SPH&HS, UMASS Amherst 1 Sampling, WLS, and Mixed Models Festschrift to Honor Professor Gary Koch Ed...

SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 11

Sampling, WLS, and Sampling, WLS, and Mixed Models Mixed Models

Festschrift to Honor Festschrift to Honor Professor Gary KochProfessor Gary Koch

Ed Stanek and Julio SingerEd Stanek and Julio Singer

U of Mass, Amherst, and U of Mass, Amherst, and

U of Sao Paulo, BrazilU of Sao Paulo, Brazil


Finite Population Mixed Models Finite Population Mixed Models Research GroupResearch 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?


Seasons Study UMASS WorcSeasons Study UMASS Worc


The Problem-SimplifiedThe Problem-Simplified

1 Measure of Sat. Fat on each Subject1 Measure of Sat. Fat on each Subject Assume Response Error Variance knownAssume Response Error Variance known How do we estimate Subject’s True Sat How do we estimate Subject’s True Sat

Fat intake?Fat intake?

Daisy Lily Rose


Population


PopulationSet


11

Response

4


11

Response

0


9

Response

4


11

Response

4


9

Response

4


11

Response

0


11

Response

04


Response Error ModelResponse Error Model jk R jk jkY E Y E

11 22

99 1111

0.50.5 0.50.5

1kY

k

1kP Y

R jk jE Y y

1 10y

11 22

00 44

0.50.5 0.50.5

2kY

k

2kP Y2 2y

jk j jkY y E

2varR jk jY

21 1

22 4

1j 2j


Response Error ModelResponse Error Model

j j jY y E

1 1 1Y y E 2 2 2Y y E

21 1 2

2 4

Latent Value


Mean Latent Value:Mean Latent Value:1

1 n

jj

yn

2 2 2

2 2

Y y E

E

1 1 1

1 1

Y y E

E



2 1

11E 12E 21E 22E


21 1 2

2 4


2 1

11E 12E 21E 22E


1E1Y 2Y 2E


6 4 6 4

-1 1 -2 2


-1 -29 0


Sample SpaceSample Space


9 4 11 4

9 0 11 0

9 4 11 4

9 0 11 0



j j jY E

1j 2j

9 4 11 4

9 0 11 0


Mixed Model (MM)Mixed Model (MM)

*j j jY a E

1j 2j

Random Effect

0jE a

2var ja



1 2 22

1

1

1

n

jj

yn

11E 12E 21E 22E



1 2

1

11E 12E

1E 2

21E 22E

2E*

1Y*

2Y



1 2

11E 12E 21E 22E

2E*

1Y*

2Y2 1E 1

Who Are They?? ?


6 6


-4

1 1 2 2

23 12-4 1 4

What Does it Mean?? ?

4

????


Sample Space (MM)Sample Space (MM)

3 12

9 4 11 4

9 0 11 0

1 8 3 8

1 12

Real

Artificial


MM Latent MM Latent Values?Values?

*j j j

j j

Y a E

P E

Daisy (j=1)Daisy (j=1) Rose (j=2)Rose (j=2)

1111 1010 11 44 22 22

99 1010 -1-1 44 22 22

1111 1010 11 00 22 -2-2

99 1010 -1-1 00 22 -2-2

33 22 11 1212 1010 22

11 22 -1-1 1212 1010 22

33 22 11 88 1010 -2-2

11 22 -1-1 88 1010 -2-2

*1Y

*2Y1P 2P1E 2E

Sam

ple

s


BLUPs of the MM-Latent BLUPs of the MM-Latent ValueValue

0jE a

2var ja

2varR j jE *j j j

j j

Y a E

P E

*ˆ ˆ ˆj j jP k Y 2

2 2jj

k

*

1

ˆn

j jj

w Y

jj

kw

nk

2 1ˆvar 1 jR j j j j

kP P k

nk

*j j jY a E


MSE of BLUPs for MM Latent MSE of BLUPs for MM Latent Values?Values?


