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Designs with Randomization Restrictions

Designs with Randomization Restrictions. RCBD with a complete factorial in each block A: Cooling Method B: Temperature Conduct ab experiments in each block. Designs with Randomization Restrictions. All factors are crossed. Designs with Randomization Restrictions.

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Designs with Randomization Restrictions

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  1. Designs with Randomization Restrictions • RCBD with a complete factorial in each block • A: Cooling Method • B: Temperature • Conduct ab experiments in each block

  2. Designs with Randomization Restrictions • All factors are crossed

  3. Designs with Randomization Restrictions • By convention, we assume there is not block by treatment interaction (the usual RCBD assumption) so that: • Note that this is different from “pooling”

  4. Designs with Randomization Restrictions • A similar example uses a Latin Square design • The treatment is in fact a factorial experiment • n=ab or n-1=(a-1)+(b-1)+(a-1)(b-1)

  5. Split Plot Design • Two factor experiment in which a CRD within block is not feasible • Example (observational study) • Blocks: Lake • Whole plot: Stream; Whole plot factor: lampricide • Split plot factor: Fish species • Response: Lamprey scars

  6. Split Plot Design • Agricultural Example • Block: Field • Whole Plot Factor: Tilling method • Split Plot Factor: Seed variety • Whole plot and whole plot factor are confounded • This is true at split plot level as well, though confounding is thought to be less serious

  7. Split Plot Design • One version of the model (See ex. 24.1):

  8. EMS Table--Whole Plot

  9. EMS Table--Split Plot SourceEMS

  10. Split Plot Design • Note that there are no degrees of freedom for error • Block and Block x Treatment interactions cannot be tested

  11. Split Plot Design • In an alternative formulation, SP x Block and SP x WP x Block are combined to form the Split Plot Error. Note the unusual subscript—a contrivance that yields the correct df.

  12. Split Plot Design • Yandell presents an alternate model • Useful when whole plots are replicated and no blocks are present

  13. EMS Table--Whole Plot SourceEMS

  14. EMS Table--Split Plot SourceEMS

  15. Split Plot Design • Yandell considers the cases where the whole plot and split plot factors, alternately, do not appear • Split plot factor missing—whole plot looks like RCBD (me) or CRD (Yandell); subplots are subsampled. • Whole plot factor—whole plots look like one-way random effects; subplots look like either RCBD or CRD again. • Yandell has nice notes on LSMeans in Ex. 23.4

  16. Split Split Plot Design • We can also construct a split split plot design (in the obvious way) • Montgomery example • Block: Day • Whole Plot: Technician receives batch • Split Plot: Three dosage strengths formulated from batch • Split split plot: Four wall thicknesses tested from each dosage strength formulation

  17. Split Split Plot Design • Surgical Glove Example • Block: Load of latex pellets • Whole Plot: Latex preparation method • Split Plot: Coagulant dip • Split Split Plot: Heat treatment

  18. Split Split Plot Design • A model version that facilitates testing:

  19. Split Plot Design with Covariates • This discussion is most appropriate for the nested whole plots example • Often, researchers would like to include covariates confounded with factors

  20. Split Plot Design with Covariates • Example (Observational study) • Whole Plot: School • Whole Plot Factor: School District • Split Plot Factor: Math Course • Split Plot : Class • Split Plot covariate: Teacher Rating • Whole Plot covariate: School Rating • Whole Plot Factor covariate: School District Rating • Response: % Math Proficient (HSAP)

  21. Split Plot Design with Covariates • Whole Plot Covariate • Xijk=Xik • Xijk=Xi occurs frequently in practice • Split Plot Covariate WP Covariate SP Covariate

  22. Split Plot Design with Covariates • Model

  23. Split Plot Design with Covariates • A Whole Plot covariate’s Type I MS would be tested against Whole Plot Error (with 1 fewer df because of confounding) • Split Plot Covariate is not confounded with any model terms (though it is confounded with the error term), so no adjustments are necessary

  24. Repeated Measures Design • Read Yandell 25.1-25.3 • Chapter 26 generally covers multivariate approaches to repeated measures—skip it • We will study the traditional approach first, and then consider more sophisticated repeated measures correlation patterns

  25. Repeated Measures Design • Looks like Yandell’s split plot design • The whole plot structure looks like a nested design • The split plot structure looks much the same

  26. Repeated Measures Design • Fuel Cell Example • Response: Current • Group: Control/Added H20 • Subject(Group): Daily Experimental Run or Fuel Cell • Repeated Measures Factor: Voltage

  27. Repeated Measures Design Sourcedf Group a-1 Subject(Group) a(n-1) Repeated Measures Factor t-1 Group x Repeated Measures (a-1)(t-1) Error a(t-1)(n-1) Total atn-1

  28. Repeated Measures Design • A great deal of work has been conducted on repeated measures design over the last 15 years • Non-normal data • More complex covariance structure

  29. Repeated Measures Design • Fuel Cell Example • Repeated Measures Factor: Voltage • Response: Current • Group: Control/Added H20 • Subject: Fuel Cell

  30. Repeated Measures Design

  31. Repeated Measures Design Yi=Yjm=(Yj1m,…,Yj8m)’ b=(m,T1,V1,…,V7,TV11,…,TV17)’ g=(R1(1),R2(1),R3(1),R1(2),R2(2))’

  32. Mixed Models • The general mixed model is

  33. Mixed Models • For many models we encounter, R is s2I • In repeated measures models, R can have a lot more structure. E.g., for t timepoints, an AR(1) covariance structure would be:

  34. Repeated Measures Structures • Toeplitz • Unstructured • Compound Symmetric • Banded Toeplitz

  35. Mixed Models • G almost always has a diagonal structure • Regardless of the form for R and G, we can write Yi~N(Xib,ZiGZi’+R)

  36. Mixed Models • For the entire sample we have

  37. Restricted MLE • If V=ZGZ’+R* were known, the MLE for b would be (X’V-1X)-1X’V-1Y • We would estimate the residuals as e=(I-H)Y=PY where H=X(X’V-1X)-1X’V-1. • The profile likelihood for the parameters of G and R would be based on the distribution of the residuals

  38. Restricted MLE • The Profile RMLE of the parameters of G and R would maximize:

  39. Case Study • To choose between non-hierachical models, we select the best model based on the Akaike Information Criterion (smaller is better for the second form; q=# of random effects estimated)

  40. Case Study • Autocorrelation was strong • A Toeplitz model worked best • Voltage effect, as expected, was strong • Treatment effect was marginal • Voltage x Treatment effect was strong to moderate

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