1 Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 8, 2006 National Tsing Hua University

2 Factorial experiments

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4 Fractional factorial designs regular fractional factorial designs

n = 5, p = 2

n = 5, p = 2

n = 5, p = 2

n = 5, p = 2

14 A design of size can accommodate at most two-level factors A saturated design of size can be constructed by writing all possible combinations of factors in a array, and then completing all possible component-wise products of the columns. A regular design with factors is obtained by choosing columns from the saturated design.

15 If we use capital letters to denote the factors, then we also use combinations of these letters to denote interactions A: main effect of factor A AB: interaction of factors A and B BCE: interaction of factors B, C and E etc.

16 Defining relation Defining words Defining contrasts, Defining effects

17 In the model matrix for the full model, the columns corresponding to the main effect A and interactions BD, CE, ABCDE are identical. Therefore they are completely mixed up. We say they are aliases of one another. One can estimate only one effect in each alias set, assuming that all the other effects in the same alias set are negligible.

18 7 alias sets

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20 Regular fractional factorial designs have simple alias structures: any two factorial effects are either orthogonal or completely aliased. Nonregular designs have complex alias structures that are difficult to disentangle.

Partial aliasing Nonregular design 12-run Plackett- Burman design

22 Under the previous design, a main effect is partially aliased with almost all interactions.

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24 Under the design defined by I = ABD = ACE = BCDE, the usual estimate of the main effect of A actually estimates A + BD + CE + ABCDE. This is an unbiased estimate of A if all its aliases are negligible. When some contrasts are found significant but cannot be attributed to specific effects, one has to perform follow-up experiments to resolve the ambiguity. This is called de-aliasing.

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26 Nonregular designs often have good projection properties. For example, the projection of the 12- run Plackett-Burman design onto any three factors consists of a complete 2 3 factorial and a design (Lin and Draper, Technometrics, 1992).

28 Construction of regular fractional factorial designs for estimating certain required effects Franklin and Bailey (1977, Applied Statistics) and Franklin (1985, Technometrics) proposed an algorithm for constructing regular designs under which certain required effects can be estimated.

29 The factorial effects are divided into three classes: {A i }: estimates are required, {B j }: not negligible but estimates are not required, {C k }: negligible. Find the smallest regular design under which the required effects can be estimated.

30 Six 2-level factors A, B, C, D, E, F Need to estimate all the main effects and interactions AC, AD, AF, BC, CD, CE, CF,DE All the other interactions are negligible

31 A,B,C,D: basic factors E,F: added factors

32 If all the effects of a certain set of basic factors are ineligible, then there is no need to check other sets of factors of the same size.

33 Model uncertainty; model robustness We consider the situation where little information about the relative sizes of the factorial effects is available. Hierarchical assumption: lower-order effects are more important than higher-order effects, and effects of the same order are equally important Desirable to minimize aliasing among lower-order effects Maximal resolution Minimum aberration

34 Resolution: length of the shortest defining word The design defined by I = ABD = ABCE = CDE has resolution III Resolution III: no aliasing among main effects Resolution IV: main effects are not aliased with other main effects and two-factor interactions Maximize the resolution

35 Technometrics

36 Understanding minimum aberration

37 7 alias sets

38 Cheng, Steinberg and Sun (1999, JRSS Ser. B)

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40 32-run minimum aberration designs

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42 (5, 5, 5, 5)is majorized by (4, 6, 2, 8)

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45 Estimation capacity (Sun, 1993) a measure of the capability of a design to handle and estimate different potential models involving interactions. Assume that all the 3-factor and higher order interactions are negligible, and the estimation of all the main effects is required.

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47 Design Key Patterson (1965) J. Agric. Sci. Patterson (1976) JRSS, Ser. B Bailey, Gilchrist and Patterson (1977) Biometrika Bailey (1977) Biometrika Factorial design construction, identification of effect aliasing and confounding with block factors

48 Design key I = ABD = ACE = BCDE

49 Miller (1997) The experiment is run in 2 blocks and employs 4 washers and 4 driers. Sets of cloth samples are run through the washers and the samples are divided into groups such that each group contains exactly one sample from each washer. Each group of samples is then assigned to one of the driers. Once dried, the extent of wrinkling on each sample is evaluated.

50 Treatment structure: A, B, C, D, E, F: configurations of washers a,b,c,d: configurations of dryers Block structure: 2 blocks/(4 washers ＊ 4 dryers)

51 If we use two sets of two 2-level pseudo factors to represent the rows and columns, then we need a design key

52 Block 1 Block

53 I=ABC=BDF=ABDE=abcd=ADab=…….

54 Block 1 Block

55 GenStat code factor [nvalue=32;levels=2] block,A,B,C,D,E,F,a,b,c,d & [levels=4] wash, dryer generate block,wash,dryer blockstructure block/(wash*dryer) treatmentstructure (A+B+C+D+E+F)*(A+B+C+D+E+F) +(a+b+c+d)*(a+b+c+d) +(A+B+C+D+E+F)*(a+b+c+d)

56 matrix [rows=10; columns=5; values=“ b r1 r2 c1 c2" 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0] Mkey

57 Akey [blockfactors=block,wash,dryer; Key=Mkey; rowprimes=!(10(2));colprimes=!(5(2)); colmappings=!(1,2,2,3,3)] Pdesign Arandom [blocks=block/(wash*dryer);seed=12345] PDESIGN ANOVA

58 Source of variationd.f. block stratum AD=BE=CF=ab=cd1 block.wash stratum A=BC=EF1 B=AC=DF 1 C=AB=DF1 D=BF=CE1 E=AF=CD1 F=BD=AE1

59 block.dryer stratum a1 b1 c1 d1 ac=bd1 bc=ad1

60 block.wash.dryer stratum Aa=Db1 Ba=Eb1 Ca=Fb1 Da=Ab1 Ea=Bb1 Fa=Cb1 Ac=Dd1 Bc=Ed1 Cc=Fd1 Dc=Ad1 Ec=Bd1 Fc=Cd1 Residual6 Total31

61 McLeod and Brewster (2004) Technometrics Chrome-plating process Block structure: 4 weeks/4 days/2 runs block/wholeplot/subplot Treatment structure: A ＊ B ＊ C ＊ p ＊ q ＊ r Each of the six factors has two levels

62 Hard-to-vary treatment factors A: chrome concentration B: Chrome to sulfate ratio C: bath temperature Easy-to-vary treatment factors p: etching current density q: plating current density r: part geometry

63 Example 1 Blocks: 2/4/4 Treatments: I=ABpqr, b=ABC=Cpqr A B C p q r

64 Example 2 Blocks: 2/8/2 Treatments: I=ABCpr, b=ABq=Cpqr A B C p q r

65 Example 3 Blocks: 4/4/2 Treatments: I=ABqr,b= ABC=ACpr =Cqr=BCpq=Bpr=Apq A B C p q r

66 Patterson and Bailey (1978) Applied Statistics Written for practicing statisticians “Difficulties of theory … tend to obscure the essential simplicity of design key applications.”