An extended GEE model when estimating marginal associations across multiple informants on the same outcome and formally comparing those associations in a hierarchical data set

1. Background & Motivation Statistical method Simulation Example Conclusion
Disclosure statement
Authors: Jonggyu Baek,
Brisa N. Sánchez,
Emma V. Sanchez-Vaznaugh
The authors do not have any nancial conicts of interests
The authors acknowledge grant funding from RWJF grants
S8-94446 (PI Sanchez-Vaznaugh) and ID69599 (PI: Sánchez)

2. Background Motivation Statistical method Simulation Example Conclusion
Hierarchical Multiple Informants Model (HMIM)
Jonggyu Baek, Brisa N. Sánchez,
Emma V. Sanchez-Vaznaugh
University of Michigan
04/03/2012

3. Background Motivation Statistical method Simulation Example Conclusion
Outline
1 Background Motivation
2 Review multiple informants model (MIM).
3 Method for MIM in hierarchical data
4 Simulation Study
5 Example for illustration
6 Conclusion discussion

4. Background Motivation Statistical method Simulation Example Conclusion
Childhood Obesity
Many factors impact child BMI
Child BMI
Energy Balance:
Intake(food) vs. expenditure(activity)
Behaviors (individual, family)
SES: e.g. education Built social Env.

5. Background Motivation Statistical method Simulation Example Conclusion
School environmental factors
Children spend a large amount of time in around schools.
Availability(number) of fast food
restaurants convenience stores
near schools been shown to
impact BMI (through ↑ intake of
junkfood thus disrupting energy
balance).

6. Background Motivation Statistical method Simulation Example Conclusion
Policy Interest: what is the radius of inuence?
Interest in comparing the marginal association of child's BMI
z-score with the number of FF across buers within size 1/4,
1/2, and 3/4 miles from a school.
Data: 2007 California physical tness test data, also known as
FitnessGram.

7. Background Motivation Statistical method Simulation Example Conclusion
Example
Counted the
number of FF
within buer 1/4,
1/2, and 3/4
miles from each
school.

8. Background Motivation Statistical method Simulation Example Conclusion
Policy Interest: what is the radius of inuence?
Interested in comparing the marginal association of child's
BMI z-score with the number of FF across buers within size
1/4, 1/2, and 3/4 miles from a school.
Published papers show results from separate marginal models:
E[BMIzij|FFRj1] = β01 + β11FFRj1
E[BMIzij|FFRj2] = β02 + β12FFRj2
E[BMIzij|FFRj3] = β03 + β13FFRj3
where for ith
child in jth
school,
FFRj1 : # of FF within 1/4 miles from jth
school,
FFRj2 : # of FF within 1/2 miles from jth
school,
FFRj3 : # of FF within 3/4 miles from jth
school,

9. Background Motivation Statistical method Simulation Example Conclusion
Policy Interest: what is the radius of inuence?
Diculty: how to formally compare if β11, β12, β13 are
dierent or not?
E[BMIzij|FFRj1] = β01 + β11FFRj1
E[BMIzij|FFRj2] = β02 + β12FFRj2
E[BMIzij|FFRj3] = β03 + β13FFRj3

10. Background Motivation Statistical method Simulation Example Conclusion
Statistical issues
1 Children within the same school would likely to provide
correlated outcomes.
2 Fitting separate regression models of the same outcome
ignores correlations (e.g. FFR1, FFR2, FFR3) across models.
3 Interest in the population averaged eects of covariates
comparing their eects.
4 Association parameter estimates β11, β12, β13 are correlated

11. Background Motivation Statistical method Simulation Example Conclusion
Statistical issues
1 Children within the same school would likely to provide
correlated outcomes.
2 Fitting separate regression models of the same outcome
ignores correlations (e.g. FFR1, FFR2, FFR3) across models.
3 Interest in the population averaged eects of covariates
comparing their eects.
4 Association parameter estimates β11, β12, β13 are correlated

12. Background Motivation Statistical method Simulation Example Conclusion
Statistical issues
1 Children within the same school would likely to provide
correlated outcomes.
2 Fitting separate regression models of the same outcome
ignores correlations (e.g. FFR1, FFR2, FFR3) across models.
3 Interest in the population averaged eects of covariates
comparing their eects.
4 Association parameter estimates β11, β12, β13 are correlated

13. Background Motivation Statistical method Simulation Example Conclusion
Statistical issues
1 Children within the same school would likely to provide
correlated outcomes.
2 Fitting separate regression models of the same outcome
ignores correlations (e.g. FFR1, FFR2, FFR3) across models.
3 Interest in the population averaged eects of covariates
comparing their eects.
4 Association parameter estimates β11, β12, β13 are correlated

14. Background Motivation Statistical method Simulation Example Conclusion
Our approach
Extend multiple informant models to hierarchical structure
data.
Compares marginal eects.

