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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 Yijk i j ij k i 1,, a ik jk ijk j 1,, b k 1,, n Designs with Randomization Restrictions By convention, we assume there is not block by treatment interaction (the usual RCBD assumption) so that: ik jk ijk ijk Note that this is different from “pooling” 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) 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 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 Split Plot Design One version of the model (See ex. 24.1): Yijk i k ik j ij jk ijk EMS Table--Whole Plot EMS Source 2 2 2 WholeP lot (A) b bn i (a 1) df a -1 i Block ab 2 n -1 A x Block 2 b 2 (a - 1)(n- 1) 2 EMS Table--Split Plot Source Split P lot (B) a 2 AB x Block an j2 j 2 (b 1)(n 1) a 2 2 2 n ij2 i j (a 1)(b 1) (a 1)(b 1) 2 b 1 b 1 2 B x Block AB EMS 2 (a 1)(b 1)(n 1) Split Plot Design Note that there are no degrees of freedom for error Block and Block x Treatment interactions cannot be tested 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. Yijk i k ik j ij jk (i ) Split Plot Design Yandell presents an alternate model Useful when whole plots are replicated and no blocks are present Yijk i k (i ) j ij jk (i ) EMS Table--Whole Plot Source A WholeP lot Error EMS 2 b 2 bn i2 i a 1 2 b 2 EMS Table--Split Plot Source Split P lot (B) AB Split P lot Error EMS 2 2 2 2 an j2 j b 1 n ij2 j k (a 1)(b 1) 2 2 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 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 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 Split Split Plot Design A model version that facilitates testing: Yijkl i l (i ) j ij jl (i ) k ik jk ijk kl (ij ) 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 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) Split Plot Design with Covariates Whole Plot Covariate – Xijk=Xik – Xijk=Xi occurs frequently in practice Split Plot Covariate WP Covariate X ijk X ijk X i.k X i.k SP Covariate Split Plot Design with Covariates Model Yijk i X i.k k (i ) j X ijk X i.k ij jk (i ) 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 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 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 Yikm i rm (i ) k ik km(i ) i 1, , a m 1,, n k 1,, t Repeated Measures Design Fuel Cell Example – Response: Current – Group: Control/Added H20 – Subject(Group): Daily Experimental Run or Fuel Cell – Repeated Measures Factor: Voltage Repeated Measures Design Source Group Subject(Group) Repeated Measures Factor Group x Repeated Measures Error Total df a-1 a(n-1) t-1 (a-1)(t-1) a(t-1)(n-1) atn-1 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 Cov(Yikm , Yik 'm ) Cov(rm (i ) , rm (i ) ) Corr(Yikm , Yik 'm ) 2 P 2 2 P 2 P Repeated Measures Design Fuel Cell Example – – – – Repeated Measures Factor: Voltage Response: Current Group: Control/Added H20 Subject: Fuel Cell Repeated Measures Design Y jkm T j Rm( j ) Vk TV jk jkm , j 1, 2; m(1) 1, 2,3; m( 2) 1, 2; k 1, ,8 Repeated Measures Design Yi=Yjm=(Yj1m,…,Yj8m)’ =(,T1,V1,…,V7,TV11,…,TV17)’ Mixed Models The general mixed model is i ~ N (0, R) Yi X i Z i i ~ N (0, G ) Repeated Measures Design For our example, we have no random effects (no Zi or ) separate from the repeated measures effects captured in R. X1 =X1(1) has the form (assume V8=TV18=0) 1|1|1 0 0 0 0 0 0|1 0 0 0 0 0 0 1|1|0 1 0 0 0 0 0|0 1 0 0 0 0 0 … 1|1|0 0 0 0 0 0 1|0 0 0 0 0 0 1 1|1|0 0 0 0 0 0 0|0 0 0 0 0 0 0 Mixed Models For many models we encounter, R is 2I In repeated measures models, R can have a lot more structure. E.g., for t timepoints, an AR(1) covariance structure would be: 2 1 2 R t 1 2 1 2 2 t 2 2 t 1 2 t 2 2 2 Repeated Measures Structures •Toeplitz kk ' 2 k k ' kk ' 2 kk ' •Unstructured •Compound Symmetric •Banded Toeplitz kk ' 2 2 k k ' , k k ' r kk ' 0, k k' r Mixed Models G almost always has a diagonal structure Ex : Regardless of the form for R and G, we can write Yi~N(Xi,ZiGZi’+R) 2 I a 0 0 G 0 2 I b 0 2 0 0 I ab Mixed Models For the entire sample we have Y ~ N ( X , ZGZ ' R*), where Y ' (Y1 ' , , Yn ' ) X ' ( X 1 ' ,, X n ' ) Z ' ( Z1 ' , , Z n ' ) R* I n R Restricted MLE If V=ZGZ’+R* were known, the MLE for would be (X’V-1X)-1X’V-1Y We would estimate the residuals as e=(IH)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 Restricted MLE The Profile RMLE of the parameters of G and R would maximize : 1 1 1 l (G, R) (n q ) log 2 log PVP' e'V 1e 2 2 2 T hencomput e 1 1 1 ˆ ˆ ˆ ( X ' V X ) X 'V Y 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) AIC ln Lˆ q or AIC 2 ln Lˆ 2q 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