Chapter 14, Problem 21P

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

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

b. Test the simple main effects using $\alpha =.05$to eva1uate the mean difference between males and females for each of the three treatments.

	Treatment
	I	II	III
	1	7	9
Male	2	2	11
	6	9	7	$T_{\text{male}}=\text{54}$
	$M=\text{3}$	$M=\text{6}$	$M=\text{9}$
	$T=\text{9}$	$T=\text{18}$	$T=\text{27}$
Factor A: Gender	$SS=\text{14}$	$SS=\text{26}$	$SS=\text{8}$	$N=\text{18}$
	3	10	16	$G=\text{144}$
	1	11	18	${\displaystyle \sum {\text{X}}^{\text{2}}=\text{16}0\text{8}}$
	5	15	11
Female	$M=\text{3}$	$M=\text{12}$	$M=\text{15}$
	$T=\text{9}$	$T=\text{36}$	$T=\text{45}$	$T_{\text{female}}=\text{9}0$
	$SS=\text{8}$	$SS=\text{14}$	$SS=\text{26}$

To determine

1. If the mean differences of the treatments are different.
2. Test the simple main effects to evaluate the difference between males and females for each treatment.

Answer to Problem 21P

Solution:

1. For these data, gender (Factor A) and treatment (Factor B) have significant effect. These data do not produce a significant interaction. This means, that the effect of treatment does not depend on the gender.
2. There is a significant difference between males and females in treatments II and III but not in treatment I.

Explanation of Solution

Given Info:

Formula used:

The degrees of freedoms are:

$\begin{array}{l}d{f}_{Between}=d{f}_A+d{f}_B+d{f}_{A\times B}\\ d{f}_A=I-1\\ d{f}_B=J-1\\ d{f}_{A\times B}=\left(I-1\right)\left(J-1\right)\\ d{f}_{Within}=N-IJ\\ d{f}_{Total}=N-1\end{array}$

The sums of squares are:

$\begin{array}{l}S{S}_{Total}=S{S}_A+S{S}_B+S{S}_{A\times B}+S{S}_{Within}\\ S{S}_{Within}={\displaystyle \sum SS}\\ S{S}_{Between}={\displaystyle \sum \frac{T^2}{n}}-\frac{G^2}{N}\\ S{S}_A={\displaystyle \sum \frac{T_{Rows}^2}{n_{Rows}}}-\frac{G^2}{N}\\ S{S}_B={\displaystyle \sum \frac{T_{Columns}^2}{n_{Columns}}}-\frac{G^2}{N}\\ S{S}_{A\times B}=S{S}_{Between}-S{S}_A-S{S}_B\end{array}$

The MS values needed for the F-ratios are:

$\begin{array}{l}M{S}_A=\frac{S{S}_A}{d{f}_A}\\ M{S}_B=\frac{S{S}_B}{d{f}_B}\\ M{S}_{A\times B}=\frac{S{S}_{A\times B}}{d{f}_{A\times B}}\\ M{S}_{Within}=\frac{S{S}_{Within}}{d{f}_{Within}}\end{array}$

The F-ratios are:

$\begin{array}{l}{F}_A=\frac{M{S}_A}{M{S}_{Within}}\\ F_B=\frac{M{S}_B}{M{S}_{Within}}\\ F_{A\times B}=\frac{M{S}_{A\times B}}{M{S}_{Within}}\end{array}$

Calculation:

Part a:

For a two-factor study, there are three separate hypotheses, the two main effects and the interaction.

For Factor A, the null hypothesis states that there is no difference in the scores for male vs. female. In symbols,

$H_0:{\mu }_{Male}={\mu }_{Female}$

For Factor B, the null hypothesis states that there is no difference in the scores at different treatments. In symbols,

$H_0:{\mu }_I={\mu }_{II}$

For the $A\times B$ interaction, the null hypothesis states that factor A does not depend on the levels of factor B (and B does not depend on A).

