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## Fractional Factorial Designs

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**Fractional Factorial Designs**Consider a 2k, but with the idea of running fewer than 2k treatment combinations. Example: (1) 23 design- run 4 t.c.’s written as 23-1 (1/2 of 23) (2) 25 design- run 8 t.c.’s written as 25-2 (1/4 of 25) 2k designs with fewer than 2k t.c.’s are called 2-level fractional factorial designs. (initiated by D.J. Finney in 1945).**Example: Run 4 of the 8 t.c.’s in 23: a, b, c, abc**It is clear that from the(se) 4 t.c.’s, we cannot estimate the 7 effects (A, B, AB, C, AC, BC, ABC) present in any 23 design, since each estimate uses (all) 8 t.c.’s. What can be estimated from these 4 t.c.’s?**4A = -1 + a - b + ab - c +ac - bc + abc**4BC= 1 + a - b - ab - c - ac + bc + abc Consider (4A + 4BC)= 2(a - b - c + abc) or 2(A + BC)= a - b - c + abc overall: 2(A + BC)= a - b - c + abc 2(B + AC)= -a + b - c + abc 2(C + AB)= -a - b + c + abc In each case, the 4 t.c.’s NOT run cancel out. Note: 4ABC=(a+b+c+abc)-(1+ab+ac+bc) cannot be estimated.**Had we run the other 4 t.c.’s:**1, ab, ac, bc, We would be able to estimate A - BC B - AC C - AB (generally no better or worse than with + signs) NOTE: If you “know” (i.e., are willing to assume) that all interactions=0, then you can say either (1) you get 3 factors for “the price” of 2. (2) you get 3 factors at “1/2 price.”**Suppose we run these 4:**1, ab, c, abc; We would then estimate A + B C+ ABC AC + BC two main effects together usually less desirable } In each case, we “Lose” 1 effect completely, and get the other 6 in 3 pairs of two effects. { members of the pair are CONFOUNDED members of the pair are ALIASED**With 4 t.c.’s, one should expect to get only 3**“estimates” (or “alias pairs”) - NOT unrelated to “degrees of freedom being one fewer than # of data points” or “with c columns, we get (c-1) df.” In any event, clearly, there are BETTER and WORSE sets of 4 t.c.’s out of a 23. (Better & worse 23-1 designs)**Consider a 24-1 with t.c.’s**1, ab, ac, bc, ad, bd, cd, abcd A+BCD B+ACD C+ABD AB+CD AC+BD BC+AD D +ABC Can estimate: - 8 t.c.’s - Lose 1 effect - Estimate other 14 in 7 alias pairs of 2 Note:**Prospect in fractional factorial designs is attractive if in**some or all alias pairs one of the effects is KNOWN. This usually means “thought to be zero.”**“Clean” estimates of the remaining member of the pair**can then be made. For those who believe, by conviction or via selected empirical evidence, that the world is relatively simple, 3 and higher order interactions (such as ABC, ABCD, etc.) may be announced as zero in advance of the inquiry. In this case, in the 24-1 above, all main effects are CLEAN. Without any such belief, fractional factorials are of uncertain value. After all, you could get A + BCD =0, yet A could be large +, BCD large -; or the reverse; or both zero.**Despite these reservations fractional factorials are almost**inevitable in a many factor situation. It is generally better to study 5 factors with a quarter replicate (25-2= 8) than 3 factors completely (23=8). Whatever else the real world is, it’s Multi-factored. The best way to learn “how” is to work (and discuss) some examples:**Example: 25-1: A, B, C, D, E**Step 1: In a 2k-p, we must “lose” 2p-1. Here we lose 1. Choose the effect to lose. Write it as a “Defining relation” or “Defining contrast.” I = ABDE (in the plus-minus table) Step 2: Find the resulting alias pairs: *A=BDE AB=DE ABC=CDE B=ADE AC=4 BCD=ACE C=ABCDEAD=BE BCE=ACD D=ABE AE=BD E=ABD BC=4 CD=4 CE=4 - lose 1 -other 30 in 15 alias pairs of 2 -run 16 t.c.’s 15 estimates { *AxABDE=BDE**See if they’re (collectively) acceptable.