1. Prepared by Res.Asst. Şura TOPTANCI
Two-way ANOVA
Prepared by Res.Asst. Şura TOPTANCI

2. Two-way ANOVA compares means in groups of two different factors.
Two-way ANOVA also considers interactions between the two different factors.
In a Two-Way Analysis of Variance design, two factors are used to explain the variability in the response variable. We deal with the two factors by fixing them at different levels. We refer to the two factors as factor A and factor B. If factor A has n levels and factor B has m levels, we refer to the design as an factorial design (Pearson, 2013).

3. Example
Aircraft primer paints are applied to aluminium surfaces by two methods: dipping and spraying. The purpose of the primer is to improve paint adhesion. We would like to examine the effect of paint primer type and application method on paint adhesion. Data for adhesion forces were collected for this purpose. Identify the main effects. Determine which factor has an effect on paint adhesion (α = 0.05).

4. The two factors are A: primer type and B: application method.

5. Data entry into Minitab

6. Hypothesis

7. Minitab Output Stat > ANOVA > Two Way
Analyze the data from this experiment (use α = 0.05)
P-values of primer type and application method are smaller than α = 0.05, so we can reject H0 (the null hypothesis). Both primer type and application method are significant. At least, one primer type and one application method are different at %95 confidence level. However, the p-value for interaction is greater than 0.05, so we fail to reject H0. So, we can conclude that the effect of interaction is not significant.
%95 güven seviyesinde en az bir boya tipi diğerlerinden farklıdır.
%95 güven seviyesinde iki metot birbirinden farklıdır.
%95 güven seviyesinde etkileşimin etkisi önemli değildir.
İki faktörün de bağımlı değişken yapışma kuvveti üzerinde etkisi istatistiksel olarak anlamlı ama etkileşimleri anlamlı çıkmamıştır.

8. Model Adequecy Checking
Errors must be normally distributed (Normality assumption)
The errors are independent (Independence assumption)
Errors have the constant variance (Constant variance assumption).
All assumptions must be satisfied.

9. Model Adequecy Checking
Analyze the residuals from this experiment.
1 vs 2 there is a significant difference.
2 vs 3 there is a significant difference.
1 vs 3 there is no significant difference.
Look at the normality probability plot for checking normality assumption. This plot has tails that do not fall exactly along a straight line passing through the center of the plot.. Some potential problems with normality assumption, but the deviation from normality does not appear severe.
To check constant variance assumption, look at residual versus fitted value (predicted value) graph. Residuals are randomly distributed through the range of fitted value. It shows constant variance.
Residual versus observation order graph is used for checking independence assumption. Independence assumption is satisfied.

10. Plot of residuals versus primer type
Store residuals and fitted values on the worksheet.
You can also construct plot of residuals vs. application method.

11. Main Effects Plot (Stat > Anova >Main effects plot)
Main effects plot is a plot of the MEAN value in each group. The means in groups of one factor are calculated ignoring the effect of other factors.
The line on the middle is the overall mean level. Mean is lower in dipping than in spraying. Mean is higher in primer type 2 than in primer types 1 or 3.

12. Interaction Plots
First graph shows the average adhesion force versus primer types. The second one demonstrates the average adhesion versus application method. These two graphs indicate that there is no interaction between two factors. We can also see that spraying is a superior application method and that primer type 2 is most effective type.

13. Tukey Test (Stat > Anova >General Linear Model)
Püskürtme metodu uygulandığında boya tipleri arasında fark mıdır?
When spreying method is applied, is there any difference between primer types?

14. (Stat > Anova >General Linear Model)
Tukey Test
(Stat > Anova >General Linear Model)
Construct a 95 percent confidence interval estimate on the mean difference among primer types?
1 vs 2 there is a significant difference.
2 vs 3 there is a significant difference.
1 vs 3 there is no significant difference.
0 içermeyen güven aralığı anlamlı farkı belirtir.
Primer type boya tipleri arasında fark var mıdır? 1-2 fark anlamlı 1-3 fark anlamsız 2-3 fark anlamlı

15. there is a significant difference.
Construct a 95 percent confidence interval estimate on the mean difference among primer types?
Dipping vs. Spraying
there is a significant difference.
Boya uygulama metotları arasında fark var mıdır? Fark anlamlı. 0 içermiyor..

16. Daldırma metodu uygulandığında boya tipleri arasında fark mıdır
Daldırma metodu uygulandığında boya tipleri arasında fark mıdır? Dipping te 1-2 fark anlamlı 1-3 fark anlamsız

17. 1 vs 2 there is no significant difference.
When spraying method is applied, is there any difference between primer types?
1 vs 2 there is no significant difference.
1 vs 3 there is no significant difference.
2 vs 3 there is a significant difference.
Püskürtme metodu uygulandığında boya tipleri arasında fark mıdır? Spraying te 1-2 fark anlamsız 1-3 fark anlamsız
Dipping 2-3 fark anlamlı

18. 1 vs 2 there is no significant difference.
When spraying method is applied, is there any difference between primer types?
1 vs 2 there is no significant difference.
1 vs 3 there is no significant difference.
2 vs 3 there is a significant difference.
Spraying te 2-3 fark anlamlı

19. 1 vs 2 there is no significant difference.
When spraying method is applied, is there any difference between primer types?
1 vs 2 there is no significant difference.
1 vs 3 there is no significant difference.
2 vs 3 there is a significant difference.
2-3 spraying p value anlamlı

20. Example 2

21. Analyze the data from this experiment. Use α = 0. 05
Analyze the data from this experiment. Use α = Is there any evidence that feed rate, depth of cut and their interaction impacts surface finish?
Draw main effects and interaction effect plots.
c) Use Tukey test to make comparisons among factors to determine specifically which feed rate and depth of cut differ in mean surface finish.
d) Analyze the residuals from this experiment.
Interpret all results and plots.