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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?

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 information?

SPH&HS, UMASS Amherst


Seasons study umass worc1
Seasons Study UMASS Worc information?

SPH&HS, UMASS Amherst


The problem simplified
The Problem-Simplified information?

  • 1 Measure of Sat. Fat on each Subject

  • Assume Response Error Variance known

  • How do we estimate Subject’s True Sat Fat intake?

Daisy

Lily

Rose

SPH&HS, UMASS Amherst


Population information?

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Population information?

Set

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Response information?

4

11

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Response information?

0

11

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Response information?

4

9

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Response information?

4

11

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Response information?

4

9

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Response information?

0

11

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Response information?

0

11

4

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Response error model
Response Error Model information?

SPH&HS, UMASS Amherst


Response error model1
Response Error Model information?

Latent Value

SPH&HS, UMASS Amherst


Mean latent value

Response Error Model information?

Mean Latent Value:

SPH&HS, UMASS Amherst


Response Error Model information?

SPH&HS, UMASS Amherst


Response Error Model information?

SPH&HS, UMASS Amherst


Response Error Model information?

-1

1

-2

2

-2

-1

SPH&HS, UMASS Amherst


Sample space

11 information?

9

9

11

11

9

11

9

0

4

4

0

4

4

0

0

Sample Space

  • Response Error Model

SPH&HS, UMASS Amherst


Response error model2

9 information?

11

9

11

4

4

0

0

Response Error Model

SPH&HS, UMASS Amherst


Mixed model mm
Mixed Model (MM) information?

Random Effect

SPH&HS, UMASS Amherst


Mixed model mm1
Mixed Model (MM) information?

SPH&HS, UMASS Amherst


Mixed model mm2
Mixed Model (MM) information?

SPH&HS, UMASS Amherst


Mixed model mm3
Mixed Model (MM) information?

Who Are They?

?

?

SPH&HS, UMASS Amherst


Mixed model mm4
Mixed Model (MM) information?

??

??

What Does

it Mean?

?

?

SPH&HS, UMASS Amherst


Sample space mm

8 information?

1

8

3

9

9

11

11

4

4

0

0

12

12

3

1

Sample Space (MM)

Artificial

Real

SPH&HS, UMASS Amherst


Mm latent values
MM Latent Values? information?

Samples

SPH&HS, UMASS Amherst


Blups of the mm latent value
BLUPs of the MM-Latent Value information?

SPH&HS, UMASS Amherst


Mse of blups for mm latent values
MSE of BLUPs for MM Latent Values? information?

Samples

Ave=0.986

Ave=3.768

SPH&HS, UMASS Amherst


Mse of blups vs true latent values
MSE of BLUPs vs True Latent Values? information?

Samples

Ave=32.06

Ave=32.06

SPH&HS, UMASS Amherst


Mse of blups vs true latent values1
MSE of BLUPs vs True Latent Values? information?

Samples

Ave=0.986

Ave=3.768

SPH&HS, UMASS Amherst


Finite Population Mixed Model (FPMM) information?

Population

SPH&HS, UMASS Amherst


Response error model3
Response Error Model information?

Latent Value

SPH&HS, UMASS Amherst


Finite population mixed model fpmm
Finite Population Mixed Model (FPMM) information?

SPH&HS, UMASS Amherst


Finite population mixed model fpmm1
Finite Population Mixed Model (FPMM) information?

SPH&HS, UMASS Amherst


Finite population mixed model fpmm2
Finite Population Mixed Model (FPMM) information?

SPH&HS, UMASS Amherst


Finite population mixed model fpmm3
Finite Population Mixed Model (FPMM) information?

SPH&HS, UMASS Amherst


Fpmm sample space

9 information?

11

9

11

4

4

0

0

FPMM- Sample Space …

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Fpmm sample space1

0 information?

4

9

9

FPMM- Sample Space …

11

11

0

4

SPH&HS, UMASS Amherst


Fpmm sample space2
FPMM- Sample Space … information?

4

4

-7

13

0

0

13

-7

SPH&HS, UMASS Amherst


Fpmm sample space3
FPMM- Sample Space … information?

13

13

0

4

-7

-7

4

0

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Fpmm sample space4
FPMM- Sample Space … information?