**Another option (among many others): I = ABCDE (in the plus-minus table) A=4 AB=3 B=4 AC=3 C=4 AD=3 D=4 AE=3 E=4 BC=3 BD=3 BE=3 CD=3 CE=3 DE=3 4: 4 way interaction; 3: 3 way interaction. Assume we choose I = ABDE**Next Step: Find the 2 blocks using ABDE as the confounded**effect. I II 1c a ac ab abc b bc de cde ade acde abde abcde bde bcde ad acd d cd bd bcd abd abcd ae ace e ce be bce abe abce { Same process as a Confounding Scheme • Which block to run if I = ABDE (ABDE = “+”)?**Example 2:**In a 25, there are 31 effects; with 8 runs, there are 7 df & 7 estimates available 25-2A, B, C, D, E { Must “Lose” 3; other 28 in 7 alias groups of 4**Choose the 3: Like in confounding schemes, 3rd must be**product of first 2: I = ABC = BCDE = ADE A = BC = 5 = DE B = AC = 3 = 4 C = AB = 3 = 4 D = 4 = 3 = AE E = 4 = 3 = AD BD = 3 = CE = 3 BE = 3 = CD = 3 Find alias groups: Assume we use this design.**Let’s find the 4 blocks**using ABC , BCDE , ADE as confounded effects 1 2 3 4 1 a b d abd bd ad ab bc abc c bcd acd cd abcd ac de ade bde e abe be ae abde bcde abcde cde bce ace ce abce acde • Which block should we run if I=ABC=BCDE=ADE?**Block 1: (in the plus-minus table)**• the sign for ABC = “-” • the sign for BCDE = “+” • the sign for ADE = “-” } Defining relation is: I = -ABC = BCDE = -ADE Exercise: find the defining relations for the other blocks.**Analysis 2k-p Design using MINITAB**• Create factorial design: • Stat>> DOE>> Factorial • >> Create Factorial Design • Input data values. • Analyze data: • Stat>> DOE>> Factorial • >> Analyze Factorial Design**A B C D rate-1 -1 -1 -1 45 1 -1 -1 1 100-1 1 -1 1 45**1 1 -1 -1 65-1 -1 1 1 75 1 -1 1 -1 60-1 1 1 -1 80 1 1 1 1 96 Example: 24-1 withdefining relation I=ABCD.**Analysis of Variance for rate (coded units)**Source DF Seq SS Adj SS Adj MS F Main Effects 4 1663 1663 415.7 * 2-Way Interactions 3 1408 1408 469.5 * Residual Error 0 0 0 0.0 Total 7 3071**Fractional Factorial Fit: rate versus A, B, C, D**Estimated Effects and Coefficients for rate (coded units) Term Effect Coef Constant 70.750 A 19.000 9.500 B 1.500 0.750 C 14.000 7.000 D 16.500 8.250 A*B -1.000 -0.500 A*C -18.500 -9.250 A*D 19.000 9.500**Alias StructureI + A*B*C*DA + B*C*DB + A*C*DC + A*B*DD +**A*B*CA*B + C*DA*C + B*DA*D + B*C**Fractional Factorial Fit: rate versus A, C, D**Estimated Effects and Coefficients for rate (coded units) Term Effect Coef SE Coef T P Constant 70.750 0.6374 111.00 0.000 A 19.000 9.500 0.6374 14.90 0.004 C 14.000 7.000 0.6374 10.98 0.008 D 16.500 8.250 0.6374 12.94 0.006 A*C -18.500 -9.250 0.6374 -14.51 0.005 A*D 19.000 9.500 0.6374 14.90 0.004 Analysis of Variance for rate (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 3 1658.50 1658.50 552.833 170.10 0.006 2-Way Interactions 2 1406.50 1406.50 703.250 216.38 0.005 Residual Error 2 6.50 6.50 3.250 Total 7 3071.50 { We should also include alias structure like A(+BCD) for all terms.