1. Lecture 12 Today: 4.2, 4.3-4.6 Next day: more 4.3-4.6 Assignment #4: Chapter 4 - 13 (a,b), 14, 15, 23, additional question at end of these notes Due in 2 weeks

2. Example Speedometer cables can be noisy because of shrinkage in the plastic casing material An experiment was conducted to find out what caused shrinkage Engineers started with 6 different factors: –A braiding tension –B wire diameter –C liner tension –D liner temperature –E coating material –F melt temperature

3. Example Response is percentage shrinkage per specimen There were two levels of each factor A 2 6-2 fractional factorial The purpose of such an experiment is to determine which factors impact the response

4. Example Constructing the design –Write down the 16 run full factorial –Use interaction columns to set levels of the other 2 factors Which interaction columns do we use? Table 4A.2 gives 16 run minimum aberration (MA) designs –E=ABC; F=ABD

5. Example

6. Results

7. Example Which effects can we estimate? Defining Contrast Sub-Group: I=ABCE=ABDF=CDEF Word-Length Patter: Resolution:

8. Example Effect Estimates and QQ-Plot:

9. Comments Use defining contrast subgroup to determine which effects to estimate Can use qq-plot or Lenth’s method to evaluate the significance of the effects Fractional factorial designs allow you to explore many factors in relatively few trials Trade-off run-size for information about interactions

10. Techniques for Resolving Ambiguities Suppose the experiment in the previous example was performed and the AC=BE interaction was identified as significant (in addition to the A and E main effects) Which is the important interaction AC or BE or both? Prior knowledge may indicate that one of the effects is not important Can conduct a follow-up experiment

11. Optimal Design Approach (4.4.2) Can perform a follow-up experiment to “de-alias” the AC and BE interaction, but what treatments should be run? Would like to estimate the model with all potentially significant effects –A, E, AC, BE The experiment is not completely randomized since the follow-up runs are performed only after original experiment –Include a block effect Model:

12. Optimal Design Approach (4.4.2) The best set of new trials should optimize some design criterion Should estimate the model of interest in best possible manner Already have initial (say 16) trials, so design criterion is driven by original experiment and the model D-optimality: Motivation:

13. Optimal Design Approach (4.4.2) D s -optimality:

14. Optimal Design Approach (4.4.2) Algorithm:

15. Assignment Question Suppose in the cable shrinkage example, effects A, E and AC=BE are identified as signifincat To resolve the aliasing of the interaction effects, a follow-up experiment with 4 trials is to be performed What 4 trails should be performed? Use the D-optimality criterion and report the value of D max