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 SingerU of Mass, Amherst, and U of Mass, Amherst, and
U of Sao Paulo, BrazilU of Sao Paulo, Brazil
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 22
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
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 33
How do we make up models to get better insight using limited information?
Study Design- SamplingResponse ErrorModel Assumptions
An Example: What is a subject’s saturated fat intake?
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 44
Seasons Study UMASS WorcSeasons Study UMASS Worc
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 55
Seasons Study UMASS WorcSeasons Study UMASS Worc
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 66
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
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 77
Population
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 88
PopulationSet
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 99
11
Response
4
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 1010
11
Response
0
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 1111
9
Response
4
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 1212
11
Response
4
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 1313
9
Response
4
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 1414
11
Response
0
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 1515
11
Response
04
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 1616
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
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 1717
Response Error ModelResponse Error Modelj j jY y E
1 1 1Y y E 2 2 2Y y E
21 1 2
2 4
Latent Value
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 1818
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
Response Error ModelResponse Error Model
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 1919
2 1
11E 12E 21E 22E
Response Error ModelResponse Error Model
21 1 2
2 4
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 2020
2 1
11E 12E 21E 22E
Response Error ModelResponse Error Model
1E1Y 2Y 2E
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 2121
6 4 6 4
-1 1 -2 2
Response Error ModelResponse Error Model
-1 -29 0
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 2222
Sample SpaceSample Space Response Error ModelResponse Error Model
9 4 11 4
9 0 11 0
9 4 11 49 0 11 0
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 2323
Response Error ModelResponse Error Modelj j jY E
1j 2j
9 4 11 49 0 11 0
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 2424
Mixed Model (MM)Mixed Model (MM)*j j jY a E
1j 2j
Random Effect
0jE a
2var ja
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 2525
Mixed Model (MM)Mixed Model (MM)1
2 22
1
11
n
jj
yn
11E 12E 21E 22E
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 2626
Mixed Model (MM)Mixed Model (MM)1
2
1
11E 12E
1E 2
21E 22E
2E*
1Y*
2Y
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 2727
Mixed Model (MM)Mixed Model (MM)1
2
11E 12E 21E 22E
2E*
1Y*
2Y2 1E 1
Who Are They?? ?
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 2828
6 6
Mixed Model (MM)Mixed Model (MM)-4
1 1 2 2
23 12-4 1 4
What Does it Mean?? ?
4
????
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 2929
Sample Space (MM)Sample Space (MM)
3 12
9 4 11 4
9 0 11 0
1 8 3 8
1 12
Real
Artificial
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 3030
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 2299 1010 -1-1 44 22 221111 1010 11 00 22 -2-299 1010 -1-1 00 22 -2-233 22 11 1212 1010 2211 22 -1-1 1212 1010 2233 22 11 88 1010 -2-211 22 -1-1 88 1010 -2-2
*1Y
*2Y1P 2P1E 2E
Sam
ples
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 3131
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
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 3232
MSE of BLUPs for MM Latent MSE of BLUPs for MM Latent Values?Values?
Daisy (j=1)Daisy (j=1) Rose (j=2)Rose (j=2)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 5.245.2410.810.8
441010 0.710.71 0.640.64 22 1.861.86
8.878.87 1010 1.281.28 0.520.52 22 2.192.193.133.13 22 1.281.28 11.411.4
881010 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 5.245.241.101.10 22 0.810.81 7.597.59 1010 5.795.79
1P 2P
Sam
ples
2P1P
Ave=0.986 Ave=3.768
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 3333
MSE of BLUPs vs True Latent MSE of BLUPs vs True Latent Values?Values?
Daisy (j=1)Daisy (j=1) Rose (j=2)Rose (j=2)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 5.245.2410.810.8
441010 0.710.71 0.640.64 22 1.861.86
8.878.87 1010 1.281.28 0.520.52 22 2.192.193.133.13 1010 47.147.1
9911.411.4
881010 89.889.8
441.161.16 1010 78.178.1
6611.311.3
661010 87.687.6
553.073.07 1010 47.947.9
997.717.71 1010 32.632.6
111.101.10 1010 79.179.1
887.597.59 1010 31.331.3
00
1P 2P
Sam
ples
2P1P
Ave=32.06 Ave=32.06
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 3434
MSE of BLUPs vs True Latent MSE of BLUPs vs True Latent Values?Values?
