finite population mixed models sampling, wls, and mixed...

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

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

Ed Stanek and Julio SingerEd Stanek and Julio SingerU of Mass, Amherst, and U of Mass, Amherst, and

U of Sao Paulo, BrazilU of Sao Paulo, Brazil

II ESAMP MeetingsNov 6, 2009Natal, Brazil

22

Finite Population Mixed Models Finite Population Mixed Models Research GroupResearch Group

Luz Mery Gonzalez, Columbia; Viviana Lencina, Argentina; Julio Singer, Brazil; Silvina San Martino, Argentina; Wenjun Li, US; and Ed Stanek US

BackgroundMotivation: – 2-stage cluster sample of hospitals

n Hospitals– m Appendectomy operations per hospital

– What is the average cost of an operation at a selected hospital (latent value)?

Choices:– Use average cost of m operations for selected

hospital– Use ‘shrunk’ cost- regressing to the mean for other

sample hospitals. Which should we use?


• Consider/Account for:

Study Design

Sampling

Response Error

• Model Assumptions

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

What is a subject’s saturated fat intake?

An Example


Seasons Study UMASS Seasons Study UMASS WorcWorc


Seasons Study UMASS Seasons Study UMASS WorcWorc--focus on 3 subjectsfocus on 3 subjects


The ProblemThe Problem--SimplifiedSimplifiedObserve:Observe:–– 1 Measure of 1 Measure of SFatSFat on each Subjecton each Subject

Assume: Assume: –– Response Error (RE) Variance knownResponse Error (RE) Variance known

Question: Question: –– How do we estimate SubjectHow do we estimate Subject’’s True Sat Fat intake?s True Sat Fat intake?

Begin with a Response Error Model Begin with a Response Error Model …… which leads towhich leads to……..–– Mixed ModelMixed Model–– Finite Population Mixed Model Finite Population Mixed Model

Daisy Lily Rose


PopulationPopulation


PopulationPopulationSetSet


11

Response

4


11

Response

0


9

Response

4


11

Response

4


9

Response

4


11

Response

0


11

Response……..

04


Response Error Model for SetResponse Error Model for Set( )Ejk R jk jkY Y E= +

99 1111

0.50.5 0.50.5

1kY

( )1kP Y

( )ER jk jY y=

1 10y =

00 44

0.50.5 0.50.5

2kY

( )2kP Y2 2y =

jk j jkY y E= +

( ) 2varR jk jY σ=

21 1σ =

22 4σ =

1j = 2j =


Summary Response Error ModelSummary Response Error Model

j j jY y E= +

1 1 1Y y E= + 2 2 2Y y E= +

21 1σ = 2

2 4σ =

Latent Value


ReRe--parameterized RE Modelparameterized RE Model

1

1 n

jj

yn

μ=

= ∑

2

2 2 2

2 Y y E

Eμ β= += + +

1 1 1

1 1 Y y E

Eμ β= += + +

Mean Latent ValueMean Latent Value--of what?:of what?: or the Setthe Population

1

1 N

jj

yN

μ=

= ∑SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 2020

2 μ β= + +1 μ β= + +

11E 12E21E 22E

21 1σ = 2

2 4σ =

Generating Response in the RE ModelGenerating Response in the RE Model


2 μ β= + +1 μ β= + +

11E 12E21E 22E

1E1Y 2Y 2E

Generating Response in the RE ModelGenerating Response in the RE Model


6 4 = + − +6 4 = + +

-1 1 -2 2

-1 -29 0

Generating an Observed Response Generating an Observed Response in the RE Modelin the RE Model


Sample SpaceSample Space

Response Error ModelResponse Error Model

9 4 11 4

9 0 11 0



j j jY Eμ β= + +

1j = 2j =9 4 11 4

9 0 11 0


Mixed Model (MM)Mixed Model (MM)*j j jY a Eμ= + +

1j = 2j =

Random Effect


Mixed Model (MM)Mixed Model (MM)*j j jY a Eμ= + +

1j = 2j =

( )E 0jaξ =

( ) 2var jaξ γ=

Latent Value*j j j

j j

Y a E

P E

μ= + +

= +


μ= + + μ= + +

Mixed Model (MM) in actionMixed Model (MM) in action

1 β2β ( )22

1

11

n

jj

yn

γ μ=

= −− ∑

11E 12E 21E 22E


μ= + + μ= + +

Mixed Model (MM) Mixed Model (MM) ……..

