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

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. FFFFFF FFFFFFFF FFFFF FFFFFFFF FFFFF FFFF

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

Pick one of the 72! Permutations randomly 4 treatments 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

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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 Randomized complete block design After randomization: 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.

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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!) 7 allowable 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 the largest possible 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 A αB βC γD δ B γA δD αC β C δD γA βB α D βC αB δA γ Trojan square

35 Treatment structures No structure Treatments vs. control Factorial structure

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

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

46 Choice of design Efficiency Combinatorial considerations Practical considerations

47 McLeod and Brewster (2004) Technometrics Blocked 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) Technometrics Strip-Plots Experimental objective: Investigate methods of reducing the wrinkling of clothes being laundered

51 Strip-Plots Suppose there are two treatment factors A and B, both of which require large plots. An economic way to run the experiment is to divide the experimental area into horizontal strips and vertical strips. Each level of factor A is assigned to all the plots in one row, and each level of B is assigned to all the plots in one column.

52 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.

53 Treatment structure: A, B, C, D, E, F: configurations of washers a,b,c,d: configurations of dryers

54 Block structure: 2 blocks/(4 washers ＊ 4 dryers)

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56 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)

57 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

58 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

59 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

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

61 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 Total 31

62 Seven Error Terms!! Are you kidding?? T. M. Loughin et al.

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73 Treatment structure: 3*2*3*7 Block structure: 4/((3/2/3)*7)

74 Factor [nvalues=504;levels=4] Block & [levels=3] Sv, Sr, Var, Rate & [levels=2] St, Time & [levels=7] Sw, Weed Generate Block, Sv, St, Sr, Sw Matrix [rows=4;columns=6; \ values="b1 b2 Col St Sr Row"\ 1, 0, 1, 0, 0, 0,\ 0, 0, 1, 1, 0, 0,\ 0, 0, 1, 1, 1, 0,\ 1, 1, 0, 0, 0, 1] Ckey Akey [blockfactor=Block,Sv,St,Sr,Sw; \ Colprimes=!(2,2,3,2,3,7);Colmappings=!(1,1,2,3,4,5);Key=Ckey] Var, Time, Rate, Weed Blocks Block/((Sv/St/Sr)*Sw) Treatments Var*Time*Rate*Weed ANOVA

75 Block stratum 3 Block.Sv stratum Var 2 Residual 6 Block.Sw stratum Weed 6 Residual 18 Block.Sv.St stratum Time 1 Var.Time 2 Residual 9

76 Block.Sv.Sw stratum Var.Weed 12 Residual 36 Block.Sv.St.Sr stratum Rate 2 Var.Rate 4 Time.Rate 2 Var.Time.Rate 4 Residual 36

77 Block.Sv.St.Sw stratum Time.Weed 6 Var.Time.Weed 12 Residual 54 Block.Sv.St.Sr.Sw stratum Rate.Weed 12 Var.Rate.Weed 24 Time.Rate.Weed 12 Var.Time.Rate. Weed 24 Residual 216 Total 503

78 Suppose each of units is assigned one of the levels of a certain factor. The relation between the units and the levels of can be described by an incidence matrix where

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80 Example. Consider the following assignment of four treatments to four blocks each of size three: where the numbers inside each block are treatment labels, and the numbers to the left of each block are unit labels.

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