15. Background Motivation Statistical method Simulation Example Conclusion
Multiple informants model
Jointly t regression models on same outcome for dierent
predictors (Pepe et. al., 1999; and Horton et. at.,1999). For the
outcome Yi of subject i, with predictors Xik
E[Yi|Xik] = β0k + β1kXik, for k = 1, 2, ..., K
Re-construct data
˜Yi =





Yi
Yi
...
Yi





(K×1)
, ˜Xi =





1 Xi1 0 0 · · · 0 0
0 0 1 Xi2 · · · 0 0
...
...
...
0 0 0 0 · · · 1 XiK





(K×2K)
,
βT = β01 β11 β02 β12 · · · β0K β1K (1×2K)
Use GEE method to solve for β and obtain Cov(β).

16. Background Motivation Statistical method Simulation Example Conclusion
Multiple informants model
Jointly t regression models on same outcome for dierent
predictors (Pepe et. al., 1999; and Horton et. at.,1999). For the
outcome Yi of subject i, with predictors Xik
E[Yi|Xik] = β0k + β1kXik, for k = 1, 2, ..., K
Re-construct data
˜Yi =





Yi
Yi
...
Yi





(K×1)
, ˜Xi =





1 Xi1 0 0 · · · 0 0
0 0 1 Xi2 · · · 0 0
...
...
...
0 0 0 0 · · · 1 XiK





(K×2K)
,
βT = β01 β11 β02 β12 · · · β0K β1K (1×2K)
Use GEE method to solve for β and obtain Cov(β).

17. Background Motivation Statistical method Simulation Example Conclusion
Hierarchical multiple informants model (HMIM)
Jointly t regression models of same outcome in hierarchical
data.
Combine the two basic concepts of MIM and GEE.
For ith
child in jth
school, with predictors Xjks
E[Yij|Xjk] = β0k + β1kXjk, for k = 1, . . . , K
If K = 2, then, data re-arranged as:
˜Yj =











Y1j
...
Ynj ,j
Y1j
...
Ynj ,j











, ˜Xj =











1 Xj1 0 0
...
...
...
...
1 Xj1 0 0
0 0 1 Xj2
...
...
...
...
0 0 1 Xj2











, β =




β01
β11
β02
β12





18. Background Motivation Statistical method Simulation Example Conclusion
HMIM Estimation
Working correlation structure, for k = 1, 2,
˜Vj =
φ1R1j 0
0 φ2R2j (2nj ×2nj )
, Rkj =








1 ρyk · · · ρyk
ρyk 1 · · · ρyk
...
...
...
...
ρyk ρyk · · · 1








(nj ×nj )
Obtain β from solving GEE.
J
j=1
˜XT
j
˜V−1
j (˜Yj − ˜µj) = 0

19. Background Motivation Statistical method Simulation Example Conclusion
HMIM Inference
Parameter estimates
β = (
J
j=1
˜XT
j
˜V−1
j
˜Xj)−1(
J
j=1
˜XT
j
˜V−1
j
˜Yj)
Empirical variance
Var(β) = ˜B−1˜F˜B−1,
where ˜B =
J
j=1
˜XT
j
˜V−1
j
˜Xj , ˜F =
J
j=1
˜XT
i
˜V−1
j (˜Yj − µ)(˜Yj − µ)T ˜V−1
j
˜Xj
Use Var(β) for testing H0 : β11 = β12 = β13.

20. Background Motivation Statistical method Simulation Example Conclusion
Simulation purposes
To measure improved power/eciency of HMIM estimator to that
from the independence assumption
e.g., compared between
˜VEX
j =
φ1R1j 0
0 φ2R2j
and ˜VI
j =
φI 0
0 φI
The empirical power of estimators were calculated by counting
the number of times we rejected the null hypothesis
(H0 : β11 = β12 = β13) in 1000 simulations.

21. Background Motivation Statistical method Simulation Example Conclusion
Simulation setting
Under the model: E[BMIzij|FFRjk] = β0k + β1kFFRjk, for ith
child in jth school, at distance index k = 1, 2, 3.
Set two hypotheses,
Diminishing eects of FFR on child's BMIz with distance
e.g., β12 = aβ11, β13 = 0.8aβ11 for 0 ≤ a ≤ 1.
Threshold eects of FFR on child's BMIz with distance
e.g., β12 = aβ11, β13 = aβ11 for 0 ≤ a ≤ 1.
Note: a controls distances among regression paramters

22. Background Motivation Statistical method Simulation Example Conclusion
Simulation setting
To keep similar features of our motivating data set, we only
generated outcome variable given our motivating data and
assumed regression parameters.
962,018 children in 6,362 clusters
The coecient of variation (CV) in cluster sizes ≈ 1.
true β11 was used from the estimate(= 0.0234) in our
motivating data set.

