II.2 Four Factors in Eight Runs

1. II.2 Four Factors in Eight Runs • Introduction • Confounding • Confounding/Aliasing • Alias Structure • Examples and Exercises • A Demonstration of the Effects of Confounding

2. II.2 Four Factors in Eight Runs: IntroductionFigure 2 - 23 Design Signs Table andFour Factors in Eight Runs Design Matrix • Let’s Compare • 23 Design Signs Table • Four Factors in Eight Runs Design Matrix

3. II.2 Four Factors in Eight Runs: Introduction Exercise-Four Factors in Eight Runs Signs Table • To compute estimates, create columns for a signs table by multiplying columns as before; some are done for you.

4. II.2 Four Factors in Eight Runs: Introduction Exercise Solution-Four Factors in Eight Runs Signs Table • The completed signs table is below

5. II.2 Four Factors in Eight Runs: IntroductionExercise-Four Factors in Eight Runs Signs Table Solution • By plan the column for D = column for ABC, so we say that for this design "D=ABC." Also, we can see from above • A = BCD • B = ACD • C = ABD • AB = CD • AC = BD • BC = AD • I = ABCD • Where "I" is a column of ones.

6. II.2 Four Factors in Eight Runs: Confounding • If we use the signs table to estimate D, what we really get is an estimate of D + ABC. (Exactly the same estimate we’d get if we had done a full 24 Design, computed D and ABC and added them.) • The two effects are “stuck” together; hence, we say they are confounded with each other (on purpose here). • Similarly, in this design, • A is confounded with BCD • B is confounded with ACD • AB is confounded with CD • ETC!

7. II.2 Four Factors in Eight Runs: Confounding • To Illustrate • We want to know if method 1 is better than method 2 for a task. Ann does method 1, Dan does method 2. If Ann’s results are better, is it because method 1 is better than method 2? Or, is Ann better than Dan? Or, is it both? The factor worker is confounded with the factor method . We can’t separate their effects. • Confounding can sometimes be a very dumb thing to do (but not always).

8. II.2 Four Factors in Eight Runs: Confounding • When we get the data and compute “D”, the result is really an estimate of D + ABC. • So, (another new word coming - duck!) “D” is a false name for the estimate - an alias. When two effects are confounded, we say they are aliases of each other. • The Alias Structure (also called the confounding structure) of the design is this table you’ve already seen (rearranged here): • I = ABCD • A = BCD • B = ACD • C = ABD • D = ABC • AB = CD • AC = BD • BC = AD

9. II.2 Four Factors in Eight Runs: An ExampleRevisit Examples 2 and 4 of Part I • Response y: Throughput (KB/sec) • The Original Experiment was a 24 Design (16 Runs) • Four Factors: A, B, C, D, performance tuning parameters such as • number of buffers • size of unix inode tables for file handling • Two Levels • In Example 2 an 8 Run Design with only Three Factors was Considered for Illustrative Purposes. The Numbers were Rounded Off for Ease of Calculation • Original Data Was In Tenths and Involved Four Factors • The Estimate of the Three-way Interaction ABC was also Estimating the Effect of D. (D and ABC are confounded/aliased.)

10. II.2 Four Factors in Eight Runs: An ExampleRevisit Examples 2 and 4 of Part I • ABCD = I determines runs in half fraction • D = ABC for these runs(Complementary half fraction is determinedby ABCD = -I, or D = -ABC, for these runs)

11. II.2 Four Factors in Eight Runs: An ExampleDesign Matrix

12. II.2 Four Factors in Eight Runs: An ExampleSigns Table • Use Eight Run Signs Table to Estimate Effects • Factor D is Assigned to the Last Column, ABC • Use Alias Structure to Determine What These Quantities Are Estimating

13. II.2 Four Factors in Eight Runs: An ExampleNormal Probability Plot • Effect A+BCD is Statistically Significant

14. II.2 Four Factors in Eight Runs: An ExampleInterpretation • Response y: Throughput (KB/sec) • Assuming BCD is negligible, you should choose A Hi (A = +) to maximize y Caution: ASSUME

15. II.2 Four Factors in Eight RunsU-Do-It Exercise: Violin Example • For the Violin Data, Pretend That a Half-fraction of the Full 24 Was Run. For your convenience, the violin data and signs table is on the next slide as well as an eight run signs table with the aliasing structure that determines the half-fraction • Find the Levels of Factors A, B, C and D that Would Have Been Run • Pick out the observed y’s for these runs. Enter these into an eight-run response table and compute the observed effects. • Compare these effects to those which were computed from the full 24

16. II.2 Four Factors in Eight RunsU-Do-It Exercise: Violin Example - Signs Tables

17. II.2 Four Factors in Eight RunsU-Do-It Exercise: Violin Solution - Completed Design Matrix • The recommended runs used for the half-fraction would assign D to column 7 of an eight run signs table • The completed eight run design matrix (with runs rearranged to standard order) is shown below • The completed signs table is shown on the next page

18. II.2 Four Factors in Eight RunsU-Do-It Exercise: Violin Solution - Completed Signs Tables • The completed signs table is below • The responses that go in standard order on A, B, C in the half-fraction are runs 1, 10, 11, 4, 13, 6, 7, and 16 (in standard order) • The half-fraction we used corresponds to those runs in the sixteen run experiment when ABCD = I

19. II.2 Four Factors in Eight RunsU-Do-It Exercise: Violin SolutionTable Comparing Estimated Effects • A table comparing estimated effect: • There is strong agreement between the two results. With the half-fraction, we would have come to essentially the same conclusions as the full 24, with half the data (and half the work.)

20. II.2 Four Factors in Eight RunsSome Notation • 24-1 is Shorthand for a Half Fraction of the 24 Design (Four Factors in Eight Runs) • The 4 stands for four factors • 24-1 = (24)(2-1 ) = half of the 24 experiment • 2k-p is Shorthand for k Factors in 2k-p Runs • k stands for number of factors • The 2-p stands for the fractionation of the 2k experiment(p=1 for a half fraction, p=2 for a quarter fraction, etc)