hierarchical models for pooling: a case study in air pollution epidemiology
DESCRIPTION
Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology. Francesca Dominici. NMMAPS. National Morbidity and Mortality Air Pollution Study (NMMAPS) Daily data on cardiovascular/respiratory mortality in 10 largest cities in U.S. Daily particulate matter (PM10) data - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/1.jpg)
Term 4, 2005 BIO656 Multilevel Models 1
Hierarchical Models for Pooling: A Case Study in Air Pollution
Epidemiology
Francesca Dominici
![Page 2: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/2.jpg)
Term 4, 2005 BIO656 Multilevel Models 2
NMMAPS
• National Morbidity and Mortality Air Pollution Study (NMMAPS)
• Daily data on cardiovascular/respiratory mortality in 10 largest cities in U.S.
• Daily particulate matter (PM10) data• Log-linear regression estimate relative risk of
mortality per 10 unit increase in PM10 for each city
• Estimate and statistical standard error for each city
cccc
![Page 3: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/3.jpg)
Term 4, 2005 BIO656 Multilevel Models 3
![Page 4: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/4.jpg)
Term 4, 2005 BIO656 Multilevel Models 4
Relative Risks* for Six Largest Cities
City RR Estimate (% per 10 micrograms/ml
Statistical Standard Error
Statistical
Variance
Los Angeles 0.25 0.13 .0169
New York 1.4 0.25 .0625
Chicago 0.60 0.13 .0169
Dallas/Ft Worth 0.25 0.55 .3025
Houston 0.45 0.40 .1600
San Diego 1.0 0.45 .2025
Approximate values read from graph in Daniels, et al. 2000. AJE
![Page 5: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/5.jpg)
Term 4, 2005 BIO656 Multilevel Models 5
*
*
*
**
*
-2-1
01
23
4City-specific MLEs for Log Relative Risks
Pe
rce
nt C
ha
ng
e
![Page 6: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/6.jpg)
Term 4, 2005 BIO656 Multilevel Models 6
Notation
c
cˆ
![Page 7: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/7.jpg)
Term 4, 2005 BIO656 Multilevel Models 7
Sources of Variation
)()()ˆ(
ˆ
ˆ
ccc
ccc
cc
ccc
eVardVarVar
ed
d
e
![Page 8: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/8.jpg)
Term 4, 2005 BIO656 Multilevel Models 8
0
0 0
0 0 0
-2-1
01
23
4
City-specific MLEs for Log Relative Risks (*) and True Values (o)
city
Pe
rce
nt C
ha
ng
e
![Page 9: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/9.jpg)
Term 4, 2005 BIO656 Multilevel Models 9
0
0 0
0 0 0
-2-1
01
23
4
City-specific MLEs for Log Relative Risks (*) and True Values (o)
city
Pe
rce
nt C
ha
ng
e
*
*
*
**
*
![Page 10: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/10.jpg)
Term 4, 2005 BIO656 Multilevel Models 10
*
*
*
**
*
-2-1
01
23
4
City-specific MLEs for Log Relative Risks
city
Pe
rce
nt C
ha
ng
e
![Page 11: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/11.jpg)
Term 4, 2005 BIO656 Multilevel Models 11
*
*
*
**
*
-2-1
01
23
4
City-specific MLEs for Log Relative Risks
city
Pe
rce
nt C
ha
ng
e
![Page 12: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/12.jpg)
Term 4, 2005 BIO656 Multilevel Models 12
Notation
![Page 13: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/13.jpg)
Term 4, 2005 BIO656 Multilevel Models 13
Estimating Overall Mean
• Idea: give more weight to more precise values
• Specifically, weight estimates inversely proportional to their variances
![Page 14: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/14.jpg)
Term 4, 2005 BIO656 Multilevel Models 14
Estimating the Overall Mean
![Page 15: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/15.jpg)
Term 4, 2005 BIO656 Multilevel Models 15
Calculations for Empirical Bayes Estimates
City Log RR
(bc)
Stat Var
(vc)
Total
Var
(TVc)
1/TVc wc
LA 0.25 .0169 .0994 10.1 .27
NYC 1.4 .0625 .145 6.9 .18
Chi 0.60 .0169 .0994 10.1 .27
Dal 0.25 .3025 .385 2.6 .07
Hou 0.45 .160 ,243 4.1 .11
SD 1.0 .2025 .285 3.5 .09
Over-all
0.65 37.3 1.00
= .27* 0.25 + .18*1.4 + .27*0.60 + .07*0.25 + .11*0.45 + 0.9*1.0 = 0.65
Var = 1/Sum(1/TVc) = 0.164^2
![Page 16: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/16.jpg)
Term 4, 2005 BIO656 Multilevel Models 16
