91Ó°ÊÓ

(a) Denote with \({\bar X_i}\), the average of measurements obtained when factor A is held at level i

with \({\bar X_{ \cdot j}}\)the average of measurements obtained when factor B is held at level j

and with \({\bar X_{..}}\)the grand mean

Observed values are denoted with small \(x\)instead of big X. The notations without line over X are just the sums.

The sum of squares are given by

With degrees of freedom,

\(\begin{aligned}{*{20}{c}}{d{f_T} = IJ - 1}\\{d{f_A} = I - 1}\\{d{f_B} = J - 1}\\{d{f_E} = (I - 1)(J - 1).}\end{aligned}\)

In every sum of squares there are differences

\(\begin{aligned}{*{20}{c}}{{x_{ij}} - {{\bar x}_{..}}}\\{{{\bar x}_{i.}} - {{\bar x}_{..}}}\\{{{\bar x}_{.j}} - {{\bar x}_{..}}}\end{aligned}\)

so, look at those two and see what they transform to after the given transformation

\(\begin{aligned}{*{20}{c}}{{y_{ij}} = {x_{ij}} + d.}\\{{y_{ij}} - {{\bar y}_{..}}}\end{aligned}\)

.

Finally, the third difference, analogously is

\({\bar y_{.j}} - {\bar y_{..}} = {\bar x_{ \cdot j}} - {\bar x_{..}}\)

Obviously, from the mentioned, the sums of squares are

For the sum of squares errors, in the same manner you can prove that

\({y_{ij}} - {\bar y_{i.}} - {\bar y_{.j}} + {\bar y_{..}} = {x_{ij}} - {\bar x_{i.}} - {\bar x_{.j}} + {\bar x_{..}}\)

(b)As mentioned, in every sum of squares there are differences

\(\begin{aligned}{*{20}{c}}{{x_{ij}} - {{\bar x}_ \ldots }}\\{{{\bar x}_{i.}} - {{\bar x}_ \ldots }}\\{{{\bar x}_{ \cdot j}} - {{\bar x}_{ \ldots .}}}\end{aligned}\)

so, look at those two and see what they transform to after the given transformation

\({y_{ij}} = c \cdot {x_{ij}}\)

\({y_{ij}} - {\bar y_{..}}\)

Finally, the third difference, analogously is

\({\bar y_{.j}} - {\bar y_{..}} = c \cdot \left( {{{\bar x}_{.j}} - {{\bar x}_{..}}} \right).\)

Obviously, from the mentioned, the sum of squares are

For the sum of squares errors, in the same manner you can prove that

\({y_{ij}} - {\bar y_{i.}} - {\bar y_{.j}} + {\bar y_{..}} = c \cdot \left( {{x_{ij}} - {{\bar x}_{i.}} - {{\bar x}_{.j}} + {{\bar x}_{..}}} \right)\)

\(\begin{aligned}{l}{F_{Ax}} = \frac{{\frac{1}{{I - 1}} \cdot SSA}}{{\frac{1}{{(I - 1)(J - I)}} \cdot SSE}} = \frac{{\frac{1}{{I - 1}} \cdot \sum \sum {{\left( {{{\bar x}_{i.}} - {{\bar x}_{..}}} \right)}^2}}}{{{{\left. {\frac{1}{{(I - 1)(J - 1)}} \cdot \sum \sum \left( {{x_{ij}} - {{\bar x}_{i.}} - {{\bar x}_{.j}} + {{\bar x}_{..}}} \right)} \right)}^2}}}\\ = \frac{{\frac{1}{{I - 1}} \cdot {c^2} \cdot \sum \sum {{\left( {{{\bar x}_{i.}} - {{\bar x}_{..}}} \right)}^2}}}{{{{\left. {\frac{1}{{(I - 1)(J - 1)}} \cdot {c^2} \cdot \sum \sum \left( {{x_{ij}} - {{\bar x}_{i.}} - {{\bar x}_{.j}} + {{\bar x}_{..}}} \right)} \right)}^2}}}\\ = \frac{{\frac{1}{{I - 1}} \cdot \sum \sum {{\left( {c \cdot \left( {{{\bar x}_{i.}} - {{\bar x}_{..}}} \right)} \right)}^2}}}{{{{\left. {\frac{1}{{(I - 1)(J - 1)}} \cdot \sum \sum \left( {c \cdot \left( {{x_{ij}} - {{\bar x}_{i.}} - {{\bar x}_{.j}} + {{\bar x}_{..}}} \right)} \right)} \right)}^2}}}\\ = \frac{{\frac{1}{{I - 1}} \cdot \sum \sum {{\left( {{{\bar y}_{i.}} - {{\bar y}_{..}}} \right)}^2}}}{{{{\left. {\frac{1}{{(I - 1)(J - 1)}} \cdot \sum \sum \left( {{y_{ij}} - {{\bar y}_{i.}} - {{\bar y}_{.j}} + {{\bar y}_{..}}} \right)} \right)}^2}}} = {F_{Ay}}\end{aligned}\)

It has been proved that \({F_{Ax}} = {F_{Ay}};{\rm{\;thus, the\;}}F\)statistic is the same for both data sets.

\(\begin{aligned}{l}{F_{Bx}} = \frac{{\frac{1}{{I - 1}} \cdot SSB}}{{\frac{1}{{(I - 1)(J - 1)}} \cdot SSE}} = \frac{{\frac{1}{{J - 1}} \cdot \sum \sum {{\left( {{{\bar x}_{.j}} - {{\bar x}_{..}}} \right)}^2}}}{{{{\left. {\frac{1}{{(I - 1)(J - 1)}} \cdot \sum \sum \left( {{x_{ij}} - {{\bar x}_{i.}} - {{\bar x}_{.j}} + {{\bar x}_{..}}} \right)} \right)}^2}}}\\ = \frac{{\frac{1}{{J - 1}} \cdot {c^2} \cdot \sum \sum {{\left( {{{\bar x}_{.j}} - {{\bar x}_{..}}} \right)}^2}}}{{{{\left. {\frac{1}{{(I - 1)(J - 1)}} \cdot {c^2} \cdot \sum \sum \left( {{x_{ij}} - {{\bar x}_{i.}} - {{\bar x}_{.j}} + {{\bar x}_{..}}} \right)} \right)}^2}}}\\ = \frac{{\frac{1}{{j - 1}} \cdot \sum \sum {{\left( {c \cdot \left( {{{\bar x}_{.j}} - {{\bar x}_{..}}} \right)} \right)}^2}}}{{{{\left. {\frac{1}{{(I - 1)(J - 1)}} \cdot \sum \sum \left( {c \cdot \left( {{x_{ij}} - {{\bar x}_{i.}} - {{\bar x}_{.j}} + {{\bar x}_{..}}} \right)} \right)} \right)}^2}}}\\ = \frac{{\frac{1}{{J - 1}} \cdot \sum \sum {{\left( {{{\bar y}_{.j}} - {{\bar y}_{..}}} \right)}^2}}}{{{{\left. {\frac{1}{{(I - 1)(J - 1)}} \cdot \sum \sum \left( {{y_{ij}} - {{\bar y}_{i.}} - {{\bar y}_{.j}} + {{\bar y}_{..}}} \right)} \right)}^2}}} = {F_{By}}\end{aligned}\)

It has been proved that \({F_{Bx}} = {F_{By}}{\rm{; thus, the\;}}F\)statistic is the same for both data sets.

The transformation