91Ó°ÊÓ

The given values are \({\rm{\;(where\;}}E(MSABC)\)is modified here,perhaps there is mistake in the book)

a)The classic F ratios should not work here because the C is random factor.

\(\frac{{E(MSABC)}}{{E(MSE)}} = \frac{{{\sigma ^2} + L \cdot \sigma _{ABC}^2}}{{{\sigma ^2}}} = 1 + \frac{{\sigma _{ABC}^2}}{{{\sigma ^2}}}\)

Which is equal to 1 when \(\sigma _{ABC}^2 = 0\)and is larger than 1 when \(\sigma _{ABC}^2 > 0\)which indicates that the following F ratio is appropriate

\({F_{ABC}} = \frac{{MSABC}}{{MSE}}\)

For testing

\({H_0}:\sigma _{ABC}^2 = 0{\rm{\;versus\;}}{H_a}:\sigma _{ABC}^2 > 0.{\rm{\;}}\)

Similarly from

\(\frac{{E(MSC)}}{{E(MSE)}} = \frac{{{\sigma ^2} + I \cdot J \cdot L \cdot \sigma _C^2}}{{{\sigma ^2}}} = 1 + \frac{{I \cdot J \cdot L \cdot \sigma _C^2}}{{{\sigma ^2}}}\)

which is equal to 1 when \(\sigma _C^2 = 0\)and is larger than 1 when \(\sigma _{ABC}^2 > 0\) which indicates F ratio is appropriate

\({F_C} = \frac{{MSC}}{{MSE}}\)

For testing

\({H_0}:\sigma _C^2 = 0{\rm{\;versus\;}}{H_a}:\sigma _C^2 > 0.{\rm{\;}}\)

From

which is equal to 1 when \(\gamma _{ij}^{AB} = 0,{\rm{\;for all\;}}i,j,{\rm{\;which indicates that the following\;}}\)indicates F ratio is appropriate.

\({F_{AB}} = \frac{{MSAB}}{{MSABC}}\)

For testing

\({H_0}:\gamma _{ij}^{AB} = 0{\rm{\;for all\;}}i,j,{\rm{\;versus\;}}{H_a}:{\rm{\;at least one\;}}{\gamma _ - }i{j^{AB}} \ne 0.{\rm{\;}}\)

From

which is equal to 1 \({\rm{\;when\;}}{\alpha _1} = {\alpha _2} = \ldots = {\alpha _I} = 0\)which indicates F ratio is appropriate.

\({F_A} = \frac{{MSA}}{{MSAC}}\)

For testing

\({H_0}:{\rm{\;all\;}}{\alpha _i} = 0{\rm{\;versus\;}}{H_a}:{\rm{\;at least one\;}}{\alpha _ - }i \ne 0.\)

b)The given sum of squares are given,all expect the total sum of squares SST

\(\begin{aligned}{*{20}{c}}{SSA = 14,318.24;}\\{SSB = 9656.40;}\\{SSC = 2270.22;}\\{SSAB = 3408.93;}\\{SSAC = 1442.58;}\\{SSBC = 3096.21;}\\{SSABC = 2832.72}\\{SSE = 8655.60.}\end{aligned}\)

The following table with corresponding values

\(SST = SSA + SSB + SSC + SSAB + SSAC + SSBC + SSABC + SSE\)

By the fundamental identity,value of the missing values SST is

\(\begin{aligned}{*{20}{c}}{SST = SSA + SSB + SSC + SSAB + SSAC + SSBC + SSABC + SSE}\\{ = 14,318.24 + 9656.4 + 2270.22 + 3408.93 + }\\{ + 1442.58 + 3096.21 + 2832.72 + 8655.60}\\{ = 45,680.9.}\end{aligned}\)

The degrees of freedom are

\(\begin{aligned}{*{20}{c}}{d{f_T} = IJKL - 1 = 2 \cdot 4 \cdot 3 \cdot 2 - 1 = 47}\\{d{f_E} = IJK(L - 1) = 2 \cdot 4 \cdot 3 \cdot (2 - 1) = 24;}\\{d{f_A} = I - 1 = 2 - 1 = 1;}\\{d{f_B} = J - 1 = 4 - 1 = 3;}\\{d{f_C} = K - 1 = 3 - 1 = 2;}\end{aligned}\)