1β 2β

1β

11E 12E

1E 2β

21E 22E

2E*1Y *

2Y


μ= + + μ= + +

Mixed Model (MM) Mixed Model (MM) …….??.??

1 β2β

11E 12E 21E 22E

2E*1Y *

2Y2β 1E 1 β

Who Are They?? ?


6 = + +6 = + +

Mixed Model (MM)Mixed Model (MM)

-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


MMMM--Latent Values?Latent Values? *j j jY P E= +

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

33 22 11 1212 1010 22

11 22 --11 1212 1010 22

33 22 11 88 1010 --22

11 22 --11 88 1010 --22

1111 1010 11 44 22 22

99 1010 --11 44 22 22

1111 1010 11 00 22 --22

99 1010 --11 00 22 --22

*1Y *

2Y1P 2P1E 2E

Sam

ples


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

*j j jY P E= +


33 22 11 1212 1010 22

11 22 --11 1212 1010 22

33 22 11 88 1010 --22

11 22 --11 88 1010 --22

1111 1010 11 44 22 22

99 1010 --11 44 22 22

1111 1010 11 00 22 --22

99 1010 --11 00 22 --22

*1Y *

2Y1P 2P1E 2E

Sam

ples


*j j jY P E= +


33 22 11 1212 1010 22

11 22 --11 1212 1010 22

33 22 11 88 1010 --22

11 22 --11 88 1010 --22

1111 1010 11 44 22 22

99 1010 --11 44 22 22

1111 1010 11 00 22 --22

99 1010 --11 00 22 --22

*1Y *

2Y1P 2P1E 2E

Sam

ples

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


BLUPsBLUPs of the MMof the MM--Latent ValueLatent Value( )E 0jaξ =

( ) 2var jaξ γ=

( ) 2varR j jE σ=( )*

j j j

j j

Y a E

P E

μ= + +

= +

( )*ˆ ˆ ˆj j jP k Yμ μ= + −2

2 2jj

k γγ σ

=+

*

1

ˆn

j jj

w Yμ=

=∑ jj

kw

nk=

( ) 2 1ˆvar 1 jR j j j j

kP P k

nkξ σ−⎛ ⎞

− = +⎜ ⎟⎝ ⎠



3.13.1 22 0.810.81 11.511.5 1010 2.192.191.21.2 22 1.151.15 11.411.4 1010 1.861.863.13.1 22 0.710.71 7.77.7 1010 5.245.241.11.1 22 1.281.28 7.67.6 1010 5.795.7910.910.9 1010 1.281.28 4.44.4 22 5.795.798.98.9 1010 0.710.71 4.34.3 22 5.245.2410.810.8 1010 1.151.15 0.640.64 22 1.861.868.98.9 1010 0.810.81 0.520.52 22 2.192.19

1P 2P

Sam

ples

1P

MSE of MSE of BLUPsBLUPs for MMfor MM--Latent ValuesLatent Values

2P

Ave=0.986 Ave=3.768

( )2

1 1P P− ( )2

2 2P P−


MSE of MSE of BLUPsBLUPs ||P=yP=yDaisy (j=1)Daisy (j=1) Rose (j=2)Rose (j=2)

10.9010.90 1010 0.810.81 4.414.41 22 5.795.798.938.93 1010 1.151.15 4.294.29 22 5.245.2410.8410.84 1010 0.710.71 0.640.64 22 1.861.868.878.87 1010 1.281.28 0.520.52 22 2.192.19

1P 2P

Sam

ples

2P1P

MSE=Ave=0.986 MSE=Ave=3.768

( )( ) ( ) ( ) ( )* *

*

2 2 22 2 2 2

1

2 1ˆ ˆE | 1 1n

jR j j j j j jj j

j

kP P k w y k

nkσ μ σ

=

⎡ ⎤⎡ ⎤ −⎛ ⎞− = = − + − + +⎢ ⎥⎢ ⎥⎜ ⎟⎜ ⎟ ⎢ ⎥⎢ ⎥⎝ ⎠⎣ ⎦ ⎣ ⎦

∑P y

( )2

1 1P P− ( )2

2 2P P−


Population

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



1 1 1Y y E= +3 3 3Y y E= +

21 1σ = 2

3 4σ =

Latent Values s s

s s

Y y EEμ β

= +

= + +

2 2 2Y y E= +

22 100σ =

1 10y = 2 3y =3 2y =

1

1 N

ss

yN

μ=

= ∑

Accounting for Sampling

isU

11 12 1

21 22 2

1 2

N

NI

n n nN

U U UU U U

U U U

⎛ ⎞⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎝ ⎠

U

*

1

N

Ii is ss

Y U Y=

=∑

Indicator random variable, 1 if ithSelected sample subject is subject “s”