Daisy (j=1)Daisy (j=1) Rose (j=2)Rose (j=2)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 5.245.2410.810.8
441010 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
ples
2P1P
Ave=0.986 Ave=3.768
* *
*
2 2 22 2 2 2
1
2 1ˆ ˆ| 1 1n
jR j j j j j jj j
j
kE P P k w y k
nk
P y
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 3535
Population Finite Population Mixed Model (FPMM)Finite Population Mixed Model (FPMM)
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 3636
Response Error ModelResponse Error Model
1 1 1Y y E 3 3 3Y y E
21 1 2
3 4
Latent Value
s s s
s s
Y y EE
2 2 2Y y E
22 100
1 10y 2 3y
3 2y
1
1 N
ss
yN
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 3737
Finite Population Mixed Model Finite Population Mixed Model (FPMM)(FPMM)
1 3 22
1
11
N
ss
yN
2
*Ii i iY b E
1
11E 12E1i 2i
3
21E 22E
1E 1E*1IY*1IY *
1IY*1IY
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 3838
Finite Population Mixed Model Finite Population Mixed Model (FPMM)(FPMM)
1 32
*Ii i iY b E
1
11E 12E1i 2i
3
21E 22E
1E1E*1IY *
1IY*1IY
*1IY
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 3939
Finite Population Mixed Model Finite Population Mixed Model (FPMM)(FPMM)
1 32
*Ii i iY b E
1
11E 12E1i 2i
1E*1IY *
1IY*1IY2 *
1IY
21E 22E
1E
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 4040
Finite Population Mixed Model Finite Population Mixed Model (FPMM)(FPMM)
1 32
*Ii i iY b E
1i 2i
*1IY *
1IY 2 *1IY
21E 22E
1E3*1IY
21E 22E
1E
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 4141
FPMM- Sample Space …FPMM- Sample Space …
9 4 11 4
9 0 11 0
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 4242
FPMM- Sample Space …FPMM- Sample Space …
0 11 4 11
0 9 4 9
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 4343
FPMM- Sample Space …FPMM- Sample Space …
-7 4 13 4
-7 0 13 0
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 4444
FPMM- Sample Space …FPMM- Sample Space …
0 13 4 13
0 -7 4 -7
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 4545
FPMM- Sample Space …FPMM- Sample Space …
-7 11 13 11
-7 9 13 9
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 4646
FPMM- Sample Space …FPMM- Sample Space …
9 13 11 13
9 -7 11 -7
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 4747
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
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 4848
FPMM- BLUPs of Realized Latent FPMM- BLUPs of Realized Latent ValuesValues
1 32
*Ii i iY b E
1i 2i
*1IY *
1IY
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 4949
1 32
*Ii i iY b E
1
11E 12E1i 2i
3
21E 22E
*1IY*1IY *
1IY*1IY
FPMM- BLUPs of Realized Latent FPMM- BLUPs of Realized Latent ValuesValues
1E 2E
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 5050
22
1
11
N
ss
yN
*Ii i iY b E
FPMM- BLUPs of Realized Latent FPMM- BLUPs of Realized Latent ValuesValues
0p iE b
2varp ib 2varR i iE
2varpR iE 2 2
1
1varN
pR i ss
EN
1 32
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 5151
*Ii i iY b E
FPMM- BLUPs of Realized Latent FPMM- BLUPs of Realized Latent ValuesValues
*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 yn n
Sample Sequenc
e
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 5252
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
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 5353
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
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 5454
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 0 134 130 -74 -7
-7 413 4-7 013 09 411 49 011 0
1 123 121 8 3 8Artificia
l
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 5555
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 0 134 130 -74 -7
-7 413 4-7 013 09 411 49 011 0
1 123 121 8 3 8
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 5656
Sample Space (MM)Sample Space (MM)
3 12
9 4 11 4
9 0 11 0
1 8 3 8
1 12
Real
Artificial
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 5757
Sample Space (MM)Sample Space (MM)
1 123 121 8 3 8
Real
Artificial
9 411 49 011 0
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 5858
MSE of BLUPs vs True Latent MSE of BLUPs vs True Latent Values?Values?
Daisy (j=1)Daisy (j=1) Rose (j=2)Rose (j=2)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 5.245.2410.810.8
441010 0.710.71 0.640.64 22 1.861.86
8.878.87 1010 1.281.28 0.520.52 22 2.192.193.133.13 1010 47.147.1
9911.411.4
881010 89.889.8
441.161.16 1010 78.178.1
6611.311.3
661010 87.687.6
553.073.07 1010 47.947.9
997.717.71 1010 32.632.6
111.101.10 1010 79.179.1
887.597.59 1010 31.331.3
00
1P 2P
Sam
ples
2P1P
Ave=32.06 Ave=32.06
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 5959
Mixed Model (MM)Mixed Model (MM)1
2
1
11E 12E
1E 2
21E 22E
2E*
1Y*
2Y
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 6060
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
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 6161
Finite Population Mixed Model Finite Population Mixed Model (FPMM)(FPMM)
1 32
*Ii i iY b E
1i 2i
*1IY *
1IY 2 *1IY
21E 22E
1E3*1IY
21E 22E
1E
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 6262
FPMM- Sample Space …FPMM- Sample Space …
-7 413 4-7 013 0
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 6363
FPMM- Sample Space …FPMM- Sample Space …
0 134 130 -74 -7
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 6464
FPMM- Sample Space …FPMM- Sample Space …
0 114 110 94 9
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 6565
FPMM- Sample Space …FPMM- Sample Space …
9 411 49 011 0
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 6666
FPMM- Sample Space …FPMM- Sample Space …
-7 1113 11-7 913 9
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 6767
11E 12E
21E 22E
21E 22E
1E 1E1E
*1IY
*1IY
*1IY
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 6868
*2 IY *
1 IY
Finite Population Mixed Model Finite Population Mixed Model (FPMM)(FPMM)
1 3 22
1
11
N
ss
yN
2
*Ii i iY b E
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 6969
11E 12E
11E 12E 11E 12E
11E
1E
21E 22E
21E 22E
21E 22E
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 7070
Sample SpaceSample Space Response Error ModelResponse Error Model
4
9 11 0
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 7171
Sample SpaceSample Space Response Error ModelResponse Error Model
4
9 11 0
SPH&HS, UMASS AmherstSPH&HS, UMASS Amherst 7272
Response Error Response Error ModelModel
2 2 2
2 2 Y y E
E
1 1 1
1 1 Y y E
E
