bayesian distributed lag models: estimating effects of particulate matter air pollution on daily...
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Bayesian Distributed Lag Models:Estimating Effects of Particulate Matter
Air Pollution on Daily Mortality
Leah J. Welty, PhDDepartment of Preventive Medicine
Northwestern University
Joint work with Scott Zeger, Roger Peng, Francesca Dominici
Johns Hopkins Department of Biostatistics
Extreme Air Pollution Events
London Fog, December 1952When Smoke Ran Like Water, D. Davis, Perseus Books © 2002
Time Series: Daily Deaths & Daily Air Pollution
Current Air Quality Standards: Adverse Health Effects?
Energy Information Administration United States Dept. of Energy
United States EPA
Time Series Models for Mortality on Air Pollution
log (expected mortality) ~
long term trends + season + weather + pollution + day of week
• Poisson regression
• Smooth functions of time, season, and weather
• GLM/GAM with natural/smoothing splines
Distributed Lag Models (DLMs)
• These are time series models with lagged exposure variables as predictors.
• The acute health effects from pollution exposure may take a day or more to develop.
• The time from exposure to event may vary among individuals.
• Compared to DLMs, models with single day exposures may under or over estimate the risk of mortality.
Distributed Lag Models
• Allow response to depend on exposure over several days
Effect of yesterday’s pollution on today’s mortality
“Total effect” = % increase in daily mortality associated w/ 1 unit increase in PM on previous days
Distributed Lag Functions
Effect of unit increase in pollution 7 days ago on today’s mortality
lag i
Estimating Distributed Lag Models
• Allow response to depend on exposure over several days
Difficult to estimate since x’s are correlated
Example distributed lag functions
Yesterday and day before constrained to have same effect
Constrained Distributed Lag Models
• Unconstrained
• Constrained
- Step function constraints
- Polynomial
- Spline
-Other
How do we set up constraints that are consistent with the ‘true’ distributed lag function?
Air Pollution Distributed Lag Function:Prior Knowledge
• Constraint is application of prior knowledge
• Prior knowledge
Acute risk varies smoothly as function of lag
Acute risk goes to zero as time from exposure increases
• Not part of polynomial or spline constraints
i
i
0 2 4 6 8 10 12 14
-0.0002
-0.0001
0
0.0001
0.0002
max likelihood (-0.00038)natural spline (-0.00042)smoothing spline (-0.00038)smoothing spline (-0.00038)
Estimation Problems (Chicago, PM10)
Bayesian Distributed Lag Models: Outline
• Propose Bayesian Dist Lag Model (BDLM)Approximate posterior distribution
Gibbs sampler implementation
• Compare to other constraints (spline, poly)
• Apply to Chicago PM & Mortality
• BDLMs in context of smoothing
1. No knowledge of early lag effects
2. Lag effects must eventually go to zero
3. Lag effects tend to zero smoothly
Bayesian Constraints for DLMs
Prior on distributed lag coefficients
Construct as to reflecti
Constructing Distributed Lag Prior
1. No knowledge of early lag effects
2. Lag effects must eventually go to zero
Large Variances → Small Variances
3. Lag effects tend to zero smoothly
Uncorrelated → Correlated
eta1 = -0.35
eta
2 =
-0
.37
0 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 14
eta1 = -0.2
0 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 14
eta1 = -0.05
0 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 14
eta
2 =
-0
.18
5
0 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 14 0 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 14 0 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 14
eta
2 =
0
0 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 14 0 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 14 0 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 140 2 4 6 8 10 12 14
Fast Variances → 0 Slow
Le
ss
Co
rre
lati
on
Mo
re
Simulated Dist Lag Functions from Prior
Bayesian Model
If the likelihood for the distributed lag coefficients is normal, then
Bayesian Model: Hyperparameters
If the likelihood for the distributed lags is normal, then this mixture is a mixture of normals
Posterior distribution for η determined from data (smoothness of DL function estimated from data)
Simulation Study
• BDLM vs.unconstrained, maximum likelihoodpolynomial of degree 4p-spline estimated via GCV p-spline estimated via REML
• 25 different ‘true’ distributed lag functionssome consistent with prior knowledge
some not -- e.g. the dist lag fun a non-zero constant
• 500 outcome series for each function
Comparing BDLM to Common Methods
black = true DL function
white = estimated DL function
gray = 95% posterior/confidence bands
Comparing BDLM to Common Methods
black = truth DL function
white = estimated DL function
gray = 95% posterior/confidence bands
Comparing BDLM to Common Methods: MSEs*
Distributed Lag
Function Bayes Poly GCV REML Bayes Poly GCV REML
* Expressed as a percent of the MSE for the MLE (unconstrained) Distributed Lag Model
