Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006

1. Design & Analysis of Multistratum Randomized Experiments Ching-Shui Cheng Dec. 7, 2006 National Tsing Hua University

2. Randomization models for designs with simple orthogonal block structures where are the treatment effects.

3. has spectral form where is the orthogonal projection matrix onto the eigenspace of with eigenvalue Each of these eignespaces is called a stratum.

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

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.

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

8. ANOVA table for an orthogonal design

9. Completely randomized design

17. Block design (b/k)

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

24. Three replications of 23 Treatment structure: 2*2*2 Block structure: 3/8

26. 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

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

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

30. Block stratum 3Block.Sv stratumVar 2Residual 6Block.Sw stratumWeed 6Residual 18 Block.Sv.St stratumTime 1Var.Time 2Residual 9

31. Block.Sv.Sw stratumVar.Weed 12Residual 36 Block.Sv.St.Sr stratumRate 2Var.Rate 4Time.Rate 2Var.Time.Rate 4Residual 36

32. Block.Sv.St.Sw stratumTime.Weed 6Var.Time.Weed 12Residual 54Block.Sv.St.Sr.Sw stratumRate.Weed 12Var.Rate.Weed 24Time.Rate.Weed 12Var.Time.Rate. Weed 24Residual 216Total 503

33. has spectral form where is the orthogonal projection matrix onto the eigenspace of with eigenvalue Each of these eignespaces is called a stratum.

34. Normal equation

41. Incomplete block designs

43. Consider the model with fixed block effects: To eliminate the nuisance parameters in , we need to project onto :

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

45. is the orthogonal projection matrix onto

47. Balanced incomplete block designs (BIBD)