\(\begin{aligned}{*{20}{c}}{d{f_{AB}} = (I - 1)(J - 1) = (2 - 1) \cdot (4 - 1) = 3}\\{d{f_{AC}} = (I - 1)(K - 1) = (2 - 1) \cdot (3 - 1) = 2}\\{d{f_{BC}} = (J - 1)(K - 1) = (4 - 1) \cdot (3 - 1) = 6}\\{d{f_{ABC}} = (I - 1)(J - 1)(K - 1) = (2 - 1) \cdot (4 - 1) \cdot (3 - 1) = 6.}\end{aligned}\)

The mean squares are

\(\begin{aligned}{*{20}{c}}{MSA = \frac{1}{{d{f_A}}} \cdot SSA = \frac{1}{1} \cdot 14,318.24 = 14,318.24}\\{MSB = \frac{1}{{d{f_B}}} \cdot SSB = \frac{1}{3} \cdot 9656.4 = 3218.80}\\{MSC = \frac{1}{{d{f_C}}} \cdot SSC = \frac{1}{2} \cdot 2270.22 = 1135.11}\\{MSAB = \frac{1}{{d{f_1}}} \cdot SSAB = \frac{1}{3} \cdot 3408.93 = 1136.31}\end{aligned}\)

\(\begin{aligned}{*{20}{c}}{MSAC = \frac{1}{{d{f_{AC}}}} \cdot SSAC = \frac{1}{2} \cdot 1442.58 = 721.29}\\{MSBC = \frac{1}{{d{f_{BC}}}} \cdot SSBC = \frac{1}{6} \cdot 3096.21 = 516.04}\\{MSABC = \frac{1}{{d{f_{ABC}}}} \cdot SSABC = \frac{1}{6} \cdot 2832.72 = 472.12}\\{MSE = \frac{1}{{d{f_E}}} \cdot SSE = \frac{1}{{24}} \cdot 8655.60 = 360.65}\end{aligned}\)

The corresponding f values a\({\rm{\$ }} \setminus {\rm{\;textbf\{ as proved in\;}}(a)\} {\rm{\$ , by symmetry,\;}}\) are

\(\begin{aligned}{*{20}{c}}{{f_A} = \frac{{MSA}}{{MSAC}} = \frac{{14,318.24}}{{721.29}} = 19.85}\\{{f_B} = \frac{{MSB}}{{MSBC}} = \frac{{3218.80}}{{516.04}} = 6.24}\\{{f_C} = \frac{{MSC}}{{MSE}} = \frac{{1135.11}}{{360.65}} = 3.15}\\{{f_{AB}} = \frac{{MSAB}}{{MSABC}} = \frac{{1136.31}}{{472.12}} = 2.41}\end{aligned}\)

\(\begin{aligned}{*{20}{c}}{{f_{AC}} = \frac{{MSAC}}{{MSABC}} = \frac{{721.29}}{{472.12}} = 2.00}\\{{f_{BC}} = \frac{{MSBC}}{{MSE}} = \frac{{516.04}}{{360.65}} = 1.43}\\{{f_{ABC}} = \frac{{MSABC}}{{MSE}} = \frac{{13.53}}{{360.65}} = 1.31}\end{aligned}\)

The P value can be computed using a software.

\({P_A} = P\left( {F > {f_A}} \right) = P(F > 19.85) = 0.047\)

P-value is the area under the \({F_{I - 1,(I - 1)(J - 1)}}{\rm{\;curve to the right of the test statistic value\;}}{f_A}{\rm{.\;}}\)

\({P_B} = P\left( {F > {f_B}} \right) = P(F > 6.24) = 0.034\)

P-value is the area under the \({F_{J - 1,(J - 1)(K - 1)}}{\rm{\;curve to the right of the test statistic value\;}}{f_B}{\rm{.\;}}\)

\({P_C} = P\left( {F > {f_C}} \right) = P(F > 3.15) = 0.061\)

P-value is the area under the \({F_{K - 1.IJK(L - 1)}}{\rm{\;curve to the right of the test statistic value\;}}{f_C}{\rm{.\;}}\)

\({P_{AB}} = P\left( {F > {f_{AB}}} \right) = P(F > 2.41) = 0.165\)

P-value is the area under the\({F_{(I - 1)(J - 1),(I - 1)(J - 1)(K - 1)}}{\rm{\;curve to the right of the test statistic value\;}}{f_{AB}}{\rm{.\;}}\) \({P_{AC}} = P\left( {F > {f_{AC}}} \right) = P(F > 2.00) = 0.216\)

P-value is the area under the \(F(I - 1)(K - 1),(I - 1)(J - 1)(K - 1){\rm{\;curve to the right of the test statistic value\;}}{f_{AC}}\)

\({P_{BC}} = P\left( {F > {f_{BC}}} \right) = P(F > 1.43) = 0.244\)

P-value is the area under the \({F_{(J - 1)(K - 1),IJK(L - 1)}}{\rm{\;curve to the right of the test statistic value\;}}{f_{BC}}{\rm{.\;}}\)

\({P_{ABC}} = P\left( {F > {f_{ABC}}} \right) = P(F > 1.31) = 0.291\)

P-value is the area under the

\({F_{(I - 1)(J - 1)(K - 1),IJK(L - 1)}}{\rm{\;curve to the right of the test statistic value\;}}{f_{ABC}}\)

To complete the ANOVA table,the left values are the critical values.

