1. 1 Design & Analysis of Multi-Stratum Randomized Experiments Ching-Shui Cheng June 5, 2008 National SunYat-sen University

2. 2 Randomization model, Null ANOVA, Orthogonal designs

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4. 4 Randomization model for designs with simple block structures where are the treatment effects. Nelder (1965a) gives simple rules for determining.

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6. 6 Block designs (b/k) The covariance matrix has three different entries: a common variance and two covariances depending on whether the corresponding pair of observations are in the same block or not.

7. 7 A model of this form also arises from the following mixed- effects model: Random block effects

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9. 9 has spectral form where is the orthogonal projection matrix onto the eigenspace of with eigenvalue The eigenspaces are independent of the values of the variances and covarinces: co-spectral. Each of these eignespaces is called a stratum.

10. 10 Completely randomized design

11. 11 Block design (b/k)

12. 12 Row-column design ( )

13. 13 Total sum of squares

14. 14 Total variability

15. 15 Suppose (in which case does not contain treatment effects, and therefore measures variability among the experimental units) It can be shown that th stratum variance Null ANOVA

16. 16 One error term Complete randomization

17. 17 b/k: s=2

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19. 19 Two error terms

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22. 22 Three error terms

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24. 24 Nelder (1965a) gave simple rules for determining the degrees of freedom, projections onto the strata and the sums of squares in the null ANOVA for any simple block structure.

25. 25 Null ANOVA

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28. 28 McLeod and Brewster (2004) Technometrics

29. 29 From the d.f. identity, we can write down a yield identity which gives projections to all the strata. For convenience, we index each unit by multi-subscripts, and as before, dot notation is used for averaging. The following is the rule given by Nelder: Expand each term in the d.f. identity as a function of the n's; then to each term in the expansion corresponds a mean of the y's with the same sign and averaged over the subscripts for which the corresponding n's are absent.

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32. 32 Miller (1997) Technometrics

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34. 34 There are 8 nontrivial strata

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38. 38 Projections of the data vector onto different strata are uncorrelated between and homoscedastic within.

39. 39 Estimates computed in different strata are uncorrelated. Estimate each treatment contrast in each of the strata in which it is estimable, and combine the uncorrelated estimates from different strata. Simple analysis results when the treatment contrasts are estimable in only one stratum.

40. 40 Designs such that falls entirely in one stratum are called orthogonal designs. Examples: Completely randomized designs Randomized complete block designs Latin squares

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42. 42 Completely randomized design

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44. 44 Complete block designs The two factors T and B satisfy the condition of proportional frequencies.

45. 45 Under a row-column design such that each treatment appears the same number of times in each row and the same number of times in each column (such as a Latin square),

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47. 47 ANOVA for an orthogonal design

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56. 56 Three replications of 2 3 Treatment structure: 2*2*2 Block structure: 3/8

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58. 58 Incomplete blocks The condition of proportional frequencies cannot be satisfied. has nontrivial projections in both the inter- and intra-block strata. Hence there is treatment information in both strata. Interblock estimate Intrablock estimate Recovery of interblock information

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64. 64 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

65. 65 Treatment structure: 2 2 Block structure: 6/2 Information for the main effects is in the intrablock stratum and information for the interaction is in the interblock stratum

66. 66 ANOVA df Interblock AB 1 Residual 4 Intrablock A 1 B 1 Residual 4 Total11

67. 67 ANOVA for 3/2/2 df Replicates 2 Interblock AB 1 Residual 2 Intrablock A 1 B 1 Residual 4 Total11

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69. 69 Treatment structure: 3*4 Block structure: 3/3/4

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72. 72 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.

73. 73 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)

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75. 75 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

76. 76 block.dryer stratum a1 b1 c1 d1 ac=bd1 bc=ad1

77. 77 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

78. 78 Blocks Block/((Sv/St/Sr)*Sw) 4/((3/2/3)*7) Treatments Var*Time*Rate*Weed 3*2*3*7

79. 79 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

80. 80 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

81. 81 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

82. 82 From the archive of S-news email list: “I'm having trouble with what I'm sure is a very simple problem, i.e., specifying the correct syntax for a simple split plot design. To check my understanding I'm using the cake data set from Table 7.5 of Cochran & Cox...... I can't figure out what error term I need to specify.....”

83. 83 “Speaking as an absolutely committed devotee of Splus, I think that the Splus syntax for analyzing experimental designs containing random effects absolutely SUCKS!!! ….. I am firmly of the opinion that talking about ‘split plots’ as such is counter-productive, confusing, and antiquated. The ***right*** way to think about such problems is simply to decide which effects are random and which are fixed. Then think about expected mean squares, and choose the denominator of your F- statistic so that under H_0 the ratio of the expected mean squares is 1 ….”

84. 84 “It's a cross-Atlantic culture clash. ….. This divide-and-conquer idea of separating the data using linear functions with homogeneous error variances, that is, the spectral decomposition of the variance matrix, I find immensely simplifying and natural. ….. The contrary notion of putting everything together in one big anova table with interactions between random and fixed effects standing in for what really are error terms and pondering which one goes on the bottom of which other is to lose the plot entirely. ……” ---- Bill Venables

85. 85 “The great advantage of the western Pacific view of specifying an experiment is that the commands follow from the design of the experiment. There is no need to know anything about fixed or random effects, or denominators of F ratios. Experimenters know what their treatments are, and they know how they randomised their experiment. That’s all they need to specify the model, and the correct analysis pops out whether it was a split plot, confounded factorial, Latin square, or whatever. ….. ”

86. 86 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.”