Software in R
beta.hat <-c(0.25,1.4,0.50,0.25,0.45,1.0)
se <- c(0.13,0.25,0.13,0.55,0.40,0.45)
NV <- var(beta.hat) - mean(se^2)
TV <- se^2 + NV
tmp<- 1/TV
ww <- tmp/sum(tmp)
v.alphahat <- sum(ww)^{-1}
alpha.hat <- v.alphahat*sum(beta.hat*ww)
![Page 17: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/17.jpg)
Term 4, 2005 BIO656 Multilevel Models 17
Two Extremes
• Natural variance >> Statistical variances– Weights wc approximately constant = 1/n– Use ordinary mean of estimates regardless of
their relative precision
• Statistical variances >> Natural variance– Weight each estimator inversely proportional to
its statistical variance
![Page 18: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/18.jpg)
Term 4, 2005 BIO656 Multilevel Models 18-4 -3 -2 -1 0 1
-0.5
0.0
0.5
1.0
1.5
Sensitivity of Inferences to Natural Variance
Log2(Natural Variance)
Ave
rag
e R
ela
tive
Ris
k
![Page 19: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/19.jpg)
Term 4, 2005 BIO656 Multilevel Models 19
Estimating Relative Risk for Each City
• Disease screening analogy– Test result from imperfect test– Positive predictive value combines prevalence
with test result using Bayes theorem• Empirical Bayes estimator of the true value for a
city is the conditional expectation of the true value given the data )ˆ|( ccE
![Page 20: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/20.jpg)
Term 4, 2005 BIO656 Multilevel Models 20
Empirical Bayes Estimate
c
ccc VarVar
ˆ
)()|~
(
))var((
ˆ)1(ˆ)ˆ|(~
cc
ccccc
NVNV
E
)()|~
( ccc VarVar
![Page 21: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/21.jpg)
Term 4, 2005 BIO656 Multilevel Models 21
Calculations for Empirical Bayes Estimates
City Log RR
Stat Var
(vc)
Total
Var
(TVc)
1/TVc wc RR.EB se
RR.EB
LA 0.25 .0169 .0994 10.1 .27 .83 0.32 0.17
NYC 1.4 .0625 .145 6.9 .18 .57 1.1 0.14
Chi 0.60 .0169 .0994 10.1 .27 .83 0.61 0.11
Dal 0.25 .3025 .385 2.6 .07 .21 0.56 0.12
Hou 0.45 .160 ,243 4.1 .11 .34 0.58 0.14
SD 1.0 .2025 .285 3.5 .09 .29 0.75 0.13
Over-all
0.65 1/37.3=
0.027
37.3 1.00 0.65 0.16
cc c~
![Page 22: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/22.jpg)
Term 4, 2005 BIO656 Multilevel Models 22
*
*
*
**
*
-2-1
01
23
4
City-specific MLEs for Log Relative Risks
city
Pe
rce
nt C
ha
ng
e
![Page 23: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/23.jpg)
Term 4, 2005 BIO656 Multilevel Models 23
*
*
*
**
*
-2-1
01
23
4
City-specific MLEs (Left) and Empirical Bayes Estimates (Right)
city
Pe
rce
nt C
ha
ng
e
![Page 24: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/24.jpg)
Term 4, 2005 BIO656 Multilevel Models 24
* *** * *
0.0 0.5 1.0 1.5 2.0
Shrinkage of Empirical Bayes Estimates
Percent Increase in Mortality
o oooo o
Maximum likelihood estimates
Empirical Bayes estimates
![Page 25: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/25.jpg)
Term 4, 2005 BIO656 Multilevel Models 250.0 0.2 0.4 0.6 0.8 1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
Sensitivity of Empirical Bayes Estimates
Natural Variance
Re
lative
Ris
k
c
![Page 26: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/26.jpg)
Term 4, 2005 BIO656 Multilevel Models 26
Key Ideas
• Better to use data for all cities to estimate the relative risk for a particular city– Reduce variance by adding some bias
– Smooth compromise between city specific estimates and overall mean
• Empirical-Bayes estimates depend on measure of natural variation– Assess sensivity to estimate of NV
![Page 27: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/27.jpg)
Term 4, 2005 BIO656 Multilevel Models 27
Daily time series of air pollution, mortality and weather in Baltimore 1987-1994
![Page 28: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/28.jpg)
Term 4, 2005 BIO656 Multilevel Models 28
90 Largest Locations in the USA
![Page 29: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/29.jpg)
Term 4, 2005 BIO656 Multilevel Models 29
Statistical Methods
• Semi-parametric regressions for estimating associations between day-to-day variations in air pollution and mortality controlling for confounding factors