MSEMSE MSEMSE

10.910.900

1010 0.810.81 4.414.41 22 5.795.79

8.938.93 1010 1.151.15 4.294.29 22 5.245.24

10.810.844

1010 0.710.71 0.640.64 22 1.861.86

8.878.87 1010 1.281.28 0.520.52 22 2.192.19

3.133.13 22 1.281.28 11.411.488

1010 2.192.19

1.161.16 22 0.710.71 11.311.366

1010 1.861.86

3.073.07 22 1.151.15 7.717.71 1010 5.245.24

1.101.10 22 0.810.81 7.597.59 1010 5.795.79

1P 2P

Sam

ple

s

2P1P

Ave=0.986 Ave=3.768


MSE of BLUPs vs True Latent MSE of BLUPs vs True Latent Values?Values?


MSEMSE MSEMSE

10.910.900

1010 0.810.81 4.414.41 22 5.795.79

8.938.93 1010 1.151.15 4.294.29 22 5.245.24

10.810.844

1010 0.710.71 0.640.64 22 1.861.86

8.878.87 1010 1.281.28 0.520.52 22 2.192.19

3.133.13 1010 47.147.199

11.411.488

1010 89.889.844

1.161.16 1010 78.178.166

11.311.366

1010 87.687.655

3.073.07 1010 47.947.999

7.717.71 1010 32.632.611

1.101.10 1010 79.179.188

7.597.59 1010 31.331.300

1P 2P

Sam

ple

s

2P1P

Ave=32.06 Ave=32.06




MSEMSE MSEMSE

10.910.900

1010 0.810.81 4.414.41 22 5.795.79

8.938.93 1010 1.151.15 4.294.29 22 5.245.24

10.810.844

1010 0.710.71 0.640.64 22 1.861.86

8.878.87 1010 1.281.28 0.520.52 22 2.192.19

1P 2P

Sam

ple

s

2P1P

Ave=0.986 Ave=3.768

* *

*

2 2 22 2 2 2

1

2 1ˆ ˆ| 1 1

nj

R j j j j j jj jj

kE P P k w y k

nk

P y


Population

Finite Population Mixed Model (FPMM)Finite Population Mixed Model (FPMM)



1 1 1Y y E 3 3 3Y y E

21 1 2

3 4

Latent Value

s s s

s s

Y y E

E

2 2 2Y y E

22 100

1 10y 2 3y

3 2y

1

1 N

ss

yN


Finite Population Mixed Model Finite Population Mixed Model (FPMM)(FPMM)

