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

Designs with Randomization Restrictions All factors are crossed All factors are crossed

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

Designs with Randomization Restrictions A similar example uses a Latin Square design A similar example uses a Latin Square design The treatment is in fact a factorial experiment The treatment is in fact a factorial experiment n=ab or n-1=(a-1)+(b-1)+(a-1)(b-1) 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 Two factor experiment in which a CRD within block is not feasible Example (observational study) 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 Agricultural Example –Block: Field –Whole Plot Factor: Tilling method –Split Plot Factor: Seed variety Whole plot and whole plot factor are confounded 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 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): One version of the model (See ex. 24.1):

EMS Table--Whole Plot

EMS Table--Split Plot SourceEMS

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

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

EMS Table--Whole Plot SourceEMS

EMS Table--Split Plot SourceEMS

Split Plot Design Yandell considers the cases where the whole plot and split plot factors, alternately, do not appear 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 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) We can also construct a split split plot design (in the obvious way) Montgomery example 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 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: A model version that facilitates testing:

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

Split Plot Design with Covariates Example (Observational study) 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 Whole Plot Covariate –X ijk =X ik –X ijk =X i occurs frequently in practice Split Plot Covariate Split Plot Covariate SP Covariate WP Covariate

Split Plot Design with Covariates Model Model

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) 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 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 Read Yandell Chapter 26 generally covers multivariate approaches to repeated measures—skip it 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 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 Looks like Yandell’s split plot design –The whole plot structure looks like a nested design –The split plot structure looks much the same

Repeated Measures Design Fuel Cell Example Fuel Cell Example –Response: Current –Group: Control/Added H 2 0 –Subject(Group): Daily Experimental Run or Fuel Cell –Repeated Measures Factor: Voltage

Repeated Measures Design Sourcedf Groupa-1 Subject(Group)a(n-1) Repeated Measures Factort-1 Group x Repeated Measures(a-1)(t-1) Errora(t-1)(n-1) Totalatn-1

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

Repeated Measures Design Fuel Cell Example Fuel Cell Example –Repeated Measures Factor: Voltage –Response: Current –Group: Control/Added H 2 0 –Subject: Fuel Cell

Repeated Measures Design

Y i =Y jm =(Y j1m,…,Y j8m ) ’  =( ,T 1,V 1,…,V 7,TV 11,…,TV 17 ) ’

Mixed Models The general mixed model is The general mixed model is

Repeated Measures Design For our example, we have no random effects (no Z i or  ) separate from the repeated measures effects captured in R. X 1 =X 1(1) has the form (assume V 8 =TV 18 =0) 1|1| | |1| | … 1|1| | |1| |

Mixed Models For many models we encounter, R is  2 I For many models we encounter, R is  2 I In repeated measures models, R can have a lot more structure. E.g., for t timepoints, an AR(1) covariance structure would be: In repeated measures models, R can have a lot more structure. E.g., for t timepoints, an AR(1) covariance structure would be:

Repeated Measures Structures Toeplitz Unstructured Compound Symmetric Banded Toeplitz

Mixed Models G almost always has a diagonal structure G almost always has a diagonal structure Regardless of the form for R and G, we can write Regardless of the form for R and G, we can write Y i ~N(X i ,Z i GZ i ’ +R)

Mixed Models For the entire sample we have For the entire sample we have

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

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) 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)

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