1. Design & Analysis of Multi-Stratum Randomized Experiments Ching-Shui Cheng June 4, 2008 National Sun Yat-sen University

2. Schedule • June 4 Introduction, treatment and block structures, examples • June 5 Randomization models, null ANOVA, orthogonal designs • June 6 Factorial designs • June 9 Non-orthogonal designs

3. Design of Comparative Experiments By R. A. Bailey Published in April 2008 by Cambridge University Press An older version of the draft is available at http://www.maths.qmul.ac.uk/~rab/DOEbook/

4. Nelder (1965a, b) The analysis of randomized experiments with orthogonal block structure,Proceedings of the Royal Society of London, Series A Fundamental work on the analysis of randomized experiments with orthogonal block structures In Nelder's own words, his approach is "almost unknown in the U.S." (Senn (2003), Statistical Science, p. 124).

5. Bailey (1981) JRSS, Ser. A “Although Nelder (1965a, b) gave a unified treatment of what he called ‘simple’ block structures over ten years ago, his ideas do not seem to have gained wide acceptance. It is a pity, because they are useful and, I believe, simplifying. However, there seems to be a widespread belief that his ideas are too difficult to be understood or used by practical statisticians or students.”

6. Experimental Design Planning of experiments to produce valid information as efficiently as possible

7. Comparative Experiments • Treatments 處理 Varieties of grain, fertilizers, drugs, …. • Experimental Units Plots, patients, ….

8. Design: How to assign the treatments to the experimental units Fundamental difficulty: variability among the units; no two units are exactly the same. Each unit can be assigned only one treatment. Different responses may be observed even if the same treatment is assigned to the units. Systematic assignments may lead to bias.

9. Suppose is an observation on the th unit, and is the treatment assigned to that unit. Assume treatment-unit additivity:

10. R. A. Fisher worked at the Rothamsted Experimental Station in the United Kingdom to evaluate the success of various fertilizer treatments.

11. Fisher found the data from experiments going on for decades to be basically worthless because of poor experimental design. Fertilizer had been applied to a field one year and not in another in order to compare the yield of grain produced in the two years. BUT It may have rained more, or been sunnier, in different years. The seeds used may have differed between years as well. Or fertilizer was applied to one field and not to a nearby field in the same year. BUT The fields might have different soil, water, drainage, and history of previous use.  Too many factors affecting the results were “uncontrolled.”

12. Fisher’s solution: Randomization 隨機化 • In the same field and same year, apply fertilizer to randomly spaced plots within the field. • This averages out the effect of variation within the field in drainage and soil composition on yield, as well as controlling for weather, etc.

13. Randomization prevents any particular treatment from receiving more than its fair share of better units, thereby eliminating potential systematic bias. Some treatments may still get lucky, but if we assign many units to each treatment, then the effects of chance will average out. Replications In addition to guarding against potential systematic biases, randomization also provides a basis for doing statistical inference. (Randomization model)

14. Start with an initial design Randomly permute (labels of) the experimental units Complete randomization: Pick one of the 72! Permutations randomly

15. 4 treatments Pick one of the 72! Permutations randomly Completely randomized design

16. Assume treatment-unit additivity

17. Randomization model for a completely randomized design The ’s are identically distributed is a constant for all

19. Blocking 區集化 A disadvantage of complete randomization is that when variations among the experimental units are large, the treatment comparisons do not have good precision. Blocking is an effective way to reduce experimental error. The experimental units are divided into more homogeneous groups called blocks. Better precision can be achieved by comparing the treatments within blocks.

20. After randomization: Randomized complete block design 完全區集設計

21. Wine tasting Four wines are tasted and evaluated by each of eight judges. A unit is one tasting by one judge; judges are blocks. So there are eight blocks and 32 units. Units within each judge are identified by order of tasting.

23. Block what you can and randomize what you cannot.

24. Randomization • Blocking • Replication

25. Incomplete block design 7 treatments

26. Incomplete block design Balanced incomplete block design Optimality established by J. Kiefer Randomize by randomly permuting the block labels and independently permuting the unit labels within each block. Pick one of the (7!)(3!)7allowable permutations randomly.

27. Two basic block (unit) structures • Nesting block/unit • Crossing row ＊ column

28. Two simple block structures • Nesting block/unit • Crossing row ＊ column Latin square

29. Pick one of the (4!)(4!) allowable permutations randomly.

30. Wine tasting

31. The purpose of randomization is to average out those nuisance factors that we cannot predict or cannot control, not to destroy the relevant information we have. Choose a permutation group that preserves any known relevant structure on the units. Usually take the group for randomization to be thelargestpossible group that preserves the structure to give the greatest possible simplification of the model.

32. Simple block structures (Nelder, 1965) Iterated crossing and nesting (2*3)/2, 2/(4*4), 3/2/3, …… • cover most, though not all block structures encountered in practice

33. Consumer testing A consumer organization wishes to compare 8 brands of vacuum cleaner. There is one sample for each brand. Each of four housewives tests two cleaners in her home for a week. To allow for housewife effects, each housewife tests each cleaner and therefore takes part in the trial for 4 weeks. 8 treatments Block structure:

34. Trojan square

35. Treatment structures • No structure • Treatments vs. control • Factorial structure

36. Unstructured treatments Might be interested in estimating pairwise comparisons (Treatment contrast), t: # of treatments The set of all treatment contrasts form a (t-1)- dimensional space (generated by all the pairwise comparisons. or

37. Treatments vs control

38. Factorial structure Each treatment is a combination of several factors

39. S=2, n=3:

40. Interested in contrasts representing main effects and interactions of the factors

41. Here each is coded by 1 and -1.

45. Treatment structure Block structure (unit structure) Design Randomization Analysis

46. Choice of design • Efficiency • Combinatorial considerations • Practical considerations

47. McLeod and Brewster (2004) TechnometricsBlocked split plots 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

48. Split-plot design In agricultural experiments, sometimes certain treatment factors require larger plots than others. For example, suppose two factors, four varieties of a crop and three different rates of a fertilizer, are to be investigated. While each fertilizer can be applied to a small plot, the varieties can only be applied to larger plots due to limitations on the machines for sowing seed.

49. 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

50. Miller (1997) TechnometricsStrip-Plots Experimental objective: Investigate methods of reducing the wrinkling of clothes being laundered