Statistics and Art: Sampling, Response Error, Mixed Models, Missing Data, and Inference

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Statistics and Art Sampling, Response Error,
Mixed Models, Missing Data, and Inference

• Ed Stanek

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• Outline
• Example Dose-response Models in Toxicology-
  Threshold vs Hormetic Models
• What is truth? Predict what?
• Subsets- sampling
• Prediction
• Results on Predictor of Realized Subject True
  Value
• Illustration and Dilemma
• Extension to two-stage problems
• Missing data framework
• Conclusions

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1. Example Dose-response Models - Threshold vs
Hormetic Models

• Yeast data- 2189 chemicals, 13 yeast strains, 5
  doses x 2 replications- Focus on doses below BMD

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i chemical J dose k replication
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2. What is truth? Predict what?
Population, subjects, true response Subject
Labels True Response
Population Parameters Mean Variance
Subject Deviation
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Non-Stochastic Model Index for
response Response error Assume
Response Error ModelFor each subject
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3. Subsets, Sampling

• Select n of N subjects (a subset, sample)
• Let all subsets be equally likely

Sample Mean Note difference with
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Sample as a Sequence(part of Permutation)

• Represent Positions in a Permutation
• Assume all Permutations Equally Likely

Define Sample positions Sample Mean

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Population
s2 Ed
s3 Wenjun
s1 Julio
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Position in Permutation
i1
i2
i3
s2
s3
s1
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i1
i2
Position in Permutation
i1
i2
i3
i3
s2
s1
s3
s1
s3
s2
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Position in Permutation
i1
i2
i3
s3
s1
s2
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Position in Permutation
i1
i2
i3
s3
s2
s1
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Position in Permutation
i1
i2
i3
Sample
Remainder
s1
s2
s3
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Position in Permutation
i1
i2
i3
Sample
Remainder
s2
s1
s3
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• Population size (N) is most likely gt 3
• We only see n subjects in the sample
• For example Suppose n3, and N7
• We may see

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Position in Permutation
i1
i2
i3
Sample
Remainder
i4
i
s3
s4
s5
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Position in Permutation
i1
i2
i3
Sample
Remainder
s2
s4
s7
i
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Traditional Sampling Approach

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N
Horvitz-Thompson Estimator
Missing Data
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With Response Error Model
Sample Mean

• Sample is a set

Sample is a Sequence

To represent positions
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Position in Permutation
i1
i2
i3
Sample
s1
s2
s3
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First Position in Permutation

Suppose s1,,3N

Then

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Positions in Sample Sequences

Sample
Remainder
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Basic Random Variables

Sample
Remainder
Population
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Response Error Model

Response Error Model
Finite Population Mixed Model
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Mixed Model
Mixed Model

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Properties of Basic Random Variables (N3)

Sum
Expected Value
Sum
Average
Expected Value
Average
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Sample Random Variables (n2)

Sum
Expected Value
Sum
Expected Value
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Prediction of Mean in a Simple Case No Response
Error (N3, n2)

Sample
Remainder
Note Criteria Linear Function of
sample Unbiased Smallest Mean Squared Error
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Prediction of Mean No Response Error (N3, n2)

Target
Sample Data
Realized
Best Linear Unbiased Predictor
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Prediction of a Subjects Mean in Position i with
No Resp. Error (N3, n2)

Target
Sample Data
Realized
Best Linear Unbiased Predictor
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Prediction of a Subjects Mean in Position i with
Response Error

Target
Sample Data
Realized
Best Linear Unbiased Predictor
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Prediction of Realized Random Effect Other
Examples

SRS Subject Resp. Error
SRS Position Resp. Error
Cluster Sampling Balanced
Cluster Sampling Un-Balanced
Similar form, more complicated
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Delimma

• Pooled Response Error Variance should be used for
  K (Using theoretical Results)
• Empirical example illustrates smaller MSE results
  with K depending on realized Subject -- but no
  theory!
• What should we do?.... Is there a gap in the
  framework?

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Basic Sample Random Variables

Sum
Sum
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Basic Random Variables

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More Work is needed!
Thanks