TOTAL EFFECT LAG 14
Comparing BDLM to Common Methods
• BDLM performs consistently well
Captures features of DL functionNarrower confidence bands
• When the goal is estimating the total effect
BDLM 10-15% betterBetter estimation at longer lags
Application: Chicago Mortality & PM10
log (expected mortality) ~
long term trends + season + weather + pollution + day of week
everything else: trends, season, weather, dow
Distributed Lag for PM10 & Mortality Chicago: Necessary Extensions
• Two problems
1. Likelihood Poisson → No closed form posterior
2. Prior on β?
• Two approaches
1. Pretend MLEs normal; ignore β uncertainty
2. Gibbs sampler
Gibbs Sampler
Rather than estimating the full model all at once:
Alternate between:
1. Sampling θ (normal approximate as proposal), using last β in offset:
2. Sampling β (random walk Metropolis, flat prior), using last θ in offset:
Posterior Mean & 95% Posterior Region PM10 on Mortality Chicago 1987-2000
Using last 4000 of 5000 iterations of Gibbs sampler
Joint Posterior for Hyperparameters η PM10 on Mortality Chicago 1987-2000
Using last 4000 of 5000 iterations of Gibbs sampler
Fast Variances → 0 Slow
Mo
re
Co
rrel
atio
n
Les
s
Estimation Method Comparison: Normal Approx vs Gibbs Sampler
Using last 4000 of 5000 iterations of Gibbs sampler
Sensitivity of BDLM for Chicago to value of σ and prior on η
Altering σ Altering η
BDLMs in Context of Smoothing
P-spline approach to smoothing distributed lag coefficients:
To estimate p-spline, minimize over :
Prior on dist lag coefficients → penalty matrix:
penalty
BDLM in the Context of Smoothing
• Equivalent results from BDLMs & specific p-spline
use reasonably flexible spline basis
• P-spline penalties related to jumps in 3rd derivative
Difficult to relate to biological or prior knowledge
BDLMs transparent method for eliciting P-spline penalties consistent w/ objective function
Conclusions: BDLMs
• Introduce flexible Bayesian DLM– Incorporates prior knowledge– Degree of smoothness of DLM est from data
• Simulation Study– smaller MSEs than comparable methods
• Relates to P-splines for DLMs– BDLM analogous to specific p-spline– Method of eliciting a prior distribution
Conclusions: PM10 and Mortality Chicago 1987-2000
• Largest effect lag day 3 10 μg/m3 increase in PM10 three days previous associated with 0.17% increase daily mortality
• Total effect -0.21% (-0.86, 0.41) increase mortality
• Normal approximation and Gibbs sampler similar results
Large # daily deaths (~116), t = 1, … 5114
Less agreement for shorter time series, less normally distributed outcomes
Future Directions: BDLMs
• Naturally extends to additional hierarchy
Peng, Dominici, Welty. “Estimating the time course of hospitalization risk associated with air pollution using a Bayesian hierarchical distributed lag model.” in press JRSS-C
• Missing data in exposure time series
Many cities air pollution 1/3 or 1/6 days
Bayesian Distributed Lag Models:
Leah J. Welty, Roger D. Peng, Scott L. Zeger, and Francesa Dominici, “Bayesian Distributed
Lag Models: Estimating Effects of Particulate Matter Air Pollution on Daily Mortality”
Biometrics [Epub ahead of print April 2008]
Roger Peng, Francesca Dominici, and Leah J. Welty, “A Bayesian hierarchical distributed lag
model for estimating the time course of hospitalization risk association with particulate matter
air pollution” JRSS-C (in press)
NMMAPS (1987-2000) Data & Programs:
Roger D. Peng, Leah J. Welty, and Aidan McDermott, "The National Morbidity, Mortality, and
Air Pollution Study Database in R" (2004).
http://www.ihapss.jhsph.edu/data/NMMAPS/R/
Contact:
Constructing Prior: More Details
3. Lag effects tend to zero smoothly
X has desired correlation structure
--
-
Comparing BDLM to Common Methods