Question 9.15: Refer to Example 9.14. Construct an ANOVA table. For each ef...
Refer to Example 9.14. Construct an ANOVA table. For each effect and interaction, test the null hypothesis that it is equal to 0. Which factors, if any, seem most likely to have an effect on the outcome?
| ANOVA table | |||||
| Source | DF | SS | MS | F | P |
| Material | 1 | 2352.0 | 2352.0 | 10.45 | 0.012 |
| Design | 1 | 3072.0 | 3072.0 | 13.65 | 0.006 |
| Interaction | 1 | 2700.0 | 2700.0 | 12.00 | 0.009 |
| Error | 8 | 1800.0 | 225.00 | ||
| Total | 11 | 9924.0 | |||
The ANOVA table follows. The sums of squares for the effects and interactions were computed by using Equation (9.44). The error sum of squares was computed by applying Equation (9.43) to the data in Example 9.14. Each F statistic is the quotient of the mean square with the mean square for error. Each F statistic has 1 and 16 degrees of freedom.
\text{Sum of squares for an effect}=\frac{K(\text{contrast})^2}{8} (9.44)
SSE=(K-1) \sum\limits_{i=1}^8 s_i^2 (9.43)
\begin{array}{lrrrrrc}\hline \text{Source}& \text{Effect}& \text{DF}& \begin{array}{r}\text{Sum of}\\\text{Squares}\end{array}& \begin{array}{r}\text{Mean}\\\text{Square}\end{array}& F & P \\\hline A & 3.525 & 1 & 74.540 & 74.540 & 7.227 & 0.016 \\B & 4.316 & 1 & 111.776 & 111.776 & 10.838 & 0.005 \\C & 0.146 & 1 & 0.128 & 0.128 & 0.012 & 0.913 \\A B & 4.239 & 1 & 107.798 & 107.798 & 10.452 & 0.005 \\A C & 1.628 & 1 & 15.893 & 15.893 & 1.541 & 0.232 \\B C & -0.198 & 1 & 0.235 & 0.235 & 0.023 & 0.882 \\A B C & 0.906 & 1 & 4.927 & 4.927 & 0.478 & 0.499 \\\text{Error}& & 16 & 165.020 & 10.314 & & \\\text{Total}& & 23 & 480.316 & & & \\\hline\end{array}
The main effects of factors A and B, as well as the AB interaction, have small P-values. This suggests that these effects are not equal to 0 and that factors A and B do affect the outcome. There is no evidence that the main effect of factor C, or any of its interactions, differ from 0. Further experiments might focus on factors A and B. Perhaps a two-way ANOVA would be conducted, with each of the factors A and B evaluated at several levels, to get more detailed information about their effects on the outcome.