5. Blocking and Confounding
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1 5. Blocking and Confounding Hae-Jin Choi School of Mechanical Engineering, Chung-Ang University 1
2 Why Blocking? Blocking is a technique for dealing with controllable nuisance variables Sometimes, it is impossible to perform all 2k factorial experiments under homogeneous condition Blocking technique is used to make the treatments are equally effective across many situation 2
3 What is Blocking? Each set of non-homogeneous conditions define a block and each replicate is run in one of blocks. If there are n replicates of the design, then each replicate is a block Each replicate is run in one of the blocks (time periods, batches of raw material, etc.) Runs within the block are randomized 3
4 Blocking a Replicated Design Consider the example from Section 6-2; k = 2 factors, n = 3 replicates This is the usual method for calculating a block sum of squares Concentration (A) Chemical Processing Catalyst (B) Filtration rate (response) SS Blocks Bi y... = å - i= =
5 ANOVA for the Blocked Design 5
6 Confounding In may case, it is impossible to perform a complete replicate of a factorial design in one block Block size smaller than the number of treatment combinations in one replicate. Confounding is a design technique for arranging experiments to make high-order interactions to be indistinguishable from(or confounded with) blocks. 6
7 Confounding in the 2k factorial Design With two factors and two blocks 1 A = [ ab + a - b - ( 1 )] 2 1 B = [ ab + b - a - ( 1 )] 2 1 AB = [ ab + ( 1 ) - a - b ] 2 A and B are Unaffected by blocks. One plus and one minus from each block -> block effect is cancelled out AB is Confounded with blocking Same sign from each block -> block effect is not cancelled out 7
8 Confounding in the 2k factorial Design With two factors and two blocks 1 A = [ ab + a - b - ( 1 )] 2 1 B = [ ab + b - a - ( 1 )] 2 1 AB = [ ab + ( 1 ) - a - b ] 2 A and B are Unaffected by blocks. One plus and one minus from each block -> block effect is cancelled out AB is Confounded with blocking Same sign from each block -> block effect is not cancelled out 8
9 Confounding in the 2k factorial Design With three factors and two blocks 9
10 How to assign the blocks in 2k factorials? Confound with High-order Interaction term 10
11 Other method for construct the blocks Linear combination with L= a 1 x 1 +a 2 x a k x k where x i = level of the i th factor a i = the exponent appearing on the i th factor in the effect to be confounded A a1 B a2 C a3 Example Confounded with ABC in 2 3 Factorial Design (a1=1, a2=1, a3=1) (1) : L = 1(0) + 1(0) + 1(0) = 0 -> Block 1 a : L = 1(1) + 1(0) + 1(0) = 1 -> Block 2 ac : L = 1(1) + 1(0) + 1(1) = 2 = 0 -> Block 1 abc : L= 1(1) + 1(0) + 1(0) = 3 = 1 -> Block 2 11
12 Example of an Unreplicated 2 k Design (repeated) 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. The factors are A = temperature, B = pressure, C = mole ratio, D= stirring rate A 2 4 factorial was used to investigate the effects of four factors on the filtration rate of a resin Experiment was performed in a pilot plant 12
13 The Table of + & - Signs Confound with interaction effect ABCD 13
14 ABCD is Confounded with Blocks Observations in block 1 are reduced by 20 units this is the simulated block effect 14
15 Effect Estimates Block (ABCD) = original ABCD - 20 = = Or Block (ABCD) = ӯ block1 - ӯ block2 15
16 The ANOVA The ABCD interaction (or the block effect) is not considered as part of the error term The reset of the analysis is unchanged from the original analysis 16
17 Without blocking, what happen?? Now the first eight runs (in run order) have filtration rate reduced by 20 units 17
