1. 1 Design & Analysis of Multi-Stratum Randomized Experiments Ching-Shui Cheng June 6, 2008 National Sun Yat-sen University

2. 2 Factorial designs

3. 3

4. 4 Fractional factorial designs regular fractional factorial designs

5. 5 11111 111 1 1 1 1 11 1 1 11 11 n = 5, p = 2

6. 6 11111 111 1 1 1 1 11 1 1 11 11 n = 5, p = 2

7. 7 11111 111 1 1 1 1 11 1 1 11 11 n = 5, p = 2 2 5-2 regular fractional factorial design

8. 8 11111 111 1 1 1 1 11 1 1 11 11 n = 5, p = 2

9. 9 11111 111 1 1 1 1 11 1 1 11 11

10. 10 1111111 111 1 1 1 1 11 11 1 1 1 1 11 1 111

11. 11 1111111 111 1 1 1 1 11 11 1 1 1 1 11 1 111

12. 12 1111111 111 1 1 1 1 11 11 1 1 1 1 11 1 111

13. 13 1111111 111 1 1 1 1 11 11 1 1 1 1 11 1 111

14. 14 A design of size can accommodate at most two-level factors A saturated design of size can be constructed by writing down 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. 15 If we use capital letters to denote the factors, then we 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. 16 Defining relation Defining words Defining contrasts, Defining effects

17. 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. 18 7 alias sets

19. 19

20. 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.

21. 21 1 1 1 1 1 1 1 111111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Partial aliasing Nonregular design 12-run Plackett- Burman design

22. 22

23. 23 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.

24. 24 Each main effect is partially aliased with almost all interactions. Complex partial aliasing

25. 25 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 2 3-1 design (Lin and Draper, Technometrics, 1992).

26. 26 1 1 1 1 1 1 1 111111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

27. 27 Selection 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.

28. 28 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.

29. 29 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

30. 30 A,B,C,D: basic factors E,F: added factors

31. 31 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.

32. 32 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

33. 33 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

34. 34 Technometrics

35. 35 Understanding minimum aberration

36. 36 7 alias sets g=7, f=2, m 1 =m 2 =2

37. 37 Cheng, Steinberg and Sun (1999, JRSS Ser. B)

38. 38

39. 39 32-run minimum aberration designs

40. 40

41. 41 (5, 5, 5, 5)is majorized by (4, 6, 2, 8)

42. 42

43. 43

44. 44 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.

45. 45

46. 46 In general, there may be information for treatment contrasts in more than one stratum. Analysis is still simple if the space of treatment contrasts can be decomposed as, where each, consisting of treatment contrasts of interest, is entirely in one stratum. Orthogonal designs

47. 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. 48 Patterson and Bailey (1978) Applied Statistics Written for practicing statisticians “Difficulties of theory … tend to obscure the essential simplicity of design key applications.”

49. 49 Design key I = ABD = ACE = BCDE

50. 50 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.

51. 51 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)

52. 52 If we use two sets of two 2-level pseudo factors to represent the rows and columns, then we need a design key

53. 53 I=ABC=BDF=ABDE=abcd=ADab=……. Blocking contrast: AD Transpose of design key matrix

54. 54 Block 1 Block 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 1 10 0 0 1 1 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 1 0 0 0 0 0 0 0 1 1 1 10 0 0 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 00 1 1 1 0 0 0 1 1 0 0 1 1 0 1 1 0 0 1 10 1 1 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 0 0 1 0 0 1 1 0 1 1 0 1 0 0 0 01 0 1 0 1 0 0 1 1 0 1 0 1 1 0 1 0 0 1 11 0 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 0 01 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 1 0 0 0 0 01 1 0 0 0 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 1 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 1 1 0 0 0 1 1 0 0 1

55. 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. 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. 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. 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. 59 block.dryer stratum a1 b1 c1 d1 ac=bd1 bc=ad1

60. 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. 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. 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. 63 4/4/2, 2 WP factors & 5 SP factors

64. 64 4/4/2, 3 WP factors & 3 SP factors A 0 1 0 1 0 B 1 0 1 0 0 C 1 0 1 1 0 p 0 0 1 0 1 q 1 0 1 1 1 r 0 1 0 0 1