Design and Analysis of Experiments

Transcription

1 Design and Analysis of Experiments Part VII: Fractional Factorial Designs Prof. Dr. Anselmo E de Oliveira anselmo.quimica.ufg.br anselmo.disciplinas@gmail.com

2 2 k : increasing k the number of runs required for a complete replicate of the design outgrows the resources of most experimenters Design redundancy and the number of effects Binomial coefficient (combinations without repetition) n k = n! k! n k! where n is the number of things to choose from, and we choose k of them (no repetition, order doesn't matter)

3 2 7 full factorial design: mean = 1 main effects (n = 7, k = 1) twofactor interactions (n = 7, k = 2) threefactor interactions (n = 7, k = 3) n = 7, k = 4 n = 7, k = 5 n = 7, k = 6 n = 7, k = 7 There are only 28 (721) degrees of freedom associated with effects that are likely to be of major interest. The remaining 99 are associated with threefactor and higher interactions. 7 1 = = = = = effects

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5 If the experimenter can reasonably assume that certain highorder interactions are negligible: fractional factorial design Screening experiments The successful use of fractional factorial designs is based on three key ideas: The sparsity of effects principle: when there are several variables, the system or process is likely to be driven primarily by some of the main effects and loworder interactions The projection property: fractional factorial designs can be projected into larger designs in the subset of significant factors Sequential experimentation: it is possible to combine the runs of two (or more) fractional factorials to construct sequentially a larger design to estimate the factor effects and interactions of interest

6 k = 3 Two levels Four runs Onehalf fraction of a 2 3 design = 4 treatment combinations design

7 generator run ABC run I A B C ABC run ABC

8 ABC = generator A design is formed by selecting only those treatment combinations that have a plus in the ABC column I is also always plus I = ABC is the defining relation

9 run I A B C AB AC BC ABC 2 a 3 b 5 c 8 abc main effects interaction effects

10 It is impossible to differentiate between A and BC B and AC C and AB A = l A l BC B = l B l AC C = l C l AB aliases l A A BC l B B AC l C C AB

11 I = ABC A I = A ABC = A 2 BC como A 2 = I A = BC Similarly, B I = B ABC = AB 2 C B = AC and C I = C ABC = ABC 2 C = AB The onehalf fraction with I = ABC is the principal fraction

12 Using the other halffraction: I = ABC run I A B C AB AC BC ABC 1 (1) 4 ab 6 ac 7 bc l A A BC l B B AC l C C AB Thus, when we estimate A, B, and C with this particular fraction, we are really estimating A BC, B AC, and C AB In practice, it does not matter which fraction is actually used. Both factions belong to the same family

13 Construction of the HalfFraction 1. Write down a basic design consisting of the runs for a full 2 k1 design 2. Add the kth factor by identifying its 2 2 run A B plus and minus levels with the plus and minus signs of the highest order interaction ABC K ; I = ABC A B C = AB 2 31 ; I = ABC A B C = AB

14 Design Resolution In general, the resolution of a design is one more than the smallest order interaction that some main effect is confounded (aliased) with. If some main effects are confounded with some 2level interactions, the resolution is 3. The onehalf fraction of the 2 3 design with the defining 3 1 relation I = ABC (or I = ABC) is a 2 III design For most practical purposes, a resolution 5 design is excellent and a resolution 4 design may be adequate. Resolution 3 designs are useful as economical screening designs.

15 Projection of Fractions into Factorials Any fractional design of resolution R contains complete factorial designs in any subset of R 1 factors If an experimenter has several factors of potential interest but believes that only R 1 of them have important efects, then a fractional factorial design of resolution R is the appropriate choice of design

16 Example: Pilot Plant Filtration Rate Experiment A chemical product is produced in a pressure vessel. A factorial experiment is carried out in the pilot plant to study the factors thought to influence the filtration rate of this product. A = temperature B = pressure C = concentration of formaldehyde D = stirring rate response: filtration rate in gal/h 2 4 full factorial design (16 runs)

17 run y (1) 45 a 71 b 48 ab 65 c 68 ac 60 bc 80 abc 65 d 43 ad 100 bd 45 abd 104 cd 75 acd 86 bcd 70 abcd 96 A = C = D = AC = AD =

