Chapter 13, Problem 25P

The following data are from a two-factor study examining the effects of two treatment conditions on males and females.

a.    Use an ANOVA with α = .05 for all tests to evaluate the significance of the main effects and the interaction.

b. Compute η² to measure the size of the effect for each main effect and the interaction.

  Factor B: Treatment

	B1	B2	B3
Male	3	1	10
	1	4	10
	1	8	14
	6	6	7	TROW1 =90 ROWI
	4	6	9
	M= 3	M = 5	M = 10
	T = 15	T=25	T= 50
	SS = 18	SS =28	SS = 26	N = 30
Factor A:	0	2	1	G = 120
Gender	2	7	I	ΣX2 = 860
	0	2	1
	0	2	6	TROW2 = 30
Female	3	2	1
	M = 1	M = 3	M = 2
	T =5	T =15	T =10
	SS =8	SS =20	SS = 20

T_CO4.1 = 20 T_CO4.2 = 40 T_CO4.3 = 60

To determine

Using an ANOVA with $\alpha =0.05$ for all tests to evaluate the significance of the main effects and the interaction.

Answer to Problem 25P

For factor A treatment has significant effect means there is difference in performance of male and female. For factor B treatment has significant effect means there is difference in treatment with $B_1,B_2,B_3$. The interaction indicates that the differences among treatment as not the same for male as they one for females.

Explanation of Solution

Given info:

The following data are given in the question:

	Factor B
		$B_1$	$B_2$	$B_3$
Factor A	Male	3	1	10	$T_{row1}=90$
		1	4	10
		1	8	14
		6	6	7
		4	6	9
		$\begin{array}{l}M=3\\ T=15\\ SS=18\end{array}$	$\begin{array}{l}M=5\\ T=25\\ SS=28\end{array}$	$\begin{array}{l}M=10\\ T=50\\ SS=26\end{array}$	$N=30$
	Female	0	2	1	$\begin{array}{c}G=120\\ \sum X^2=860\\ T_{row2}=30\end{array}$
		2	7	1
		0	2	1
		0	2	6
		3	2	1
		$\begin{array}{l}M=1\\ T=5\\ SS=8\end{array}$	$\begin{array}{l}M=3\\ T=15\\ SS=20\end{array}$	$\begin{array}{l}M=2\\ T=10\\ SS=20\end{array}$
		Total = 20	Total = 40	Total = 60

Calculation:

Total variability:

Total sum of squareis calculated as:

$\begin{array}{l}S{S}_{\text{total}}={\displaystyle \sum X^2-\frac{G^2}{N}}\\ =860-\frac{{\left(120\right)}^2}{30}\\ =380\end{array}$

Total degree of freedomis calculated as:

$\begin{array}{c}d{f}_{\text{total}}=N-1\\ =30-1\\ =29\end{array}$

Within treatment:

Sum of square within treatment is calculated as:

$\begin{array}{l}S{S}_{\text{within}\text{treatments}}={\displaystyle \sum S{S}_{\text{each}\text{treatments}}}\\ =18+28+26+8+20+20\\ =120\end{array}$

Degree of freedom within treatmentis calculated as:

$\begin{array}{l}d{f}_{\text{within}\text{treatments}}=\left(4+4+4\right)+\left(4+4+4\right)\\ =24\end{array}$

Between treatment variability:

Sum of square between treatmentsis calculated as:

$\begin{array}{l}S{S}_{\text{between}\text{treatment}}=S{S}_{\text{total}}-S{S}_{\text{within}}\\ =380-120\\ =260\end{array}$

Degree of freedom between treatments is calculated as:

$\begin{array}{c}d{f}_{\text{between}\text{treatment}}=\text{number of cells }-1\\ =6-1\\ =5\end{array}$

For factor A, the row totals are 90 and 30, and each total was obtained by adding nothing.

$\begin{array}{c}S{S}_{\text{A}}={\displaystyle \sum \frac{T_{\text{row}}^2}{n_{\text{row}}}-\frac{G^2}{N}}\\ =\frac{{90}^2}{15}+\frac{{30}^2}{15}-\frac{120\times 120}{30}\\ =540+60-480\\ =120\end{array}$

Degree of freedom for factor Ais calculated as:

$\begin{array}{c}d{f}_{\text{A}}=\text{number of row}-1\\ =2-1\\ =1\end{array}$

For factor B, sum of square is calculated as:

$\begin{array}{c}S{S}_{\text{B}}={\displaystyle \sum \frac{T_{\text{col}}^2}{n_{\text{col}}}-\frac{G^2}{N}}\\ =\frac{{20}^2}{10}+\frac{{40}^2}{10}+\frac{{60}^2}{10}-\frac{{\left(120\right)}^2}{30}\\ =80\end{array}$

Degree of freedom for factor B is calculated as:

$\begin{array}{c}d{f}_{\text{B}}=\text{number of columns}-1\\ =3-1\\ =2\end{array}$

Interaction between factors A and B denoted as $A\times B$, sum of square for interaction between factors is calculated as:

$\begin{array}{c}S{S}_{\text{A×B}}=S{S}_{\text{between}\text{treatment}}-S{S}_{\text{A}}-S{S}_{\text{B}}\\ =260-120-80\\ =60\end{array}$

Degree of freedom for interaction between factors is calculated as:

$\begin{array}{c}d{f}_{\text{A×B}}=d{f}_{\text{between}\text{treatment}}-d{f}_{\text{A}}-d{f}_{\text{B}}\\ =5-1-2\\ =2\end{array}$

Two factor ANOVA consists of three separate hypothesis test with 3 separate F-ratios.

