# Example of Balanced ANOVA

A manufacturing engineer ran an experiment to determine how several conditions affect the thickness of a coating substance. Three different operators ran the experiment twice. Each operator measured the thickness twice for each time and setting.

Because the design is balanced, the analyst uses balanced ANOVA to determine whether time, operator, and machine setting affect coating thickness.

1. Open the sample data, CoatingThickness.MTW.
2. Choose Stat > ANOVA > Balanced ANOVA.
3. In Responses, enter Thickness.
4. In Model, enter Time Operator Setting Time*Operator Time*Setting Operator*Setting.
5. In Random factors, enter Operator.
6. Click Results.
7. Select Display expected mean squares and variance components.
8. In Display means corresponding to the terms, enter Time Setting Time*Setting.
9. Click OK in each dialog box.

## Interpret the results

Minitab displays a list of factors, with their type (fixed or random), number of levels, and values. The Analysis of Variance table displays p-values for all of the terms in the model. The low p-values for setting and all of the interaction effects indicate that these terms are significant at the 0.05 level.

Setting is a fixed factor and this main effect is significant. This result indicates that the mean coating thickness is not equal for all machine settings.

Time*Setting is an interaction effect that involves two fixed factors. This interaction effect is significant which indicates that the relationship between each factor and the response depends on the level of the other factor. In this case, you should not interpret the main effects without considering the interaction effect.

The means table shows how the mean thickness varies by each level of time (morning and evening), each machine setting, and by each combination of time and machine setting. Setting is statistically significant and the means differ between the machine settings. However, because the Time*Setting interaction term is also statistically significant, do not interpret the main effects without considering the interaction effects. For example, the table for the interaction term shows that with a setting of 44, time 2 is associated with a thicker coating. However, with a setting of 52, time 1 is associated with a thicker coating.

Operator is a random factor and all interactions that include a random factor are considered to be random. If a random factor is significant, you can conclude that the factor contributes to the amount of variation in the response. Operator is not significant at the 0.05 level, but the interaction effects that include operator are significant. These interaction effects indicate that the amount of variation that operator contributes to the response depends on the value of both time and machine setting.

### ANOVA: Thickness versus Time, Operator, Setting

Factor Information Factor Type Levels Values Time Fixed 2 1, 2 Operator Random 3 1, 2, 3 Setting Fixed 3 35, 44, 52
Analysis of Variance for Thickness Source DF SS MS F P Time 1 9.0 9.00 0.29 0.644 Operator 2 1120.9 560.44 4.28 0.081 x Setting 2 15676.4 7838.19 73.18 0.001 Time*Operator 2 62.0 31.00 4.34 0.026 Time*Setting 2 114.5 57.25 8.02 0.002 Operator*Setting 4 428.4 107.11 15.01 0.000 Error 22 157.0 7.14 Total 35 17568.2 x Not an exact F-test.
Model Summary S R-sq R-sq(adj) 2.67140 99.11% 98.58%
Error Terms for Tests Variance Error Expected Mean Square for Each Source component term Term (using unrestricted model) 1 Time 4 (7) + 6 (4) + Q[1, 5] 2 Operator 35.789 * (7) + 4 (6) + 6 (4) + 12 (2) 3 Setting 6 (7) + 4 (6) + Q[3, 5] 4 Time*Operator 3.977 7 (7) + 6 (4) 5 Time*Setting 7 (7) + Q[5] 6 Operator*Setting 24.994 7 (7) + 4 (6) 7 Error 7.136 (7) * Synthesized Test.
Error Terms for Synthesized Tests Synthesis of Source Error DF Error MS Error MS 2 Operator 5.12 130.9747 (4) + (6) - (7)
Means Time N Thickness 1 18 67.7222 2 18 68.7222
Setting N Thickness 35 12 40.5833 44 12 73.0833 52 12 91.0000
Time*Setting N Thickness 1 35 6 40.6667 1 44 6 70.1667 1 52 6 92.3333 2 35 6 40.5000 2 44 6 76.0000 2 52 6 89.6667
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