18 design with I = ABCD, 2 IV Main effects: run A B C D = ABC y 45 (1) 100 ad 45 bd 65 ab 75 cd 60 ac 80 bc 96 abcd A. I = A. ABCD A = A 2 BCD A = BCD B.I = B.ABCD B = AB 2 CD B = ACD C.I = C.ABCD C = ABC 2 D C = ABD D.I = D.ABCD D = ABCD 2 D = ABC Each main effect is aliased with a threefactor interaction

19 twofactor interactions: AB.I = AB.ABCD AB = A 2 B 2 CD AB = CD AC.I = AC.ABCD AC = A 2 BC 2 D AC = BD AD.I = AD.ABCD AD = A 2 BCD 2 AD = BC Every twofactor interaction is aliased with another twofactor interaction 2 3 design = 7 effects o o o 3 main 3 secondorder 1 thirdorder 2 41 design = 7 effects o o 4 main 3 secondorder

20 y 45 (1) 100 ad 45 bd 65 ab 75 cd 60 ac 80 bc 96 abcd main effect: A Twofactor interaction: AB

21 l A = 19 l B = 1.5 l C = 14 l D = 16.5 l AB = 1 l AC = 18.5 l AD = full design A = 21,625 C = 9,875 D = 14,625 AC = 18,125 AD = 16,625 Because factor B is not signficant (l B ), we drop it from consideration.

22 4 1 This 2 IV design can be projected into a single replicate of the 2 3 design in factors A, C, and D y 45 (1) 100 ad 45 bd 65 ab 75 cd C () () AC interaction: A(): concentration has a large positive effect A() : concentration has a very small effect AD interaction: A(): stirring rate has a very small effect A() : stirring rate has a large positive effect 60 ac 80 bc 96 abcd () 45 () () A 65 () D

23 > library(frf2) > design<frf2(8, randomize = FALSE, factor.names = c("a", "B", "C","D"), default.levels = c(1, 1)) > y<c(45,100,45,65,75,60,80,96) > design<add.response(design=design,response=y) > design A B C D y class=design, type= FrF2 > design.lm < lm(y~a*b*c*d,data=design) > design.mean<design.lm$coefficients[1] > design.effects<design.lm$coefficients[1]*2 > design.mean (Intercept) > design.effects A1 B1 C1 D1 A1:B1 A1:C B1:C1 A1:D1 B1:D1 C1:D1 A1:B1:C1 A1:B1:D NA NA NA NA NA A1:C1:D1 B1:C1:D1 A1:B1:C1:D1 NA NA NA

24 > cubeplot(design.lm, "A", "D", "C")

25 > MEPlot(design)

26 > IAPlot(design)

27 Regression Model y = β 0 β A A β C C β D D β AD AC β AD AD y = β 0 β 1 x 1 β 3 x 3 β 4 x 4 β 13 x 1 x 3 β 14 x 1 x 4 y = x x x x 1x x 1x 4 y = x 1 7x x x 1 x 3 9.5x 1 x 4 >> x1=1:.1:1; >> x3=x1; x4=x1; >> [X1,X3,X4]=meshgrid(x1,x3,x4); >> Y= *X17*X38.25*X49.25*X1.*X39.5*X1.*X4; >> slice(x1,x3,x4,y,[1. 1.],[1. 1.],[1. 1.]) >> xlabel("x1"); >> ylabel("x3"); >> zlabel("x4"); >> colorbar on

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29 2 kp Fractional Design 2 k p runs = 1 2p fraction of 2k full design p = 2: 2 k 2 = 1 = 1 fraction of p independent generators The defining relation consists of all columns that are equal to the identity colums, I Ex: k = 6, p = generators: I = ABCE I = BCDF I = ADEF 6 2 E = ABC F = BCD 2 IV Main effect: A A. I = A. ABCE = A. BDF = A. ADEF A = BCE = ABDF = DEF Interaction effect: AB AB. I = AB. ABCE = AB. BDF = AB. ADEF AB = CE = ADF = BDEF

30 run A B C D E = ABC F = BCD

31 Generators Summary tables of useful fractional factorial designs