SPH&HS, UMASS Amherst SPH&HS, UMASS Amherst 1 Sampling, WLS, and Mixed Sampling, WLS, and Mixed Models Models Ed Stanek and Julio Singer Ed Stanek and Julio Singer U of Mass, Amherst, and U of Mass, Amherst, and U of Sao Paulo, Brazil U of Sao Paulo, Brazil II ESAMP Meetings Nov 6, 2009 Natal, Brazil 2 Finite Population Mixed Models Finite Population Mixed Models Research Group Research Group Luz Mery Gonzalez, Columbia; Viviana Lencina, Argentina; Julio Singer, Brazil; Silvina San Martino, Argentina; Wenjun Li, US; and Ed Stanek US Background Motivation: – 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? SPH&HS, UMASS Amherst SPH&HS, UMASS Amherst 4 • 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 SPH&HS, UMASS Amherst SPH&HS, UMASS Amherst 5 Seasons Study UMASS Seasons Study UMASS Worc Worc SPH&HS, UMASS Amherst SPH&HS, UMASS Amherst 6 Seasons Study UMASS Seasons Study UMASS Worc Worc- focus on 3 subjects focus on 3 subjects
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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?
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 44
• 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
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 55
Seasons Study UMASS Seasons Study UMASS WorcWorc
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 66
Seasons Study UMASS Seasons Study UMASS WorcWorc--focus on 3 subjectsfocus on 3 subjects
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The ProblemThe Problem--SimplifiedSimplifiedObserve:Observe:–– 1 Measure of 1 Measure of SFatSFat on each Subjecton each Subject
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
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PopulationPopulation
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PopulationPopulationSetSet
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 1010
11
Response
4
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 1111
11
Response
0
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 1212
9
Response
4
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 1313
11
Response
4
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 1414
9
Response
4
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 1515
11
Response
0
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11
Response……..
04
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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 =
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 1818
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
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 1919
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
Comparison of MMComparison of MM--BLUP and BLUP and FPMMFPMM--BLUPBLUP
MM-BLUP FPMM-BLUP
Target Random Variable
MM-Latent Value
Latent Value
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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=
=∑
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 5858
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
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 5959
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
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 6060
Bigger Sample (n=3) Population (N=4)
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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
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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
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 6363
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
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 6464
n=3, What is Lilyn=3, What is Lily’’s Latent value?s Latent value?
CombinationsCombinations4
43
Nn
⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 6565
n=3, What is Lilyn=3, What is Lily’’s Latent value?s Latent value?192 Sample Points192 Sample Points
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Select one sequenceSelect one sequence
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Select one sequence, Select one sequence, Observe Sample PointObserve Sample Point
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FPMMFPMM--Average MSE of Predictor over Average MSE of Predictor over PermutationsPermutations
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11 11 11 111311-7
Ave MSEAve MSE
5.05.0
16.2 16.2 FPMMFPMM
4.64.6
MMMM
XX
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 7070
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
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
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 7272
ConclusionsConclusionsPopulation
11 11 11 111311-7
Sample Space
FPMM-BLUP( )*
Ii IiP Y k Y Y= + −Design BasedDesign Based
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 7373
ConclusionsConclusionsPopulation
11 11 11 111311-7
Design BasedDesign Based FPMM-BLUP
( )*Ii IiP Y k Y Y= + −
Evaluate Performance Conditional on the
Sample
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 7474
ConclusionsConclusions
111311-7
Model BasedModel BasedMM-BLUP
( )*ˆ ˆ ˆj j jP k Yμ μ= + −
Conceptual “Priors”
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 7575
ConclusionsConclusions
111311-7
Model BasedModel BasedMM-BLUP
( )*ˆ ˆ ˆj j jP k Yμ μ= + −
Evaluate Performance Conditional on the
Sample
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 7676
ConclusionsConclusions
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
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 7777
ThanksThanks
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 7878
Any thoughts? Any thoughts? Next steps?Next steps?Questions?Questions?