11

11

-7

13

9

9

13

-7

SPH&HS, UMASS Amherst


Fpmm sample space5
FPMM- Sample Space … information?

13

13

9

11

-7

-7

11

9

SPH&HS, UMASS Amherst


Fpmm sample space6

11 information?

4

0

9

11

9

9

9

0

4

4

0

13

13

11

4

11

4

13

13

0

11

11

11

-7

0

-7

13

13

4

11

4

0

-7

-7

-7

0

-7

9

13

11

4

13

9

9

0

-7

-7

FPMM- Sample Space

All sample points are Potentially Observable

SPH&HS, UMASS Amherst


Fpmm blups of realized latent values
FPMM- BLUPs of Realized Latent Values information?

SPH&HS, UMASS Amherst


Fpmm blups of realized latent values1
FPMM- BLUPs of Realized Latent Values information?

SPH&HS, UMASS Amherst


Fpmm blups of realized latent values2
FPMM- BLUPs of Realized Latent Values information?

SPH&HS, UMASS Amherst


Fpmm blups of realized latent values3
FPMM- BLUPs of Realized Latent Values information?

Sample Sequence

SPH&HS, UMASS Amherst


Comparison of mm blup and fpmm blup
Comparison of MM-BLUP and FPMM-BLUP information?

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 information?

MM-BLUP

FPMM-BLUP

Predictor

SPH&HS, UMASS Amherst


Comparison of fpmm blup and mm blup sample space

12 information?

3

8

1

11

0

9

9

4

11

9

4

9

0

0

4

12

8

1

3

4

11

13

13

13

4

13

11

11

0

11

11

-7

0

-7

13

11

4

4

13

0

-7

-7

0

-7

-7

9

13

13

4

11

9

-7

9

0

-7

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

Artificial

SPH&HS, UMASS Amherst


To compare focus on this sample space

12 information?

3

8

1

9

9

11

11

4

0

9

9

4

4

0

0

12

8

1

3

4

11

13

13

13

4

11

13

0

11

11

11

-7

-7

0

13

4

11

13

4

-7

0

-7

-7

0

-7

9

13

4

11

13

9

-7

9

-7

0

To Compare, Focus on…THIS Sample Space

SPH&HS, UMASS Amherst


Sample space mm1

8 information?

1

8

3

9

9

11

11

4

4

0

0

12

12

3

1

Sample Space (MM)

Artificial

Real

SPH&HS, UMASS Amherst


Sample space mm2

12 information?

3

8

1

9

11

11

9

4

4

0

0

12

8

1

3

Sample Space (MM)

Artificial

Real

SPH&HS, UMASS Amherst


Mse of blups vs true latent values2
MSE of BLUPs vs True Latent Values? information?

Samples

Ave=32.06

Ave=32.06

SPH&HS, UMASS Amherst


Mixed model mm5
Mixed Model (MM) information?

SPH&HS, UMASS Amherst


Blups of the mm latent value1
BLUPs of the MM-Latent Value information?

SPH&HS, UMASS Amherst


Finite population mixed model fpmm4
Finite Population Mixed Model (FPMM) information?

SPH&HS, UMASS Amherst


Fpmm sample space7

4 information?

4

-7

13

0

0

13

-7

FPMM- Sample Space …

SPH&HS, UMASS Amherst


Fpmm sample space8

13 information?

13

0

4

-7

-7

4

0

FPMM- Sample Space …

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Fpmm sample space9

0 information?

4

9

9

11

11

0

4

FPMM- Sample Space …

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Fpmm sample space10

9 information?

11

9

11

4

4

0

0

FPMM- Sample Space …

SPH&HS, UMASS Amherst


Fpmm sample space11

11 information?

11

-7

13

9

13

9

-7

FPMM- Sample Space …

SPH&HS, UMASS Amherst


SPH&HS, UMASS Amherst information?


Finite population mixed model fpmm5
Finite Population Mixed Model (FPMM) information?

SPH&HS, UMASS Amherst


SPH&HS, UMASS Amherst information?


Sample space1

9 information?

11

4

0

Sample Space

  • Response Error Model

SPH&HS, UMASS Amherst


Sample space2

9 information?

11

4

0

Sample Space

  • Response Error Model

SPH&HS, UMASS Amherst


Response error model4
Response Error Model information?

SPH&HS, UMASS Amherst


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