1. Stat 470-6 Today: 2-way ANOVA (Section 2.3)…2.3.1 and 2.3.2; Transformation of the response

2. Two-Way ANOVA One-way ANOVA considered impact of 1 factor with k levels (e.g. meat packaging example) Two-way ANOVA considers the impact of 2 factors with I and J levels respectively Have possible treatments for each replicate of the experiment If have n replicates, the the experiment has observations

3. Example: An experiment was run to understand the impact of two factors (Table speed and Wheel grit size) on the the strength of the ceramic material (bonded S i nitrate). (Jahanmir, 1996, NIST) Each factor has two levels (coded -1 and +1 respectively) The experiment was repeated 2 times

4. Data

5. Model:

6. Hypotheses

7. Running the Experiment Two-Way ANOVA Model is appropriate for experiments performed as completely randomized designs That is, we list the treatments (e.g., 1-8 in the ceramics example) and assign treatments to experimental units in random order The trials are in random order

8. ANOVA Table

9. Return to Ceramic Data

10. Interaction Plot

11. ANOVA Table

12. Residuals Must still do residual analysis

13. What would happen if the experiment was unreplicated (l =1)? What could we do to address this?

14. Multi-Way (or N-Way) ANOVA (Section 2.4) Can extend model to more that 2 factors Approach is the same

15. Experiment Situation Have N factors The experiment is performed as a completely randomized design Assumptions:

16. Transformations (Section 2.5) Often one will perform a residual analysis to verify modeling assumptions…and at least one assumption fails A defect that can frequently arise in non-constant variance This can occur, for example, when the data follow a non-normal, skewed distribution The F-test in ANOVA is only slightly violated In such cases, a variance stabalizing transformation may be applied

17. Transformations Several transformations may be attemted: –Y * =

18. Transformations Analyze the data on the Y * scale, choosing the transformation where: –The simplest model results, –There are no patterns in the residuals –One can interpret the transformation

19. Example An engineer wishes to study the impact of 4 factors on the rate of advance of a drill. Each of the 4 factors (labeled A-D) were studied at 2 levels

20. Example Would like to fit an N-way ANOVA to these data (main effects and 2- factor interactions only) Model:

21. Example