\begin{aligned}{l}E\left( {SSTr} \right) = E\left( {\sum\limits_{i = 1}^I \, \frac{1}{{{J_i}}}x_i^2 - \frac{1}{n}{x^2}..} \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = E\left( {\sum\limits_{i = 1}^I \, \frac{1}{{{J_i}}}{{\left( {J_i^2.{{\mathop x\limits^\_ }_{i.}}} \right)}^2} - \frac{1}{n}{{\left( {n\,\,.\mathop x\limits^\_ ..} \right)}^2}} \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = E\left( {\sum\limits_{i = 1}^I {\,{J_i}.\mathop x\limits^\_ _{i.}^2\,} - n\,\,.\mathop x\limits^\_ _{...}^2} \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \sum\limits_{i = 1}^I {\,{J_i}E} \left( {\mathop x\limits^\_ _{.i\,.}^2} \right) - n\,\,.\,E\left( {\mathop x\limits^\_ _{..}^2} \right)\end{aligned}\)

\(\begin{aligned}{l} = \mathop \sum \limits_{i = 1}^I {J_i}\,\left( {V\,\left( {{{\mathop x\limits^\_ }_{i.}}} \right) + \left( {E\left( {{{\mathop x\limits^\_ }_{i.}}} \right)} \right)_{..}^2} \right){\rm{ }} - \,\,n\,\,.\,\left( {V\,\left( {{{\mathop x\limits^\_ }_{..}}} \right) + \left( {E\left( {{{\mathop x\limits^\_ }_{..}}} \right)} \right)_{..}^2} \right)\\ = \mathop \sum \limits_{i = 1}^I {J_i}\,\left( {\frac{{{\sigma ^2}}}{{{J_i}}} + \mu _i^2} \right){\rm{ }} - \,\,n\,\,.\,\left( {\frac{{{\sigma ^2}}}{n} + {{\left( {\frac{1}{n}\;\,\,.\,\mathop \sum \limits_{i = 1}^I {J_i}.{\mu _i}\,} \right)}^2}} \right)\\\, = \mathop \sum \limits_{i = 1}^I {J_i}.\frac{{{\sigma ^2}}}{{{J_i}}} + \mathop \sum \limits_{i = 1}^I {J_i}.\mu _i^2 - n\,.\frac{{{\sigma ^2}}}{n} - n\,.\,\frac{1}{{{n^2}}}\,.\,{\left( {\mathop \sum \limits_{i = 1}^I {J_i}.{\mu _i}} \right)^2}\\ = I \cdot {\sigma ^2} + \mathop \sum \limits_{i = 1}^I {J_i} \cdot {\rm{ }}{(\mu + {\alpha _i})^2} - {\sigma ^2} - \frac{1}{n}\; \cdot \left( {\mathop \sum \limits_{i = 1}^I {J_i}\,{{(\mu + {\alpha _i})}^2}} \right)\end{aligned}\)

\(\begin{aligned}{l} = {\rm{ }}(I - {\rm{ }}1){\rm{ }} \cdot {\sigma ^2} + \mathop \sum \limits_{i = 1}^I {J_i}{\mu ^2} + {\rm{ }}2{\rm{ }} \cdot \mu \mathop \sum \limits_{i = 1}^I {J_i}{\alpha _i} + \mathop \sum \limits_{i = 1}^I {J_i}\alpha _{^i}^2 - \frac{1}{n} \cdot (\mu \cdot \mathop \sum \limits_{i = 1}^I {J_i} + \mathop \sum \limits_{i = 1}^n {J_i}{\alpha _i})\\\, = {\rm{ }}(I - {\rm{ }}1){\sigma ^2} + {\mu ^2} \cdot n + {\rm{ }}2\mu \cdot {\rm{ }}0{\rm{ }} + \mathop \sum \limits_{i = 1}^I {J_i}\alpha _{^i}^2 - \frac{1}{n}\; \cdot {\rm{ }}{(n \cdot \mu + 0)^2}\\\, = (I - {\rm{ }}1){\sigma ^2} + n{\mu ^2} + \mathop \sum \limits_{i = 1}^I {J_i}\alpha _{^i}^2 - n{\mu ^2}\\\, = (I - {\rm{ }}1){\sigma ^2} + \mathop \sum \limits_{i = 1}^I {J_i}\alpha _{^i}^2\end{aligned}\)

From which, the expected value of MSTr is

E(MSTr) =\(\frac{1}{{I - 1}}\,\, \cdot \,\,\left( {(I - {\rm{ }}1){\sigma ^2} + \mathop \sum \limits_{i = 1}^I {J_i}\alpha _{^i}^2} \right)\)

Here the final result is,

E(MSTr) =\(\frac{1}{{I - 1}}\,\, \cdot \,\,\left( {(I - {\rm{ }}1){\sigma ^2} + \mathop \sum \limits_{i = 1}^I {J_i}\alpha _{^i}^2} \right)\)