$H_0:\text{factor A does not depend on the levels of factor B }\left(\text{and B does not depend on A}\right)$

The degrees of freedoms are:

$\begin{array}{l}d{f}_A=I-1=2-1=1\\ d{f}_B=J-1=3-1=2\\ d{f}_{A\times B}=\left(I-1\right)\left(J-1\right)=\left(1\right)\left(2\right)=2\\ d{f}_{Between}=d{f}_A+d{f}_B+d{f}_{A\times B}=1+2+2=5\\ d{f}_{Within}=N-IJ=18-\left(2\right)\left(3\right)=12\\ d{f}_{Total}=N-1=18-1=17\end{array}$

The sums of squares are:

$\begin{array}{c}S{S}_{Within}={\displaystyle \sum SS}\\ =14+26+8+8+14+26\\ =\text{96}\end{array}$ $\begin{array}{c}S{S}_{Between}={\displaystyle \sum \frac{T^2}{n}}-\frac{G^2}{N}\\ =\left[\frac{{\left(9\right)}^2}{3}+\frac{{\left(18\right)}^2}{3}+\frac{{\left(27\right)}^2}{3}+\frac{{\left(9\right)}^2}{3}+\frac{{\left(36\right)}^2}{3}+\frac{{\left(45\right)}^2}{3}\right]-\frac{{144}^2}{18}\\ =1512-1152\\ =360\end{array}$ $\begin{array}{c}S{S}_A={\displaystyle \sum \frac{T_{Rows}^2}{n_{Rows}}}-\frac{G^2}{N}\\ =\left[\frac{{54}^2}{9}+\frac{{90}^2}{9}\right]-\frac{{144}^2}{18}\\ =1224-1152\\ =72\end{array}$ $\begin{array}{c}S{S}_B={\displaystyle \sum \frac{T_{Columns}^2}{n_{Columns}}}-\frac{G^2}{N}\\ =\left[\frac{{18}^2}{6}+\frac{{54}^2}{6}+\frac{{72}^2}{6}\right]-\frac{{180}^2}{60}\\ =\text{1404}-1152\\ =252\end{array}$ $\begin{array}{c}S{S}_{A\times B}=S{S}_{Between}-S{S}_A-S{S}_B\\ =360-72-252\\ =36\\ S{S}_{Total}=S{S}_{Between}+S{S}_{Within}\\ =360+96\\ =456\end{array}$

The MS values needed for the F-ratios are:

$\begin{array}{l}M{S}_A=\frac{S{S}_A}{d{f}_A}=\frac{72}{1}=72\\ M{S}_B=\frac{S{S}_B}{d{f}_B}=\frac{252}{2}=126\\ M{S}_{A\times B}=\frac{S{S}_{A\times B}}{d{f}_{A\times B}}=\frac{36}{2}=18\\ M{S}_{Within}=\frac{S{S}_{Within}}{d{f}_{Within}}=\frac{96}{12}=8\end{array}$

The F-ratios are:

$\begin{array}{l}{F}_A=\frac{M{S}_A}{M{S}_{Within}}=\frac{72}{8}=9.00\\ F_B=\frac{M{S}_B}{M{S}_{Within}}=\frac{126}{8}=15.75\\ F_{A\times B}=\frac{M{S}_{A\times B}}{M{S}_{Within}}=\frac{18}{8}=2.25\end{array}$

The two factor ANOVA table is shown below:

Source	SS	df	MS	F
Between Treatments	360	5
Factor $A$	72	1	72	9.00
Factor $B$	252	2	126	15.75
$A\times B$ Interaction	36	2	18	2.25
Within Treatments	96	12	8
Total	456	17

From F table, at $\alpha =0.05$, the critical values are:

$\begin{array}{l}F-critica{l}_A=F_{.05,1,12}=4.75\\ F-critica{l}_B=F_{.05,2,12}=3.88\\ F-critica{l}_{A\times B}=F_{.05,2,12}=3.88\end{array}$

Part b:

We test the simple main effect of factor A for each level of factor B.