\(\begin{aligned}{*{20}{c}}{{F_{0.01,1,2}} = 98.50}\\{{F_{0.01,3,6}} = 9.78}\\{{F_{0.01,2,24}} = 5.61}\\{{F_{0.01,3,6}} = 9.78}\end{aligned}\)

\(\begin{aligned}{*{20}{c}}{{F_{0.01,2,6}} = 10.92}\\{{F_{0.01,6,24}} = 3.67}\\{{F_{0.01,6,24}} = 3.67}\end{aligned}\)

Finally,ANOVA table becomes

The relevant Hypotheses are

\(\begin{aligned}{*{20}{c}}{{H_{0A}}:{\alpha _1} = {\alpha _2} = \ldots = {\alpha _I} = 0{\rm{\;versus\;}}{H_{aA}}:{\rm{\;at least one\;}}{\alpha _ - }i \ne 0;}\\{{H_{0B}}:{\beta _1} = {\beta _2} = \ldots = {\beta _J} = 0{\rm{\;versus\;}}{H_{aB}}:{\rm{\;at least one\;}}{\beta _ - }j \ne 0;}\\{{H_{0C}}:\sigma _C^2 = 0{\rm{\;versus\;}}{H_{aC}}:\sigma _{ABC}^2 > 0,}\end{aligned}\)

For the interactions

\({H_{0AB}}:\gamma _{ij}^{AB} = 0{\rm{\;for all\;}}i,j{\rm{\;versus\;}}{H_{aAB}}:{\rm{\;at least one\;}}{\gamma _ - }i{j^{AB}} \ne 0{\rm{;\;}}\)

\({H_{0AC}}:\gamma _{ij}^{AC} = 0{\rm{\;for all\;}}i,j{\rm{\;versus\;}}{H_{aAC}}:{\rm{\;at least one\;}}{\gamma _ - }i{j^{AC}} \ne 0{\rm{;\;}}\)

\({H_{0BC}}:\gamma _{ij}^{BC} = 0{\rm{\;for all\;}}i,j{\rm{\;versus\;}}{H_{aBC}}:{\rm{\;at least one\;}}{\gamma _ - }i{j^{BC}} \ne 0{\rm{,\;}}\)

And for the interactions ABC

\({H_{0ABC}}:\sigma _{ABC}^2 = 0\quad {\rm{\;versus\;}}{H_{aABC}}:\sigma _{ABC}^2 > 0\)

\({\rm{\;When testing hypotheses\;}}{H_{0A}}{\rm{\;versus\;}}{H_{aA}}{\rm{,\;}}\)the test statistics value is

\({f_A} = \frac{{MSA}}{{MSAC}},\)

P-value is the area under the \({F_{I - 1,(I - 1)(J - 1)}}{\rm{\;curve to the right of the test statistic value\;}}{f_A}{\rm{.\;}}\)

\({\rm{\;When testing hypotheses\;}}{H_{0B}}{\rm{\;versus\;}}{H_{aB}}\), the test statistics value is

\({f_B} = \frac{{MSB}}{{MSBC}},\)

P-value is the area under the \({F_{J - 1,(J - 1)(K - 1)}}{\rm{\;curve to the right of the test statistic value\;}}{f_B}{\rm{.\;}}\)

the test statistics value is

\({f_C} = \frac{{MSC}}{{MSE}}\)

P-value is the area under the \({F_{K - 1.IJK(L - 1)}}{\rm{\;curve to the right of the test statistic value\;}}{f_C}{\rm{.\;}}\)

For factor A

\(\begin{aligned}{*{20}{c}}{{F_{0.01,1,2}} = 98.5 > 19.85 = {f_A}}\\{{P_A} = 0.047 > 0.01 = \alpha ,}\end{aligned}\)

\({\rm{\;do not reject null hypothesis\;}}{H_{0A}}\)

\({\rm{\;at given significance level\;}}\alpha {\rm{. Factor (main effect)\;}}\)A is not statiscally significant.