s s sY Eμ β= + +

1

N

i is ss

b U β=

=∑*

1

N

i is ss

E U E=

=∑

*Ii i iY b Eμ= + +*I n I Iμ= + +Y 1 U β U E



1 β3β ( )22

1

11

N

ss

yN

γ μ=

= −− ∑2 β

* *Ii i iY b Eμ= + +

1 β

11E 12E1i = 2i =

3β

21E 22E

μ= + + 1E 1E*1IY *1IY μ= + +*

1IY *1IY

1,...,i n=



1β 3β2β

1 β

11E 12E1i = 2i =

3β

21E 22E

μ= + + 1E1E*

1IY μ= + +*1IY*

1IY *1IY




1 β3β2 β

1 β

11E 12E1i = 2i =

μ= + + 1E*1IY μ= + +*

1IY *1IY2 β*

1IY

21E 22E

1E




1β 3β2β

1i = 2i =

μ= + +*1IY μ= + +*

1IY 2 β*1IY

21E 22E

1E3β

*1IY

21E 22E

1E


4545

FPMMFPMM-- Sample Space Sample Space ……

9 4

9 0 11 0

11 4



0 11 4 11

0 9 4 9



-7 4 13 4

-7 0 13 0SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 4848


0 13 4 13

0 -7 4 -7



-7 11 13 11

-7 9 13 9SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 5050


9 13 11 13

9 -7 11 -7


FPMMFPMM-- Sample Space Sample Space

11 1311139 -711 -7

-7 1113 11-7 913 9

9 411 49 011 0

0 114 110 9 4 9

0 134 130 -74 -7

-7 413 4-7 013 0

All sample points are Potentially Observable


FPMMFPMM-- BLUPsBLUPs of Realized Latent Valuesof Realized Latent Values

1β 3β2β

1i = 2i =

μ= + +*1IY μ= + +*

2IY



1 β3β2 β

1 β

11E 12E1i = 2i =

3β

21E 22E

μ= + +*1IY *1IY μ= + +*

1IY*2IY


1E 2E



( )22

1

11

N

ss

yN

γ μ=

= −− ∑

*Ii i iY b Eμ= + +


( )E 0p ib =

( ) 2varp ib γ=( ) 2varR i iE σ=

( ) 2varpR iE σ=( ) 2 2

1

1varN

pR i ss

EN

σ σ=

= = ∑

1β 3β2β



( )*Ii IiP Y k Y Y= + −

2

2 2k γγ σ

=+

*

1

1 n

Iii

Y Yn =

= ∑

( )* *

*

Ii i i

Ii i

Y b E

P E

μ= + +

= +

( ) ( )2

ˆvar 1 1pR Ii IiP P k nnσ

⎡ ⎤− = + −⎣ ⎦

( ) ( ) ( ) ( ) ( ) ( )22 22 21 1ˆ | 2 2 1 1pR Ii Ii j h j hhMSE P P I n k k k k yn n

σ σ μ⎡ ⎤− = − + + − + − −⎣ ⎦

Sample Sequence

2 2

1

1 n

h jjn

σ σ=

= ∑1

1 n

h jj

yn

μ=

= ∑SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 5656

Comparison of MMComparison of MM--BLUP and BLUP and FPMMFPMM--BLUPBLUP

MM-BLUP FPMM-BLUP

Target Random Variable

MM-Latent Value

Latent Value


Comparison of MMComparison of MM--BLUP and BLUP and FPMMFPMM--BLUPBLUP

MM-BLUP FPMM-BLUP

Predictor ( )*Ii IiP Y k Y Y= + −

*

1

1 n

Iii

Y Yn =

= ∑

2

2 2k γγ σ

=+

( )*ˆ ˆ ˆj j jP k Yμ μ= + −

2

2 2jj

k γγ σ

=+

*

1

ˆn

j jj

w Yμ=

=∑


Comparison of FPMMComparison of FPMM--BLUP and BLUP and MMMM--BLUPBLUP--Sample SpaceSample Space