18 The interpretation is harder; not as easy to identify the large effects One important interaction is not identified (AD) Failing to block when we should have causes problems in interpretation the result of an experiment and can mask the presence of real factor effects 18
19 Confounding in More than Two Blocks More than two blocks (page 282) The two-level factorial can be confounded in 2, 4, 8, (2 p, p > 1) blocks For four blocks, select two effects to confound, automatically confounding a third effect See example, page 282 Choice of confounding schemes non-trivial; see Table 7.9, page
20 General Advice About Blocking When in doubt, block Block out the nuisance variables you know about, randomize as much as possible and rely on randomization to help balance out unknown nuisance effects Measure the nuisance factors you know about but can t control It may be a good idea to conduct the experiment in blocks even if there isn't an obvious nuisance factor, just to protect against the loss of data or situations where the complete experiment can t be finished 20
21 6. Fractional Factorial Designs Hae-Jin Choi School of Mechanical Engineering, Chung-Ang University 21
22 Introduction to The 2 k-p Fractional Factorial Design Motivation for fractional factorials is obvious; as the number of factors becomes large enough to be interesting, the size of the designs grows very quickly Emphasis is on factor screening; efficiently identify the factors with large effects There may be many variables (often because we don t know much about the system) Almost always run as unreplicated factorials 22
23 Why do Fractional Factorial Designs Work? The sparsity of effects principle There may be lots of factors, but few are important System is dominated by main effects, low-order interactions The projection property Every fractional factorial contains full factorials in fewer factors Sequential experimentation Can add runs to a fractional factorial to resolve difficulties (or ambiguities) in interpretation 23
24 The One-Half Fraction of the 2 k Notation: because the design has 2 k /2 runs, it s referred to as a 2 k-1 Consider a really simple case, the Note that I =ABC 24
25 The One-Half Fraction of the 2 3 For the principal fraction, notice that the contrast for estimating the main effect A is exactly the same as the contrast used for estimating the BC interaction. This phenomena is called aliasing and it occurs in all fractional designs Aliases can be found directly from the columns in the table of + and - signs 25
26 Projection of Fractional Factorials Every fractional factorial contains full factorials in fewer factors The flashlight analogy A one-half fraction will project into a full factorial in any k 1 of the original factors 26
27 Aliasing in the One-Half Fraction of the 2 3 A = BC, B = AC, C = AB (or me = 2fi) Aliases can be found from the defining relation I = ABC by multiplication ABC is called the generator. AI = A(ABC) = A 2 BC = BC BI =B(ABC) = AC CI = C(ABC) = AB 27
28 Aliasing in the One-Half Fraction of the 2 3 Main effect 1 A = [ a -b- c + abc] 2 1 B = - a + b- c + abc 2 1 C = -a - b + c + abc 2 [ ] [ ] Two factor interaction effect 1 BC = [ a -b- c + abc] 2 1 AC = - a + b- c + abc 2 1 AB = -a - b + c + abc 2 [ ] [ ] Alias structure of effects [ A] A + BC, [ B] B + AC, [ C] C + AB 28
29 The Alternate Fraction of the I = -ABC is the defining relation Implies slightly different aliases: A = -BC, B= -AC, and C = -AB Both designs belong to the same family, defined by I = ± ABC [ A]' A - BC, [ B]' B - AC, [ C]' C - AB 29
30 Design Resolution Resolution III Designs: me = 2fi (i.e., main effect = 2 factor interaction) 3 1 example Resolution IV Designs: 2fi = 2fi example Resolution V Designs: 2fi = 3fi example 2 IIĪ IV V 30
31 Construction of a One-half Fraction 31
