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 \(\alpha=.05 .\) $$\begin{array}{ccccc}\hline & \multicolumn{4}{c} {\text { Levels of Factor A }} \\\\\cline { 2 - 5 } \text { Levels of } & & & & \\\\\text { Factor B } & \mathrm{A} & \mathrm{B} & \mathrm{C} & \text { Total } \\\\\hline 1 & 5,7 & 9,7 & 4,6 & 38 \\\2 & 8,7 & 12,13 & 7,10 & 57 \\\3 & 14,11 & 8,9 & 12,15 & 69 \\\\\hline \text { Total } & 52 & 58 & 54 & 164 \\\\\hline\end{array}$$

Short answer

Answer: No, there is not a significant interaction between factors A and B, and there is no significant effect of either factor A or B on the response at a significance level of 0.05.

Step-by-step solution

Calculate cell, row, column, and grand means

Compute the mean values for each cell, row, column, and the grand mean of all data points. $$\begin{array}{ccccc}\hline & \multicolumn{4}{c} {\text { Levels of Factor A}} \\\\\cline { 2 - 5 } \text { Levels of } & & & & \\\\\text { Factor B } & \mathrm{A} & \mathrm{B} & \mathrm{C} & \text { Mean } \\\\\hline 1 & 6 & 8 & 5 & 6.33 \\\2 & 7.5 & 12.5 & 8.5 & 9.5 \\\3 & 12.5 & 8.5 & 13.5 & 11.5 \\\\\hline \text { Mean } & 8.67 & 9.67 & 9 & 9.11 \\\\\hline\end{array}$$

Calculate sums of squares

Compute the sum of squares for factor A (SSA), factor B (SSB), interaction (SSAB), and total (SST). $$\text{SSA} = (6-9.11)^2+(6.33-9.11)^2+(9.5-9.11)^2+(7.5-9.11)^2+(11.5-9.11)^2+(12.5-9.11)^2 = 2.017$$ $$\text{SSB} = (8.67-9.11)^2+(9.67-9.11)^2+(9-9.11)^2= 0.337$$ $$\text{SSAB} = (6-8.67)^2+(8-9.67)^2+(5-9)^2+(7.5-8.67)^2+(12.5-9.67)^2+(8.5-9)^2 +(12.5-8.67)^2+(8.5-9.67)^2+(13.5-9)^2 = 5.429$$ $$\text{SST} = (5-9.11)^2+(7-9.11)^2+\cdots+(12-9.11)^2+(15-9.11)^2=43.109$$

Calculate degrees of freedom

Determine the degrees of freedom for factor A (dfA), factor B (dfB), interaction (dfAB), and total (dfT). $$\text{dfA} = 3-1 = 2$$ $$\text{dfB} = 3-1 = 2$$ $$\text{dfAB} = (3-1)(3-1) = 4$$ $$\text{dfT} = 9-1 = 8$$

Calculate the mean squares

Using the sums of squares and degrees of freedom, calculate the mean square for factor A (MSA), factor B (MSB), and interaction (MSAB). $$\text{MSA} = \frac{\text{SSA}}{\text{dfA}} = \frac{2.017}{2} = 1.0085$$ $$\text{MSB} = \frac{\text{SSB}}{\text{dfB}} = \frac{0.337}{2} = 0.1685$$ $$\text{MSAB} = \frac{\text{SSAB}}{\text{dfAB}} = \frac{5.429}{4} = 1.35725$$

Calculate F statistics

Calculate the F statistics for factor A (FA), factor B (FB), and interaction (FAB). $$F_A = \frac{\text{MSA}}{\text{MSAB}}= \frac{1.0085}{1.35725} = 0.74326 $$ $$F_B = \frac{\text{MSB}}{\text{MSAB}}= \frac{0.1685}{1.35725} = 0.1241 $$

Find the critical F-values and draw conclusion

Using the F-distribution table and comparing with critical values, determine if there is a significant interaction or if factors A or B have a significant effect on the response. For \(\alpha = 0.05\), the critical values for df = (2,4) are: $$F_A^* = F_B^* = F_{0.95, 2, 4} = 6.94$$ Since \(F_A = 0.74326 < 6.94\) and \(F_B = 0.1241 < 6.94\), there isn't a significant interaction between factors A and B, nor is there a significant effect of either factor A or B on the response.

Key concepts

Sums of Squares Calculation

Understanding how to calculate sums of squares (SS) is a fundamental part of ANOVA interaction analysis. The sum of squares is a measure of variation within your data. It tells us how much the observed values differ from the mean of those values. Think of it as a way to quantify the spread of your data points.

When running an ANOVA, you’ll come across several types of SS. For factor A (SSA), it represents the variation due to the factor A levels, while for factor B (SSB), it's due to factor B levels. The SS for their interaction (SSAB) measures how much variance is explained by the combined effects of factors A and B that cannot be attributed to either factor individually. Lastly, the total SS (SST) represents the overall variation in the data.

To calculate each, subtract the mean of a particular level from each data point within that level, square the result, and sum those squared differences up. The process is akin to how you would calculate the variance, but without dividing by the total number of observations.

Mean Squares Calculation

Once we have our sums of squares, the next step in our ANOVA is to find the mean squares (MS). The mean square is essentially an adjusted version of the SS, accounting for the number of degrees of freedom associated with the source of variance. It reflects the average variation due to that source.

To calculate the mean square for a factor or interaction, divide its respective sum of squares by its degrees of freedom (df). This is analogous to finding the average, but in the context of variance. For instance, the mean square of factor A (MSA) is the SSA divided by dfA, which in our exercise was found to be \(\frac{2.017}{2} = 1.0085\). These mean squares are especially important for the next step: computing F-statistics.

F-Statistic Computation

The F-statistic is the ratio of two mean square values, and it's used to test hypotheses in ANOVA. Specifically, an F-statistic is computed by dividing the mean square of a source of variance (like a factor or an interaction) by the mean square of the error or a relevant comparison mean square. This result tells us if the variance associated with the source is statistically significant.

In our case, we calculate the F-statistics for factors A and B, as well as for the interaction effect. It's the division of the mean square of factor A (MSA) by the mean square of the interaction (MSAB), and similarly for factor B and the interaction. Our F-statistics help us decide whether factors A and B, and their interaction, are contributing significantly to the variation we see in our response variable.

Significant Effects Testing

After computing the F-statistics, the next step is to determine whether these statistics indicate significant effects. We do this by comparing our calculated F values to critical values from the F-distribution table, which vary based on our chosen alpha level (commonly set at 0.05 for a 95% confidence level) and our degrees of freedom.

If our calculated F-statistic is greater than the critical value from the table, we have evidence to reject our null hypothesis, which suggests that the factor or interaction does indeed have a statistically significant effect on our response variable. In the exercise, neither factor A nor B nor the interaction had F-statistics exceeding the critical value, meaning we did not find significant effects.