Question

Use the computing formulas to calculate the sums of squares and mean squares for the experiments described in Exercises 9-10. Enter these results into the appropriate ANOVA table and use them to find the F statistics used to test for a significant interaction between factors \(A\) and \(B\). If the interaction is not significant, test to see whether factors A or B have a significant effect on the response. Use \(\alpha=.05 .\) $$\begin{array}{cccc}\hline & \multicolumn{3}{c} {\text { Levels of Factor A }} \\\\\cline { 2 - 4 } \text { Levels of } & & & \\\\\text { Factor B } & 1 & 2 & \text { Total } \\\\\hline 1 & 2.1,2.7, & 3.7,3.2, & 23.1 \\\& 2.4,2.5 & 3.0,3.5 & \\\2 & 3.1,3.6, & 2.9,2.7, & 24.3 \\\& 3.4,3.9 & 2.2,2.5 & \\\\\hline \text { Total } & 23.7 & 23.7 & 47.4 \\\\\hline\end{array}$$

Short answer

Based on the two-way ANOVA analysis, there is a significant interaction between factors A and B. This means that the relationship between the dependent variable and the factors is not independently influenced by each factor; rather, the factors interact with each other to affect the dependent variable. Due to this interaction, it is not appropriate to examine the main effects of factors A and B independently.

Step-by-step solution

Calculate the sums of squares (SS)

First, we need to calculate the overall mean, the mean for each level, and the mean for the combinations of levels. Using the given data, we can find the following means: - Overall Mean (\(\bar{Y}_{..}\)): \(\frac{47.4}{8} = 5.925\) - Level A1 Mean (\(\bar{Y}_{1.}\)): \(\frac{23.7}{4} = 5.925\) - Level A2 Mean (\(\bar{Y}_{2.}\)): \(\frac{23.7}{4} = 5.925\) - Level B1 Mean (\(\bar{Y}_{.1}\)): \(\frac{23.7}{4} = 5.925\) - Level B2 Mean (\(\bar{Y}_{.2}\)): \(\frac{23.7}{4} = 5.925\) - Level A1B1 Mean: \(\frac{9.7}{4} = 2.425\) - Level A1B2 Mean: \(\frac{14.1}{4} = 3.525\) - Level A2B1 Mean: \(\frac{14}{4} = 3.5\) - Level A2B2 Mean: \(\frac{10.3}{4} = 2.575\) Now, we can calculate the sums of squared deviations for each level and their combinations: - Sum of Squares for Level A (SSA): \(4 \times ( (5.925 - 5.925)^2 + (5.925 - 5.925)^2 ) = 0\) - Sum of Squares for Level B (SSB): \(4 \times ( (5.925 - 5.925)^2 + (5.925 - 5.925)^2 ) = 0\) - Sum of Squares for Interaction (SSAB): \([((2.425 - 5.925)^2 + (3.525 - 5.925)^2 + (3.5 - 5.925)^2 + (2.575 - 5.925)^2)] \times 2 = 25.13\) - Sum of Squares within (SSE): \([(2.1-2.425)^2 + (2.7-2.425)^2 + \cdots+(3.9-3.5)^2 +(2.2-3.5)^2 +(2.5-3.5)^2] = 4.86\)

Calculate the mean squares for factors A, B, and interaction

Next, we calculate the mean squares for factors A, B, and their interaction: - Mean Square for Factor A (MSA): \(\frac{SSA}{df_A} = \frac{0}{(2-1)} = 0\) - Mean Square for Factor B (MSB): \(\frac{SSB}{df_B} = \frac{0}{(2-1)} = 0\) - Mean Square for Interaction (MSAB): \(\frac{SSAB}{df_{AB}} = \frac{25.13}{(1 \times 1)} = 25.13\) - Mean Square within (MSE): \(\frac{SSE}{df_{\mbox{within}}} = \frac{4.86}{(8-1)} = 0.6933\)

Calculate the F-statistics

Now, we can calculate the F-statistics to test for a significant interaction between factors A and B: - F-statistic for Interaction (FAB): \(\frac{MSAB}{MSE} = \frac{25.13}{0.6933} = 36.24\)

Determine the significance of the interaction and main effects

Using the F-statistic and the given confidence level (\(\alpha = 0.05\)), we can test for a significant interaction between factors A and B. The critical F-value for an interaction effect with \((1,6)\) degrees of freedom and a \(0.05\) significance level is \(5.99\). Since \(FAB = 36.24 > 5.99\), there is a significant interaction between factors A and B. In conclusion, there is a significant interaction between factors A and B. Therefore, the main effects of factors A and B should not be individually tested, as it would be difficult to interpret their separate effects when there is an interaction between them.

Key concepts

Sums of Squares

In ANOVA (Analysis of Variance), the term 'sums of squares' represents a critical component and refers to the total squared variation within the data. It helps in partitioning the overall variability into components associated with specific factors and their interactions. The greater the sums of squares, the greater the variation that can be attributed to that particular factor or interaction.

For instance, in our exercise, we calculate the sums of squares for different factors, such as the Factor A (SSA), Factor B (SSB), and their interaction (SSAB), as well as the error or residual (SSE). These values are essential in understanding how much each factor contributes to the total variability observed in the experimental data. In the given solution, the sums of squares for Factors A and B were zero since their means equal the overall mean, signifying no variation due to these factors alone. However, the interaction between Factors A and B (SSAB) had a substantial sum of squares, indicating a significant variability due to the interplay between these two factors.

Mean Squares

The 'mean squares' come into play after calculating the sums of squares. They are obtained by dividing the sums of squares by their respective degrees of freedom. This step normalizes the sums of squares and allows us to compare the variance due to each factor on a per-unit basis. Mean squares are used to compute the F-statistic, which we use to test hypotheses.

In our worked example, the Mean Square for Interaction (MSAB) was calculated by dividing the interaction sums of squares (SSAB) by its degrees of freedom. Since there is significant variability indicated by the non-zero SSAB, the MSAB helps to determine if this variability is greater than what could be attributed to random chance within the variation of the dataset (SSE).

F-statistic

The F-statistic is a ratio that compares the model's explained variance with the unexplained variance, essentially testing whether the group means are significantly different from each other. It plays a pivotal role in conducting hypothesis tests in ANOVA. To compute the F-statistic, we divide the mean square of a factor by the mean square of the error (MSE).

In the context of our exercise, the F-statistic for Interaction (FAB) was quite high, suggesting that the variability due to the interaction of factors A and B is significantly greater than the variability due to random chance. This finding allows us to infer that the interaction effect is indeed meaningful and should be further considered in our analysis.

Significant Interaction

A significant interaction in ANOVA indicates that the combined effect of two factors on the response variable is not simply additive; the effect of one factor depends on the level of the other factor. Determining significant interaction is crucial as it can change the interpretation of the main effects, and in such cases, it becomes less meaningful to look at these effects in isolation.

In the given example, the F-statistic for interaction was much larger than the critical F-value at the \(\alpha = 0.05\) level. This result implies a significant interaction between Factors A and B, meaning that their combined effects on the response variable are not additive and the levels of one factor affect the impact of the other factor. Consequently, this interaction needs to be considered when deciphering the results of the experiment, and the main effects of A and B should not be interpreted separately.