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Design & Analysis of Multistratum Randomized Experiments

Ching-Shui Cheng

Dec. 7, 2006National Tsing Hua University

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Randomization models for designs with simple orthogonal block structures

where are the treatment effects.

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has spectral form

where is the orthogonal projection matrix onto theeigenspace of with eigenvalue

Each of these eignespaces is called a stratum.

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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

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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.

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Designs such that falls entirely in one stratum arecalled orthogonal designs.

Examples:

Completely randomized designsRandomized complete block designsLatin squares

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ANOVA table for an orthogonal design

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Completely randomized design

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Block design (b/k)

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Complete block designs

The two factors T and B satisfy the condition of proportional frequencies.

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Three replications of 23

Treatment structure: 2*2*2

Block structure: 3/8

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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

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

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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

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Block stratum 3 Block.Sv stratumVar 2Residual 6 Block.Sw stratumWeed 6Residual 18

Block.Sv.St stratumTime 1Var.Time 2Residual 9  

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 Block.Sv.Sw stratumVar.Weed 12Residual 36

Block.Sv.St.Sr stratumRate 2Var.Rate 4Time.Rate 2Var.Time.Rate 4Residual 36

 

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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  

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has spectral form

where is the orthogonal projection matrix onto theeigenspace of with eigenvalue

Each of these eignespaces is called a stratum.

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Normal equation

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Incomplete block designs

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Consider the model with fixed block effects:

To eliminate the nuisance parameters in , we need toproject onto :

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The intrablock estimator of a treatment contrast is the sameas its least squares estimator under the model with fixed block effects.

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is the orthogonal projection matrix onto

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Balanced incomplete block designs (BIBD)

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ANOVA table for a BIBD

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Recovery of interblock information

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Non-orthogonal row-column designs

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To eliminate the nuisance parameters in and ,project onto :

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Youden square

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Optimal block designs

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Kiefer (1975)

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