11 1311139 -711 -7

-7 1113 11-7 913 9

0 114 110 9 4 9 0 134 130 -74 -7

-7 413 4-7 013 09 411 4

9 011 0

1 123 121 8 3 8Artific

ial


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

11 1311139 -711 -7

-7 1113 11-7 913 9

0 114 110 9 4 9 0 134 130 -74 -7

-7 413 4-7 013 09 411 4

9 011 0

1 123 121 8 3 8


Bigger Sample (n=3) Population (N=4)


n=3, What is Lilyn=3, What is Lily’’s Latent value?s Latent value?•• Use n=3 subject effects for MMUse n=3 subject effects for MM1 possible sample set1 possible sample set

11

4

1311

4

-7

9

0

-7 11

0

-7

11

4

1311

4

3

9

0

-711

0

3


n=3, What is Lilyn=3, What is Lily’’s Latent value?s Latent value?•• 8 sample points8 sample points

11 1311-7

11 1311-7


n=3, What is Lilyn=3, What is Lily’’s Latent value?s Latent value?•• 8x(6 permutations)=48 sample points8x(6 permutations)=48 sample points


n=3, What is Lilyn=3, What is Lily’’s Latent value?s Latent value?

CombinationsCombinations4

43

Nn

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠


n=3, What is Lilyn=3, What is Lily’’s Latent value?s Latent value?192 Sample Points192 Sample Points


Select one sequenceSelect one sequence


Select one sequence, Select one sequence, Observe Sample PointObserve Sample Point


FPMMFPMM--Average MSE of Predictor over Average MSE of Predictor over PermutationsPermutations


11 11 11 111311-7

Ave MSEAve MSE

5.05.0

16.2 16.2 FPMMFPMM

4.64.6

MMMM

XX


11 11 11 111311-7

34.3 34.3 FPMMFPMM

17.717.7

29.429.4

Ave MSEAve MSE

MMMM

XX

Summary MSE ResultsSample Set MM FPMM

Set j=1 j=2 j=3 Target MSE MSE1 Daisy Lily Rose Mean 2.667 11.6671 Daisy Lily Rose Daisy 0.9931 15.6791 Daisy Lily Rose Lily 12.3195 34.1651 Daisy Lily Rose Rose 3.7561 9.785

2 Daisy Lily Violet Mean 7.409 14.0002 Daisy Lily Violet Daisy 0.993 18.3622 Daisy Lily Violet Lily 17.765 34.311 2 Daisy Lily Violet Violet 18.929 18.487

3 Daisy Rose Violet Mean 2.464 3.3333 Daisy Rose Violet Daisy 0.994 3.647 3 Daisy Rose Violet Rose 3.540 3.304 3 Daisy Rose Violet Violet 13.563 17.224

4 Lily Rose Violet Mean 3.066 14.333 4 Lily Rose Violet Lily 4.593 16.177 4 Lily Rose Violet Rose 3.345 13.751 4 Lily Rose Violet Violet 4.147 15.027


ConclusionsConclusionsPopulation

11 11 11 111311-7

Sample Space

FPMM-BLUP( )*

Ii IiP Y k Y Y= + −Design BasedDesign Based


ConclusionsConclusionsPopulation

11 11 11 111311-7

Design BasedDesign Based FPMM-BLUP

( )*Ii IiP Y k Y Y= + −

Evaluate Performance Conditional on the

Sample


ConclusionsConclusions

111311-7

Model BasedModel BasedMM-BLUP


Conceptual “Priors”



111311-7

Model BasedModel BasedMM-BLUP


Evaluate Performance Conditional on the

Sample



To Evaluate Performance of BLUP Estimators:– For Mixed Model: Condition on P=y

i.e. MM Latent Values match subject Latent Values

– For the FPMM: Condition on the sample set

MSE for BLUPs not evaluated Correctly– Extends to WLS estimate of mean

MM-BLUP not always best 111311-7


ThanksThanks


Any thoughts? Any thoughts? Next steps?Next steps?Questions?Questions?

finite population mixed models sampling, wls, and mixed...

Documents