designs with randomization restrictions
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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
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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”
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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):
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EMS Table--Whole Plot
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EMS Table--Split Plot
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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.
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Split Plot Design
Yandell presents an alternate model
Useful when whole plots are replicated and no blocks are present
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EMS Table--Split Plot
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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:
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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
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SP Covariate
WP Covariate
Split Plot Design with Covariates
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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
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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 dfGroup a-1Subject(Group) a(n-1)Repeated Measures Factor t-1Group x Repeated Measures (a-1)(t-1)Error a(t-1)(n-1)Total 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
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Repeated Measures Design
Fuel Cell Example– Repeated Measures Factor: Voltage– Response: Current– Group: Control/Added H20– Subject: Fuel Cell
Repeated Measures Design
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Repeated Measures Design
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b=(m,T1,V1,…,V7,TV11,…,TV17)’
Mixed Models
The general mixed model is
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Repeated Measures Design
For our example, we have no random effects (no Zi or g) separate from the repeated measures effects captured in R. X1 =X1(1) has the form (assume V8=TV18=0)
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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:
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Repeated Measures Structures
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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)
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Mixed Models
For the entire sample we have
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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=Y-Yhat=Y=XB=Y-HY=(I-H)Y=PY where H=X(X’V-1X)-1X’V-1.
PY has a multivariate normal distribution that is a function of (G,R) only
Restricted MLE
The profile likelihood for the parameters of G and R would be based on the distribution of the residuals.
P has rank n-q, where q is the number of random effects; this improves the performance of the estimates of the variance components.
Restricted MLE
The Profile RMLE of the parameters of G and R would maximize :
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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)
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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
A note on fitting
REML essentially uses OLS to estimate the mean parameters, then uses these estimated mean parameters to estimate mean-zero residuals, then uses maximum likelihood to estimate variance components from the residuals. The variance component estimates are then used in GLS (more general than WLS) to re-estimate the mean parameters. This results in unbiased estimates of variance components.
A note on fitting
I’ve seen versions that plug in WLS estimates of the mean parameters assuming the variance components are known to create a likelihood for the variance components from the residuals. The mean parameters are then estimated using GLS, as above.
A note on fitting
Maximum likelihood simply estimates both the mean parameters and variance components at the same time using maximum likelihood. In either case the g are not part of the likelihood. These can be estimated after the population parameters are estimated using Bayes rule; they are called “BLUPs” for best linear unbiased predictor.
A note on fitting
PROC MIXED models repeated measures effects with a REPEATED statement, while GLIMMIX uses RANDOM _RESIDUAL_ . The default fitting method for PROC MIXED for the normal linear mixed model is METHOD=REML. PROC GLIMMIX uses METHOD=RSPL; these are equivalent. METHOD=MSPL in PROC GLIMMIX is equivalent to METHOD=ML in PROC MIXED. PROC GLIMMIX has more optimization methods available, and uses a different default optimization method from PROC MIXED.
A note on fitting
PROC GLM uses Method-of-Moments to estimate variance components. It constructs appropriate F-tests, but doesn’t build randomness of effects into tests or estimation of main effects, lsmeans, contrasts, etc. PROC MIXED has some more complex covariance structures (Kronecker-type for spatiotemporal models) that GLIMMIX lacks. Some of PROC GLIMMIX’s most useful features are not available in PROC MIXED: COVTEST, LSMESTIMATE, OUTPUT.