**From S.A.S:**1) 23 factorial (3 replicates for each of 8 cols): A L H Factor B Factor B L H L H 8,10, 24,28, 16,16, 28,18, 18 20 19 23 19,16 27,16, 16,25, 30,23, 16 17 22 25 L H C**Source DF Sum of Squares Mean Square Fvalue**PR>F Model 7 432.0000000 61.71428571 3.38 0.0206 Error 16 292.0000000 18.25000000 Corr. Total 23 724.0000000 Source DF Type I SS F Value PR>F DF PR>F A 1 73.50000000 4.03 0.0620 1 0.0620 B 1 253.50000000 13.89 0.0018 1 0.0018 C 1 24.00000000 1.32 0.2683 1 0.2683 A*B 1 6.00000000 0.33 0.5744 1 0.5744 A*C 1 13.50000000 0.74 0.4025 1 0.4025 B*C 1 37.50000000 2.05 0.1710 1 0.1710 A*B*C 1 24.00000000 1.32 0.2683 1 0.2683 “p-values”**A**AL AH A Low A High BL BH BL BH 36 69 72 51 60 63 51 78 BL BH BL BH 36 72 51 69 51 60 63 78 L CL CH D D DL DH CL CH H 2) 24-1**Source DF Sum of Squares Mean Square**Fval Model 7 1296.0000000 185.14285714 Error 0 0.0000000 0.00000000 Corr. Total 7 1296.0000000 Source DF Type I SS F Value PR>F DF A + BCD 1 220.50000000 . . 1 B + ACD 1 760.50000000 . . 1 C + ABD 1 72.00000000 . . 1 A*B + CD 1 18.00000000 . . 1 A*C + BD 1 112.50000000 . . 1 B*C + AD 1 40.50000000 . . 1 A*B*C + D 1 72.00000000 . . 1**Real World Example of 28-2:**Level FactorLH A Geography E W B Volume Cat. L H C Price Cat. L H D Seasonality NO YES E Shelf Space Normal Double F Price Normal 20% cut G ADV None Normal(IF) H Loc. Q Normal Prime Product Attributes { { Managerial Decision Variables**- E, F, G, H important**- B, C, D main effects, but not important - A “less” important - XY,X= B, C, D very important Y= E, F, G, H -EF,EG,EH,FG,FH,GH very important - all > 3fi’s = 0, except EFG, EFH, EGH, FGH**I = BCD = ABEFGH = ACDEFGH**Did objectives get met? A = 4 = 5 = 6 E,F,G,H = 4 = 5 = 6 (XY) BE = 3 = 4 = 7 .... DE,CE = 3 = 6 = 5 ... EF = 5 = 4 = 5 ... EFG = 6 = 3 = 4 Results:**An alternative:**I = ABCD = ABEFGH = CDEFGH A = 3 = 5 = 7 E,F,G,H = 5 = 5 = 5 (XY) BE = 4 = 4 = 6 .... DE,CE = 4 = 6 = 4 ... EF = 6 = 4 = 4 ... EFG = 7 = 3 = 3**Minimum Detectable Effects in 2k-pWhen we testfor**significance of an effect, we can also determine the power of the test.H0: A = 0Hl: A not = 0**By looking at power tables, we can determine the power of**the test, by specifying s, and, essentially, (what reduces to) D, the true value of the A effect (since D = [true A - 0], = true A).Here, we look at the issue from an opposite (of sorts) perspective:Given a value of a, and for a given value of b (or power = 1-b), along with N and n,N = r • 2k-p = # of data pointsn = degrees of freedom for error term,**We can determine the “MDE,” the Minimum Detectable**Effect (i.e., the minimum detectable D, so that the a and 1-b are achieved). The results are expressed in “s units,” since we don’t know s. Note: if r = 1 (no replication), there may be few (or even no) df left for error. The tables that follow assume that there are at least 3 df for an error estimate.First, a table of df, assuming that all main effects and as many 2fi’s as possible are cleanly estimated. (All 3fi’s and higher are assumed zero).*** 23 with 8 runs complete factorial 1 df for error**(ABC)** 23 replicated twice 1 + 8 (for repl.) 9 df for error***25-1 replicated twice all 15 alias rows have mains or 2fi’s, only 16 df for error (replication)*** 25 no replication there will be 16 “3fi’s or higher” effects 16 df for error TABLES ON NEXT PAGES ASSUME DF OF THIS PAGE**TABLES 11.33 – 11.36**Of course, these tables can be used either1) find MDE, given a, 1-b, or2) find power, given MDE [required] and a.**Ex 1: 25-2, main effects and some 2fi’sa) a = .05,**1-b=.90, [b = .10] MDE = 3.5s (not good!!- effect must be very large to be detectable)b) Do a 25-1, 16 runs, with a = .05, 1-b=.90, [b=.10] MDE = 2.5s (not as bad!)Maybe settle for 1-b=.75, [b=.25] MDE = 2.0s**Ex 2:27, unreplicated, = 128 runsa = .01, 1-b=.99, [b=.01]** MDE = .88s(e.g., if s estimate = .3 microns, MDE = .264 microns)Ex 3: 27-2, unreplicated, = 32 runsa = .05, 1-b=.95, [b=.05] MDE = 1.5s