For factor B

\(\begin{aligned}{*{20}{c}}{{F_{0.01,3,6}} = 9.78 > 6.24 = {f_B}}\\{{P_B} = 0.034 > 0.01 = \alpha }\end{aligned}\)

\({\rm{\;do not reject null hypothesis\;}}{H_{0B}}\)

\({\rm{\;at given significance level\;}}\alpha {\rm{. Factor (main effect)\;}}\)B is not statiscally significant.

For factor C

\(\begin{aligned}{*{20}{r}}{{F_{0.01,2,24}} = 5.61 > 3.15 = {f_C}}\\{{P_C} = 0.061 > 0.01 = \alpha }\end{aligned}\)

\({\rm{\;do not reject null hypothesis\;}}{H_{0C}}\)

\({\rm{\;at given significance level\;}}\alpha {\rm{. Factor (main effect)\;}}\)C is not statiscally significant.

When testing hypotheses \({H_{0AB}}{\rm{\;versus\;}}{H_{aAB}}\), the test statistic value is \({f_{AB}} = \frac{{MSAB}}{{MSABC}}\)and the p-value is the area under the \({F_{(I - 1)(J - 1),(I - 1)(J - 1)(K - 1)}}\)curve to the right of the test statistic value \({f_{AB}}\).

When testing hypotheses \({H_{0AC}}{\rm{\;versus\;}}{H_{aAC}}\), the test statistic value is \({f_{AC}} = \frac{{MSAC}}{{MSABC}},\) and

the p-value is the area under the \({F_{(I - 1)(J - 1),(I - 1)(J - 1)(K - 1)}}\)curve to the right of the test statistic value\({f_{AC}}\).

When testing hypotheses\({H_{0BC}}{\rm{\;versus\;}}{H_{aBC}}{\rm{,\;}}\) the test statistic value is \({f_{BC}} = \frac{{MSBC}}{{MSE}}\)and

the p-value is the area under the \({F_{(I - 1)(J - 1),(I - 1)(J - 1)(K - 1)}}\)curve to the right of the test statistic value \({f_{BC}}\)

When testing hypotheses \({H_{0ABC}}{\rm{\;versus\;}}{H_{aABC}}\), the test statistic value is \({f_{ABC}} = \frac{{MSABC}}{{MSE}}\)and

the p-value is the area under the \({F_{(I - 1)(J - 1),(I - 1)(J - 1)(K - 1)}}\)curve to the right of the test statistic value \({f_{ABC}}.\)

For interaction AB

\(\begin{aligned}{*{20}{c}}{{F_{0.01,3,6}} = 9.78 > 2.41 = {f_{AB}};}\\{{P_{AB}} = 0.165 > 0.01 = \alpha ,}\end{aligned}\)

\({\rm{\;do not reject null hypothesis\;}}{H_{0AB}}\)

\({\rm{\;at given significance level}}{\rm{. Interaction\;}}AB{\rm{\;is not statistically significant}}{\rm{.\;}}\)

For interaction AC

\(\begin{aligned}{*{20}{c}}{{F_{0.01,2,5}} = 10.92 > 2.00 = {f_{AC}}}\\{{P_{AB}} = 0.216 > 0.01 = \alpha }\end{aligned}\)

\({\rm{\;do not reject null hypothesis\;}}{H_{0AC}}\)

\({\rm{\;at given significance level}}{\rm{. Interaction\;}}AC{\rm{\;is not statistically significant}}{\rm{.\;}}\)

For interaction BC

\(\begin{aligned}{*{20}{c}}{{F_{0.01,6,24}} = 3.67 > 1.43 = {f_{BC}}}\\{{P_{BC}} = 0.244 > 0.01 = \alpha }\end{aligned}\)

\({\rm{\;do not reject null hypothesis\;}}{H_{0BC}}\)

\({\rm{\;at given significance level}}{\rm{. Interaction\;}}BC{\rm{\;is not statistically significant}}{\rm{.\;}}\)

For interaction ABC

\(\begin{aligned}{*{20}{c}}{{F_{0.01,6,24}} = 3.67 > 1.31 = {f_{ABC}}}\\{{P_{ABC}} = 0.291 > 0.01 = \alpha }\end{aligned}\)

\({\rm{\;do not reject null hypothesis\;}}{H_{0ABC}}\)

\({\rm{\;at given significance level}}{\rm{. Interaction\;}}ABC{\rm{, the three way interaction, is not statistically significant}}{\rm{.\;}}\)