23. Background Motivation Statistical method Simulation Example Conclusion
Result gures
q
q
q
q
q
q
q
q
q
q
q
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
a
Empiricalpower
a) Diminishing effects of FFR
q β
^
Ex
β
^
I
Ha : β11 = aβ11 = 0.8aβ11
q
q
q
q
q
q
q
q
q
q q
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
a
Empiricalpower
b) Threshold effects of FFR
q β
^
Ex
β
^
I
Ha : β11 = aβ11 = aβ11

24. Background Motivation Statistical method Simulation Example Conclusion
Key simulation study ndings
1 Increased power of estimators accounting for correlation within
clusters.
2 Simple approach of independence working correlation ˜VI
j huge
loss of eciency (Not shown here).
3 Valid statistical inference w/ correlated predictors
(i.e., keep 5% false positive rate).

25. Background Motivation Statistical method Simulation Example Conclusion
Example for illustration
Data: 2007 California physical tness test data, also known as
FitnessGram.
children's weight, height, grade, age, gender, race/ethnicity
BMI z-score from BMI(weight/height2) w/ age and gender
adjusted
School charateristic information was combined from the
California Department of Education's databases (CDE) and
the 2000 US Census.
Information of fast food restaurants in California was
purchased from InfoUSA

26. Background Motivation Statistical method Simulation Example Conclusion
Example for illustration
Table 3. Descriptive statistic for BMIz, FFRk, the correlations of FFRk for
k = 1, 2, 3
Var. Mean SD Corr(FFRk, FFRk )) FFR1 FFR2 FFR3
BMIz 0.744 0.374 FFR1 1 0.55 0.36
FFR1 0.233 0.679 FFR2 0.55 1 0.73
FFR2 1.150 1.758 FFR3 0.36 0.73 1
FFR3 2.709 2.915
Fitted model: E[BMIzij |FFRjk] = β0k + β1kFFRjk for k = 1, 2, 3,
where FFRj1 : # of FF within 1/4 miles from jth
school,
FFRj2 : # of FF within 1/2 miles from jth
school,
FFRj3 : # of FF within 3/4 miles from jth
school,

27. Background Motivation Statistical method Simulation Example Conclusion
Example for illustration
Table 4. The result from tted HMIM and p-values from tests
β Estimates Std. Err Hypothesis test p-value
β11 0.025 0.006 H0 : β11 = β12 = β13 0.192
β12 0.020 0.003
β13 0.017 0.002
Given H0 : β11 = β12 = β13 was not rejected, the marginal
eects of FFR measured across 1/4, 1/2, 3/4 miles are not
signicantly dierent on child's BMIz, even though the eects
look decreasing with distance.

28. Background Motivation Statistical method Simulation Example Conclusion
Conclusion discussion
Extended multiple informants model in hierarchical data setting
Improved the eciency of estimators accounting for correlation
within clusters.
Consistent regression estimates from GEE despite
misspecication of correlation structure under a block diagonal
correlation structure.
Ability to test across separate models with hierarchical data
structure.
Free of distributional assumption, but need a diagnoal
structure to gaurantee consistency

29. Background Motivation Statistical method Simulation Example Conclusion
Conclusion discussion
Extended multiple informants model in hierarchical data setting
Improved the eciency of estimators accounting for correlation
within clusters.
Consistent regression estimates from GEE despite
misspecication of correlation structure under a block diagonal
correlation structure.
Ability to test across separate models with hierarchical data
structure.
Free of distributional assumption, but need a diagnoal
structure to gaurantee consistency

30. Background Motivation Statistical method Simulation Example Conclusion
Conclusion discussion
Extended multiple informants model in hierarchical data setting
Improved the eciency of estimators accounting for correlation
within clusters.
Consistent regression estimates from GEE despite
misspecication of correlation structure under a block diagonal
correlation structure.
Ability to test across separate models with hierarchical data
structure.
Free of distributional assumption, but need a diagnoal
structure to gaurantee consistency

31. Background Motivation Statistical method Simulation Example Conclusion
Conclusion discussion
Extended multiple informants model in hierarchical data setting
Improved the eciency of estimators accounting for correlation
within clusters.
Consistent regression estimates from GEE despite
misspecication of correlation structure under a block diagonal
correlation structure.
Ability to test across separate models with hierarchical data
structure.
Free of distributional assumption, but need a diagnoal
structure to gaurantee consistency

32. Background Motivation Statistical method Simulation Example Conclusion
Conclusion discussion
Extended multiple informants model in hierarchical data setting
Improved the eciency of estimators accounting for correlation
within clusters.
Consistent regression estimates from GEE despite
misspecication of correlation structure under a block diagonal
correlation structure.
Ability to test across separate models with hierarchical data
structure.
Free of distributional assumption, but need a diagnoal
structure to gaurantee consistency

33. Background Motivation Statistical method Simulation Example Conclusion
End
Question??
Contact: jongguri@umich.edu
Thanks to Veronica Berrocal for valuable comments.