Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 8, 2006

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

2. Factorial experiments

4. Fractional factorial designs regular fractional factorial designs

5. n = 5, p = 2

6. n = 5, p = 2

7. n = 5, p = 2

8. 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

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. Nonregular design 12-run Plackett- Burman design Partial aliasing

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

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.

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 23 factorial and a 23-1 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: {Ai }: estimates are required, {Bj }: not negligible but estimates are not required, {Ck }: 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)

40. 32-run minimum aberration designs

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

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.

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. I = ABD = ACE = BCDE Design key

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 washersa,b,c,d: configurations of dryersBlock structure:2 blocks/(4 washers＊4 dryers)