1 3 22

1

1

1

N

ss

yN

2

*Ii i iY b E

1

11E 12E1i 2i

3

21E 22E

1E 1E*1IY*1IY *

1IY*1IY



1 32

*Ii i iY b E

1

11E 12E1i 2i

3

21E 22E

1E1E*1IY *

1IY*1IY

*1IY



1 32

*Ii i iY b E

1

11E 12E1i 2i

1E*1IY *

1IY*1IY2 *

1IY

21E 22E

1E



1 32

*Ii i iY b E

1i 2i

*1IY *

1IY 2 *1IY

21E 22E

1E3*1IY

21E 22E

1E


FPMM- Sample Space …FPMM- Sample Space …

9 4 11 4

9 0 11 0



0 11 4 11

0 9 4 9



-7 4 13 4

-7 0 13 0



0 13 4 13

0 -7 4 -7



-7 11 13 11

-7 9 13 9



9 13 11 13

9 -7 11 -7


FPMM- Sample Space FPMM- Sample Space

111311139 -711-7

-71113 11-7 913 9

9 411 49 011 0

0 114 110 94 9

0 134 130 -74 -7

-7 413 4-7 013 0

All sample points are Potentially Observable


FPMM- BLUPs of Realized Latent FPMM- BLUPs of Realized Latent ValuesValues

1 32

*Ii i iY b E

1i 2i

*1IY *

1IY


1 32

*Ii i iY b E

1

11E 12E1i 2i

3

21E 22E

*1IY*1IY *

1IY*1IY


1E 2E


22

1

1

1

N

ss

yN

*Ii i iY b E


0p iE b

2varp ib 2varR i iE

2varpR iE 2 2

1

1var

N

pR i ss

EN

1 32


*Ii i iY b E


*Ii IiP Y k Y Y

2

2 2k

*

1

1 n

Iii

Y Yn

*Ii i i

Ii i

Y b E

P E

2

ˆvar 1 1pR Ii IP P k nn

22 22 21 1ˆ | 2 2 1 1pR Ii Ii j h jh hMSE P P I n k k k k y

n n

Sample Sequenc

e


Comparison of MM-BLUP and Comparison of MM-BLUP and FPMM-BLUPFPMM-BLUP

MM-BLUP FPMM-BLUP

Target Rando

m Variabl

e

MM-Latent Value

Latent Value


Comparison of MM-BLUP and Comparison of MM-BLUP and FPMM-BLUPFPMM-BLUP

MM-BLUP FPMM-BLUP

Predictor

*Ii IiP Y k Y Y

*

1

1 n

Iii

Y Yn

2

2 2k

*ˆ ˆ ˆj j jP k Y

2

2 2jj

k

*

1

ˆn

j jj

w Y


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

111311139 -711-7

-71113 11-7 913 9

0 114 110 94 9 0 134 13

0 -74 -7

-7 413 4-7 013 09 411 4

9 011 0

1 123 121 8 3 8Arti

ficial


To Compare, Focus onTo Compare, Focus on…THIS Sample Space…THIS Sample Space

111311139 -711-7

-71113 11-7 913 9

0 114 110 94 9 0 134 13

0 -74 -7

-7 413 4-7 013 09 411 4

9 011 0

1 123 121 8 3 8



3 12

9 4 11 4

9 0 11 0

1 8 3 8

1 12

Real

Artificial



1 123 121 8 3 8

Real

Artificial

9 411 49 011 0




MSEMSE MSEMSE

10.910.900

1010 0.810.81 4.414.41 22 5.795.79

8.938.93 1010 1.151.15 4.294.29 22 5.245.24

10.810.844

1010 0.710.71 0.640.64 22 1.861.86

8.878.87 1010 1.281.28 0.520.52 22 2.192.19

3.133.13 1010 47.147.199

11.411.488

1010 89.889.844

1.161.16 1010 78.178.166

11.311.366

1010 87.687.655

3.073.07 1010 47.947.999

7.717.71 1010 32.632.611

1.101.10 1010 79.179.188

7.597.59 1010 31.331.300

1P 2P

Sam

ple

s

2P1P

Ave=32.06 Ave=32.06



1 2

1

11E 12E

1E 2

21E 22E

2E*

1Y*

2Y


BLUPs of the MM-Latent BLUPs of the MM-Latent ValueValue

0jE a

2var ja

2varR j jE *j j j

j j

Y a E

P E

*ˆ ˆ ˆj j jP k Y 2

2 2jj

k

*

1

ˆn

j jj

w Y

jj

kw

nk

2 1ˆvar 1 jR j j j j

kP P k

nk

*j j jY a E



1 32

*Ii i iY b E

1i 2i

*1IY *

1IY 2 *1IY

21E 22E

1E3*1IY

21E 22E

1E



-7 413 4-7 013 0



0 134 130 -74 -7



0 114 110 94 9



9 411 49 011 0



-7 1113 11-7 913 9


11E 12E

21E 22E

21E 22E

1E 1E1E

*1IY

*1IY

*1IY


*2 IY *

1 IY


1 3 22

1

1

1

N

ss

yN

2

*Ii i iY b E


11E 12E

11E 12E 11E 12E

11E

1E

21E 22E

21E 22E

21E 22E




4

9 11 0


Response Error Response Error ModelModel

2 2 2

2 2

Y y E

E

1 1 1

1 1

Y y E

E

SPH&HS, UMASS Amherst 1 Sampling, WLS, and Mixed Models Festschrift to Honor Professor Gary Koch Ed...

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