$\begin{array}{c}M{S}_{\text{within}\text{treatment}}=\frac{S{S}_{\text{within}\text{treatment}}}{d{f}_{\text{within}\text{treatment}}}\\ =\frac{120}{24}\\ =5\end{array}$

This value is same for all three F-ratios.

The numerator of the 3 F-ratios factor A, Factor B, $A\times B$ interaction. Variances for factor A is calculated as:

$\begin{array}{c}M{S}_{\text{A}}=\frac{S{S}_{\text{A}}}{d{f}_{\text{A}}}\\ =\frac{120}{1}\\ =120\end{array}$

Variances for factor Bis calculated as:

$M{S}_{\text{B}}=\frac{S{S}_{\text{B}}}{d{f}_{\text{B}}}=\frac{80}{2}=40$

Variance for $A\times B$ interaction is calculated as:

$M{S}_{\text{A×B}}=\frac{S{S}_{\text{A×B}}}{d{f}_{\text{A×B}}}=\frac{60}{2}=30$

Now F statistic for factor A is calculated as:

$\begin{array}{c}{F}_{\text{A}}=\frac{M{S}_{\text{A}}}{M{S}_{\text{within}\text{treatment}}}\\ =\frac{120}{5}\\ =24\end{array}$

F statistic for factor B is calculated as:

$\begin{array}{c}{F}_{\text{B}}=\frac{M{S}_{\text{B}}}{M{S}_{\text{within}\text{treatment}}}\\ =\frac{40}{5}\\ =8\end{array}$

F statistic for $A\times B$ interaction is calculated as:

$\begin{array}{c}{F}_{\text{A×B}}=\frac{M{S}_{\text{A×B}}}{M{S}_{\text{within}\text{treatment}}}\\ =\frac{30}{5}\\ =6\end{array}$

Decision rule:

If the absolute value of the F-ratio is greater than the table value then there is sufficient evidence to reject the null hypothesis.

F-ratio for factor A has $df\left(1,24\right)$ with $\alpha =0.05$ critical value F is 4.26.

Thus, the absolute F-ratio forfactor A is 24 which is greater than 4.26. Thus, there is enough evidence to reject that there is no mean difference for factor A. Hence, factor A  has a significant effect.

Thus, the absolute F-ratio for factor B is * which is greater than 3.40. Thus, there is enough evidence to reject that there is no mean difference for factor A. Hence, factor B has a significant effect.

Thus, the absolute F-ratio forinteraction effect is 6 which is greater than 3.40. Thus, there is enough evidence to reject that there is no mean difference for factor A. Hence, factor B has a significant effect. For these data, treatment of interaction has significant effect.

To determine

Compute: The value of ${\eta }^2$ to measure the size of effect for each main effect and the interaction.

Answer to Problem 25P

The value of ${\eta }^2$ to measure the size of effect for each main effect and the interaction is 0.5 for factor A, 0.4 for factor B and 0.33 for interaction between factors A and B.

Explanation of Solution

Given info:

The following data are given in the question:

	Factor B
		$B_1$	$B_2$	$B_3$
Factor A	Male	3	1	10	$T_{row1}=90$
		1	4	10
		1	8	14
		6	6	7
		4	6	9
		$\begin{array}{l}M=3\\ T=15\\ SS=18\end{array}$	$\begin{array}{l}M=5\\ T=25\\ SS=28\end{array}$	$\begin{array}{l}M=10\\ T=50\\ SS=26\end{array}$	$N=30$
	Female	0	2	1	$\begin{array}{c}G=120\\ \sum X^2=860\\ T_{row2}=30\end{array}$
		2	7	1
		0	2	1
		0	2	6
		3	2	1
		$\begin{array}{l}M=1\\ T=5\\ SS=8\end{array}$	$\begin{array}{l}M=3\\ T=15\\ SS=20\end{array}$	$\begin{array}{l}M=2\\ T=10\\ SS=20\end{array}$
		Total = 20	Total = 40	Total = 60

Calculation:

The size of the effect for factor A is calculated as:

${\eta }^2=\frac{S{S}_A}{S{S}_A+S{S}_{\text{within}\text{treatment}}}$

Now substitute the values calculated in part A then:

$\begin{array}{c}{\eta }^2=\frac{S{S}_A}{S{S}_A+S{S}_{\text{within}\text{treatment}}}\\ =\frac{120}{120+120}\\ =\frac{120}{240}\\ =0.5\end{array}$

The size of the effect for factor B is calculated as:

${\eta }^2=\frac{S{S}_B}{S{S}_B+S{S}_{\text{within}\text{treatment}}}$

Now substitute the values calculated in part A then:

$\begin{array}{c}{\eta }^2=\frac{S{S}_B}{S{S}_B+S{S}_{\text{within}\text{treatment}}}\\ =\frac{80}{80+120}\\ =\frac{80}{200}\\ =0.4\end{array}$

The size of the effect for interaction $A\times B$ is calculated as:

${\eta }^2=\frac{S{S}_{A\times B}}{S{S}_{A\times B}+S{S}_{\text{within}\text{treatment}}}$

Now substitute the values calculated in part A then:

$\begin{array}{c}{\eta }^2=\frac{S{S}_{A\times B}}{S{S}_{A\times B}+S{S}_{\text{within}\text{treatment}}}\\ =\frac{60}{60+120}\\ =\frac{60}{180}\\ =0.33\end{array}$

Conclusion:

The value of ${\eta }^2$ to measure the size of effect for each main effect and the interaction is 0.5 for factor A, 0.4 for factor B and 0.33 for interaction between factors A and B.