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Sampling, WLS, and Mixed Models Festschrift to Honor Professor Gary Koch. Ed Stanek and Julio Singer U of Mass, Amherst, and U of Sao Paulo, Brazil. Finite Population Mixed Models Research Group.

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sampling wls and mixed models festschrift to honor professor gary koch

Sampling, WLS, and Mixed Models Festschrift to Honor Professor Gary Koch

Ed Stanek and Julio Singer

U of Mass, Amherst, and

U of Sao Paulo, Brazil

SPH&HS, UMASS Amherst

finite population mixed models research group
Finite Population Mixed Models Research Group

Luzmery Gonzalas, Columbia; Viviana Lencina, Argentina; Julio Singer, Brazil; Silvina San Martino, Argentina; Wenjun Li, US; and Ed Stanek US

how do we make up models to get better insight using limited information

How do we make up models to get better insight using limited information?

Study Design- Sampling

Response Error

Model Assumptions

An Example: What is a subject’s saturated fat intake?

SPH&HS, UMASS Amherst

seasons study umass worc
Seasons Study UMASS Worc

SPH&HS, UMASS Amherst

the problem simplified
The Problem-Simplified
  • Observe:
    • 1 Measure of SFat on each Subject
  • Assume:
    • Response Error (RE) Variance known
  • Question:
    • How do we estimate Subject’s True Sat Fat intake?
  • Begin with a Response Error Model … which leads to….
    • Mixed Model
    • Finite Population Mixed Model

Daisy

Lily

Rose

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population

Population

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population1

Set

Population

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Response

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Response

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Response

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Response

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Response

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Response

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slide15

Response……..

0

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4

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response error model for set
Response Error Model for Set

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summary response error model
Summary Response Error Model

Latent Value

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re parameterized re model
Re-parameterized RE Model

Mean Latent Value:

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sample space

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Sample Space
  • Response Error Model

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response error model

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Response Error Model

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mixed model mm
Mixed Model (MM)

Random Effect

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mixed model mm1
Mixed Model (MM)

Latent Value

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mixed model mm in action
Mixed Model (MM) in action

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mixed model mm2
Mixed Model (MM) ….

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mixed model mm3
Mixed Model (MM) ….??

Who Are They?

?

?

SPH&HS, UMASS Amherst

mixed model mm4
Mixed Model (MM)

??

??

What Does

it Mean?

?

?

SPH&HS, UMASS Amherst

sample space mm

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Sample Space (MM)

Artificial

Real

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mm latent values
MM-Latent Values?

Samples

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what are they for daisy
What are they (for Daisy)?

Samples

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what are they for rose
What are they (for Rose)?

Samples

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blups of the mm latent value
BLUPs of the MM-Latent Value

SPH&HS, UMASS Amherst

mse of blups for mm latent values
MSE of BLUPs for MM-Latent Values

Samples

Ave=0.986

Ave=3.768

SPH&HS, UMASS Amherst

mse of blups for true latent values
MSE of BLUPs for True-Latent Values

Samples

Ave=32.06

Ave=32.06

SPH&HS, UMASS Amherst

mse of blups p y
MSE of BLUPs |P=y

Samples

Ave=0.986

Ave=3.768

SPH&HS, UMASS Amherst

slide38

Finite Population Mixed Model (FPMM)

Population

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response error model1
Response Error Model

Latent Value

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fpmm sample space

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FPMM- Sample Space …

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fpmm sample space1
FPMM- Sample Space …

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fpmm sample space2
FPMM- Sample Space …

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-7

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fpmm sample space3
FPMM- Sample Space …

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-7

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fpmm sample space4
FPMM- Sample Space …

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-7

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fpmm sample space5
FPMM- Sample Space …

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fpmm sample space6

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FPMM- Sample Space

All sample points are Potentially Observable

SPH&HS, UMASS Amherst

fpmm blups of realized latent values3
FPMM- BLUPs of Realized Latent Values

Sample Sequence

SPH&HS, UMASS Amherst

comparison of mm blup and fpmm blup
Comparison of MM-BLUP and FPMM-BLUP

MM-BLUP

FPMM-BLUP

Target Random Variable

MM-Latent Value

Latent Value

SPH&HS, UMASS Amherst

comparison of mm blup and fpmm blup1
Comparison of MM-BLUP and FPMM-BLUP

MM-BLUP

FPMM-BLUP

Predictor

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comparison of fpmm blup and mm blup sample space

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Comparison of FPMM-BLUP and MM-BLUP-Sample Space

Artificial

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to compare focus on this sample space

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To Compare, Focus on…THIS Sample Space

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n 3 what is lily s latent value
n=3, What is Lily’s Latent value?

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  • Use n=3 subject effects for MM 1 possible sample set

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n 3 what is lily s latent value1
n=3, What is Lily’s Latent value?

11

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  • 8 sample points

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n 3 what is lily s latent value2
n=3, What is Lily’s Latent value?
  • 8x(6 permutations)=48 sample points

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n 3 what is lily s latent value3
n=3, What is Lily’s Latent value?

Combinations

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n 3 what is lily s latent value4
n=3, What is Lily’s Latent value?

192 Sample Points

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select one sequence
Select one sequence

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ave mse
Ave MSE

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-7

X

5.0

MM

4.6

16.2 FPMM

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ave mse1
Ave MSE

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-7

X

29.4

MM

17.7

34.3 FPMM

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conclusions
Conclusions

FPMM-BLUP

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Population

Sample Space

Design Based

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conclusions1
Conclusions

FPMM-BLUP

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Population

Evaluate Performance Conditional on the Sample

Design Based

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conclusions2
Conclusions

MM-BLUP

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Conceptual “Priors”

Model Based

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conclusions3
Conclusions

MM-BLUP

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Evaluate Performance Conditional on the Sample

Model Based

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conclusions4
Conclusions

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  • Evaluate Performance Conditional on the Sample
  • MSE for BLUPs not evaluated Correctly
    • Extends to WLS estimate of mean
  • MM-BLUP not always best
  • Sample Design is relevant only for derivation- not performance
  • Prior Distributions relevant only for derivation- not performance

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thanks to gary
Thanks to Gary

For keeping the focus on “reality”

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