Design & Analysis of Multi-Stratum Randomized Experiments Ching-Shui Cheng June 5, 2008

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

2. Randomization model, Null ANOVA, Orthogonal designs

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

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

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. Completely randomized design

11. Block design (b/k)

12. Row-column design ( )

13. Total sum of squares

14. Total variability

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. Complete randomization One error term

17. b/k: s=2

19. Two error terms

22. Three error terms

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. Null ANOVA

28. McLeod and Brewster (2004) Technometrics

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.

32. Miller (1997) Technometrics

34. There are 8 nontrivial strata

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

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

42. Completely randomized design

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

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),

47. ANOVA for an orthogonal design