design & analysis of multistratum randomized experiments ching-shui cheng dec. 7, 2006
DESCRIPTION
Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006 National Tsing Hua University. Randomization models for designs with simple orthogonal block structures. where are the treatment effects. has spectral form - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/1.jpg)
1
Design & Analysis of Multistratum Randomized Experiments
Ching-Shui Cheng
Dec. 7, 2006National Tsing Hua University
![Page 2: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/2.jpg)
2
Randomization models for designs with simple orthogonal block structures
where are the treatment effects.
![Page 3: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/3.jpg)
3
has spectral form
where is the orthogonal projection matrix onto theeigenspace of with eigenvalue
Each of these eignespaces is called a stratum.
![Page 4: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/4.jpg)
4
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
Ad-Watch SE Professional.lnk
![Page 5: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/5.jpg)
5
![Page 6: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/6.jpg)
6
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.
![Page 7: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/7.jpg)
7
Designs such that falls entirely in one stratum arecalled orthogonal designs.
Examples:
Completely randomized designsRandomized complete block designsLatin squares
![Page 8: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/8.jpg)
8
ANOVA table for an orthogonal design
![Page 9: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/9.jpg)
9
Completely randomized design
![Page 10: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/10.jpg)
10
![Page 11: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/11.jpg)
11
![Page 12: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/12.jpg)
12
![Page 13: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/13.jpg)
13
![Page 14: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/14.jpg)
14
![Page 15: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/15.jpg)
15
![Page 16: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/16.jpg)
16
![Page 17: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/17.jpg)
17
Block design (b/k)
![Page 18: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/18.jpg)
18
![Page 19: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/19.jpg)
19
Complete block designs
The two factors T and B satisfy the condition of proportional frequencies.
![Page 20: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/20.jpg)
20
![Page 21: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/21.jpg)
21
![Page 22: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/22.jpg)
22
![Page 23: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/23.jpg)
23
![Page 24: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/24.jpg)
24
Three replications of 23
Treatment structure: 2*2*2
Block structure: 3/8
![Page 25: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/25.jpg)
25
![Page 26: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/26.jpg)
26
In general, there may be information for treatment contrastsin 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 inone stratum.
Orthogonal designs
![Page 27: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/27.jpg)
27
Blocks Block/((Sv/St/Sr)*Sw) 4/((3/2/3)*7) Treatments Var*Time*Rate*Weed 3*2*3*7
![Page 28: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/28.jpg)
28
![Page 29: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/29.jpg)
29
Factor [nvalues=504;levels=4] Block & [levels=3] Sv, Sr, Var, Rate & [levels=2] St, Time & [levels=7] Sw, WeedGenerate Block, Sv, St, Sr, SwMatrix [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] CkeyAkey [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, WeedBlocks Block/((Sv/St/Sr)*Sw)Treatments Var*Time*Rate*WeedANOVA
![Page 30: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/30.jpg)
30
Block stratum 3 Block.Sv stratumVar 2Residual 6 Block.Sw stratumWeed 6Residual 18
Block.Sv.St stratumTime 1Var.Time 2Residual 9
![Page 31: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/31.jpg)
31
Block.Sv.Sw stratumVar.Weed 12Residual 36
Block.Sv.St.Sr stratumRate 2Var.Rate 4Time.Rate 2Var.Time.Rate 4Residual 36
![Page 32: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/32.jpg)
32
Block.Sv.St.Sw stratumTime.Weed 6Var.Time.Weed 12Residual 54 Block.Sv.St.Sr.Sw stratumRate.Weed 12Var.Rate.Weed 24Time.Rate.Weed 12Var.Time.Rate. Weed 24Residual 216
Total 503
![Page 33: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/33.jpg)
33
has spectral form
where is the orthogonal projection matrix onto theeigenspace of with eigenvalue
Each of these eignespaces is called a stratum.
![Page 34: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/34.jpg)
34
Normal equation
![Page 35: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/35.jpg)
35
![Page 36: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/36.jpg)
36
![Page 37: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/37.jpg)
37
![Page 38: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/38.jpg)
38
![Page 39: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/39.jpg)
39
![Page 40: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/40.jpg)
40
![Page 41: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/41.jpg)
41
Incomplete block designs
![Page 42: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/42.jpg)
42
![Page 43: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/43.jpg)
43
Consider the model with fixed block effects:
To eliminate the nuisance parameters in , we need toproject onto :
![Page 44: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/44.jpg)
44
The intrablock estimator of a treatment contrast is the sameas its least squares estimator under the model with fixed block effects.
![Page 45: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/45.jpg)
45
is the orthogonal projection matrix onto
![Page 46: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/46.jpg)
46
![Page 47: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/47.jpg)
47
Balanced incomplete block designs (BIBD)
![Page 48: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/48.jpg)
48
![Page 49: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/49.jpg)
49
![Page 50: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/50.jpg)
50
![Page 51: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/51.jpg)
51
![Page 52: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/52.jpg)
52
![Page 53: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/53.jpg)
53
ANOVA table for a BIBD
![Page 54: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/54.jpg)
54
![Page 55: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/55.jpg)
55
![Page 56: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/56.jpg)
56
![Page 57: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/57.jpg)
57
![Page 58: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/58.jpg)
58
![Page 59: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/59.jpg)
59
![Page 60: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/60.jpg)
60
![Page 61: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/61.jpg)
61
![Page 62: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/62.jpg)
62
![Page 63: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/63.jpg)
63
![Page 64: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/64.jpg)
64
Recovery of interblock information
![Page 65: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/65.jpg)
65
![Page 66: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/66.jpg)
66
![Page 67: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/67.jpg)
67
Non-orthogonal row-column designs
![Page 68: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/68.jpg)
68
![Page 69: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/69.jpg)
69
![Page 70: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/70.jpg)
70
![Page 71: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/71.jpg)
71
To eliminate the nuisance parameters in and ,project onto :
![Page 72: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/72.jpg)
72
![Page 73: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/73.jpg)
73
![Page 74: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/74.jpg)
74
![Page 75: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/75.jpg)
75
Youden square
![Page 76: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/76.jpg)
76
![Page 77: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/77.jpg)
77
![Page 78: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/78.jpg)
78
![Page 79: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/79.jpg)
79
![Page 80: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/80.jpg)
80
Optimal block designs
![Page 81: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/81.jpg)
81
![Page 82: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/82.jpg)
82
![Page 83: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/83.jpg)
83
![Page 84: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/84.jpg)
84
![Page 85: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/85.jpg)
85
Kiefer (1975)
![Page 86: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/86.jpg)
86
![Page 87: Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006](https://reader035.vdocuments.us/reader035/viewer/2022070419/56815d8c550346895dcb98f4/html5/thumbnails/87.jpg)
87