For treatment I:

Treatments I
Male	Female
$\begin{array}{c}n=3\\ M=3\\ SS=14\\ T=9\end{array}$	$\begin{array}{c}n=3\\ M=12\\ SS=14\\ T=36\end{array}$	$\begin{array}{c}G=9+9\\ =18\\ N=3+3\\ =6\end{array}$

$\begin{array}{c}S{S}_{Between}={\displaystyle \sum \frac{T^2}{n}}-\frac{G^2}{N}\\ =\left[\frac{{\left(9\right)}^2}{3}+\frac{{\left(9\right)}^2}{3}\right]-\frac{{18}^2}{6}\\ =54-54\\ =0\\ d{f}_{Between}=I-1\\ =2-1\\ =1\\ M{S}_{Between}=\frac{S{S}_{Between}}{d{f}_{Between}}\\ =\frac{0}{1}\\ =0\end{array}$ Using $M{S}_{Within}=8$ from the original two-factor analysis, the final F-ratio is:

$\begin{array}{c}F=\frac{M{S}_{Between}}{M{S}_{Within}}\\ =\frac{0}{8}\\ =0\end{array}$ From F table, at $\alpha =0.05$, the critical value is:

$F-critical=F_{.05,1,12}=4.75$

There is no significant difference between males and females in treatment I.

For treatment II:

Treatments II
Male	Female
$\begin{array}{c}n=3\\ M=6\\ SS=26\\ T=18\end{array}$	$\begin{array}{c}n=3\\ M=12\\ SS=14\\ T=36\end{array}$	$\begin{array}{c}G=18+36\\ =54\\ N=3+3\\ =6\end{array}$

$\begin{array}{c}S{S}_{Between}={\displaystyle \sum \frac{T^2}{n}}-\frac{G^2}{N}\\ =\left[\frac{{\left(18\right)}^2}{3}+\frac{{\left(36\right)}^2}{3}\right]-\frac{{54}^2}{6}\\ =540-486\\ =54\\ d{f}_{Between}=I-1\\ =2-1\\ =1\\ M{S}_{Between}=\frac{S{S}_{Between}}{d{f}_{Between}}\\ =\frac{54}{1}\\ =54\end{array}$ Using $M{S}_{Within}=8$ from the original two-factor analysis, the final F-ratio is:

$\begin{array}{c}F=\frac{M{S}_{Between}}{M{S}_{Within}}\\ =\frac{54}{8}\\ =6.75\end{array}$ From F table, at $\alpha =0.05$, the critical value is:

$F-critical=F_{.05,1,12}=4.75$

There is significant difference between males and females in treatment II.

For treatment III:

Treatments III
Male	Female
$\begin{array}{c}n=3\\ M=9\\ SS=8\\ T=27\end{array}$	$\begin{array}{c}n=3\\ M=15\\ SS=26\\ T=45\end{array}$	$\begin{array}{c}G=27+45\\ =72\\ N=3+3\\ =6\end{array}$

$\begin{array}{c}S{S}_{Between}={\displaystyle \sum \frac{T^2}{n}}-\frac{G^2}{N}\\ =\left[\frac{{\left(27\right)}^2}{3}+\frac{{\left(45\right)}^2}{3}\right]-\frac{{72}^2}{6}\\ =918-864\\ =54\\ d{f}_{Between}=I-1\\ =2-1\\ =1\\ M{S}_{Between}=\frac{S{S}_{Between}}{d{f}_{Between}}\\ =\frac{54}{1}\\ =54\end{array}$ Using $M{S}_{Within}=8$ from the original two-factor analysis, the final F-ratio is:

$\begin{array}{c}F=\frac{M{S}_{Between}}{M{S}_{Within}}\\ =\frac{54}{8}\\ =6.75\end{array}$ From F table, at $\alpha =0.05$, the critical value is:

$F-critical=F_{.05,1,12}=4.75$

There is significant difference between males and females in treatment III.

Conclusion:

1. There is a significant main effect for gender (Factor A) and treatment (Factor B) but there is no significant interaction.
2. There is a significant difference between males and females in treatments II and III but not in treatment I.

Justification:

Since $F_A=9.00$ is greater than 4.75, there is a significant main effect for gender (Factor A).

Since $F_B=15.75$ is greater than 3.88, there is a significant main effect for treatment (Factor B).

Since $F_{A\times B}=2.25$ is less than 3.88, there is no significant interaction.