32 Resin Plant Experiment the Design 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. The factors are A = temperature, B = pressure, C = mole ratio, D= stirring rate A fractional factorial was used to investigate the effects of four factors on the filtration rate of a resin Generator I = ABCD 32
33 Resin Plant Experiment the Design 33
34 Aliasing the 2 IV 4-1 Factorial Design Resolution IV design with the generator I=ABCD Main effect is aliased with three factor interaction A=A 2 BCD=BCD; B=AB 2 CD=ACD; C=ABC 2 D=ABD; D=ABCD 2 =ABC; Two factor interaction is aliased with other two factor interaction AB=CD; AC=BD; AD=BC; 34
35 Resin Plant Experiment the Design Interpretation of results often relies on making some assumptions Ockham s razor Confirmation experiments can be important Adding the alternate fraction see page
36 Resin Plant Experiment MINITAB Results 36
37 Resin Plant Experiment MINITAB Results Zero degree of freedom for residuals y = b + b x + b x + b x + b x + b x x + b x x + b x x y = b + b x + b x + b x + b x x + b x x + e degree of freedom for residuals 37
38 Resin Plant Experiment MINITAB Results ŷ = bˆ + bˆ x + bˆ x + bˆ x + bˆ x x + bˆ x x yˆ æ ö x æ ö x æ ö x æ - = ö x x + æ ö x x ç è 2 ø èç 2 ø èç 2 ø èç 2 ø èç 2 ø
39 Resin Plant Experiment MINITAB Results yˆ æ ö x æ ö x æ ö x æ - = ö x x + æ ö x x ç è 2 ø èç 2 ø èç 2 ø èç 2 ø èç 2 ø For example the residual at x = 1, x = - 1, x = - 1, x = e = y - yˆ é æ19.00ö æ14.00ö æ16.50ö æ-18.50ö æ19.00ö ù = (1) + (- 1) + (1) + (1)( - 1) + (1)(1) ê ç 2 ç 2 ç 2 ç 2 ç 2 ë è ø è ø è ø è ø è ø ú û = =
40 Resin Plant Experiment MINITAB Results 40
41 Manufacturing Process for a Circuit Five factors in a manufacturing process for an integrated circuit were investigated in a design with the objective of improving the process yield. Select ABCDE as the generator (Resolution V design) I=ABCDE ; E=ABCD ; Every main effect is aliased with a four-factor interaction. E.g., [A] -> A+BCDE Every two factor interaction is aliased with a three-factor interaction. E.g., [AB]-> AB+CDE 41
42 Manufacturing Process MINITAB Results 42
43 Manufacturing Process MINITAB Results A, B, C, and AB are significant 43
44 Manufacturing Process MINITAB Results Selecting only A, B, C, and AB This implies 2 3 Design with 2 replicates at each experimental point 44
45 Manufacturing Process MINITAB Results ANOVA Residual analysis 45
46 Manufacturing Process MINITAB Results Interaction Plot of AB Cube Plot 46
47 The Sequential Experimentation Suppose that after running the principal fraction, the alternate fraction was also run The two groups of runs can be combined to form a full factorial an example of sequential experimentation De-aliased estimates of the effects can be obtained by adding and subtracting 1 1 ([ A] + [ A]') = ( A+ BC + A- BC) A ([ A] -[ A]') = ( A+ BC - A+ BC) BC
48 The Sequential Experimentation If it is necessary to resolve ambiguities, we can run the alternate fraction and complete 2 k design. Run 1 Run 2 48
49 Resin Plant Experiment Alternate Fraction Recall the resin plant experiment Generator I=-ABCD [ A] = 19 A+ BCD (from main fraction) 1 [ A]' = ( ) 4 = A- BCD (from alternative fraction) Main Effect of original design 1 A = ( [ A ] + [ A ]' ) =
50 The One-Quarter Fraction of the 2 k 50
51 The One-Quarter Fraction of the
52 The General 2 k-p Fractional Factorial Design 2 k-1 = one-half fraction, 2 k-2 = one-quarter fraction, 2 k-3 = one-eighth fraction,, 2 k-p = 1/ 2 p fraction Add p columns to the basic design; select p independent generators Important to select generators so as to maximize resolution, see the table in the next slide Projection a design of resolution R contains full factorials in any R 1 of the factors Effects of factors are Effect where N i Contrasti = ( N / 2) = number of observations 52
53 The General 2 k-p Design Resolution may not be sufficient Minimum abberation designs Our choice 53
54 54
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