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

II.2 Four Factors in Eight Runs: Introduction Figure 2 - 23 Design Signs Table and Four Factors in Eight Runs Design Matrix Let’s Compare 23 Design Signs Table Four Factors in Eight Runs Design Matrix Effects versus runs. Design for four factors in 8 runs is simply proposed; we’ll motivate it later. Note that D=ABC

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. Do AD, ABD, ABCD

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

II.2 Four Factors in Eight Runs: Introduction Exercise-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. I=ABCD is the design generator. D=ABC tells you how to select runs, but also how to confound.

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! Much like previous example in Lecture 16.

II.2 Confounding of two factors in 4 runs 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).

II.2 Four Factors in Eight Runs: Confounding Aliases 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

II.2 Four Factors in Eight Runs: An Example Revisit 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.) Set aside third bullet till later.

II.2 Four Factors in Eight Runs: An Example Revisit Example 2 of Part I ABCD = I determines runs in half fraction D = ABC for these runs (Complementary half fraction is determined by ABCD = -I, or D = -ABC, for these runs) Reminder—we typically do not have access to the full factorial output. Run full example in Minitab (THROUGHPUT_FAC.MPJ)

II.2 Four Factors in Eight Runs: An Example Design Matrix

II.2 Four Factors in Eight Runs: An Example Signs 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 Show how to run this design in Minitab. Does this help convince you that the method works? Compare results from the two designs.

II.2 Four Factors in Eight Runs: An Example Normal Probability Plot Effect A+BCD is Statistically Significant Not quite like full plot (C was significant, C+ABD is not).

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

II.2 Four Factors in Eight Runs U-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 Again, we usually don’t have the full table.

II.2 Four Factors in Eight Runs U-Do-It Exercise: Violin Example - Signs Tables Select runs where ABCD=I (D=ABC).

II.2 Four Factors in Eight Runs U-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 For U-Do-It, generate fractional factorial design in Minitab. Select correct runs from full violin experiment. Analyze data and compare effects and effects plots across the two experiments.

II.2 Four Factors in Eight Runs U-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

II.2 Four Factors in Eight Runs U-Do-It Exercise: Violin Solution Table 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.)

II.2 Four Factors in Eight Runs Some 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)