1. T WO W AY ANOVA W ITH R EPLICATION  Also called a Factorial Experiment.  Factorial Experiment is used to evaluate 2 or more factors simultaneously.  Replication means an independent repeat of each factor combination.  The purpose of factorial experiment is to examine: 1. The effect of factor A on the dependent variable, y. 2. The effect of factor B on the dependent variable, y along with 3. The effects of the interactions between different levels of the factors on the dependent variable, y.  Interaction exists when the effect of a level for one factor depends on which level of the other factor is present.  Advantages of Factorial Experiment over one factor at a time (one-way ANOVA) – more efficient & allow interactions to be detected.

2. The effect model for a factorial experiment can be written as:

3. There are three sets of hypothesis: 1. Factor A effect: 2. Factor B effect: 3. Interaction effect:

4.  The results obtained in this analysis are summarized in the following ANOVA table:

5. Two way Factorial Treatment Structure

6. where

8. Example: The two-way table gives data for a 2x2 factorial experiment with two observations per factor – level combination. Construct the ANOVA table for this experiment and do a complete analysis at a level of significance 0.05. Factor A Factor B Level12 129.6 35.2 47.3 42.1 212.9 17.6 28.4 22.7

9. Solution: Factor A Factor B Level12 129.6 35.2 64.8 47.3 42.1 89.4 212.9 17.6 30.5 28.4 22.7 51.1 154.2 81.6 95.3140.5235.8

10. Solution: 1. Set up hypothesis Factor A effect: Factor B effect: Interaction effect:

13. 2. Calculation (given the ANOVA table is as follows): 3. With = 0.05 we reject if : Source of Variation SSdfMSF A658.8451 46.652 B255.381 18.083 AB2120.1416 Error56.49414.1225 Total972.7157

14. 4. From ANOVA/table F, the critical and F effects are given as follow: 5. Factor A : since, thus we reject We conclude that the difference level of A effect the response Factor B : since, thus we reject We conclude that the difference level of B effect the response Interaction: since, thus we failed to reject We conclude that no interaction between factor A and factor B.

15. Exercise: In a study to determine which are the important source of variation in an industrial process, 3 measurements are taken on yield for 3 operators chosen randomly and 4 batches a raw materials chosen randomly. It was decided that a significance test should be made at the 0.05 level of significance to determine if the variance components due to batches, operators, and interaction are significant. In addition, estimates of variance components are to be computed. The data are as follows, with the response being percent by weight. Batch 1234 Operator 166.9 68.1 67.2 68.3 67.4 67.7 69.0 69.8 67.5 69.3 70.9 71.4 266.3 65.4 65.8 68.1 66.9 67.6 69.7 68.8 69.2 69.4 69.6 70.0 365.6 66.3 65.2 66.0 66.9 67.3 67.1 66.2 67.4 67.9 68.4 68.7

16. Perform the analysis of variance of this experiment at level of significance 0.05. State your conclusion