• Hierarchical Models for estimating:– national-average relative rate
– national-average exposure-response relationship
– exploring heterogeneity of air pollution effects across the country
![Page 30: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/30.jpg)
Term 4, 2005 BIO656 Multilevel Models 30
Hierarchical Models for Estimating a National Average
Relative Rate of Mortality
![Page 31: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/31.jpg)
Term 4, 2005 BIO656 Multilevel Models 31
Pooling
City-specific relative rates are pooled across cities to:
1. estimate a national-average air pollution effect on mortality;
2. explore geographical patterns of variation of air pollution effects across the country
![Page 32: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/32.jpg)
Term 4, 2005 BIO656 Multilevel Models 32
Pooling
• Implement the old idea of borrowing strength across studies
• Estimate heterogeneity and its uncertainty
• Estimate a national-average effect which takes into account heterogeneity
![Page 33: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/33.jpg)
Term 4, 2005 BIO656 Multilevel Models 33
City-specific and regional estimates
City-specific and regional estimates
![Page 34: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/34.jpg)
Term 4, 2005 BIO656 Multilevel Models 34
Spatial Model for Relative Rates
![Page 35: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/35.jpg)
Term 4, 2005 BIO656 Multilevel Models 35
Three Models
• “Three stage”- as in previous slide
• “Two stage”- ignore region effects; assume cities have exchangeable random effects
• Two stage with “spatial” correlation -city random effects have isotropic exponentially decaying autocorrelation function
![Page 36: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/36.jpg)
Term 4, 2005 BIO656 Multilevel Models 36
Estimating a national-average relative rate
Dominici, Zeger, Samet RSSA 2000
Samet, Dominici, Zeger et al. NEJM 2000
![Page 37: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/37.jpg)
Term 4, 2005 BIO656 Multilevel Models 37
Epidemiological Evidence from NMMAPS
![Page 38: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/38.jpg)
Term 4, 2005 BIO656 Multilevel Models 38
Maximum likelihood and Bayesian estimates of air pollution effects
Use only city-specific information Borrow strength across cities
Dominici, McDermott, Zeger, Samet EHP 2003
![Page 39: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/39.jpg)
Term 4, 2005 BIO656 Multilevel Models 39
Shrinkage
![Page 40: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/40.jpg)
Term 4, 2005 BIO656 Multilevel Models 40
Posterior Distribution of National Average
![Page 41: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/41.jpg)
Term 4, 2005 BIO656 Multilevel Models 41
Results Stratified by Cause of Death
![Page 42: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/42.jpg)
Term 4, 2005 BIO656 Multilevel Models 42
Regional map of air pollution effects
Partition of the United States used in the 1996 Review of the NAAQS
![Page 43: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/43.jpg)
Term 4, 2005 BIO656 Multilevel Models 43
Findings
• NMMAPS has provided at least four important findings about air pollution and mortality
1. There is evidence of an association between acute exposure to particulate air pollution and mortality
2. This association is strongest for cardiovascular and respiratory mortality
3. The association is strongest in the Northeast region of the USA
4. The exposure-response relationship is linear
![Page 44: Hierarchical Models for Pooling: A Case Study in Air Pollution Epidemiology](https://reader033.vdocuments.us/reader033/viewer/2022051401/56814746550346895db4835d/html5/thumbnails/44.jpg)
Term 4, 2005 BIO656 Multilevel Models 44
Caveats
• Used simplistic methods to illustrate the key ideas:– Treated natural variance and overall estimate as
known when calculating uncertainty in EB estimates
– Assumed normal distribution or true relative risks
• Can do better using Markov Chain Monte Carlo methods – more to come