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Bayesian Distributed Lag Models: Estimating Effects of Particulate Matter Air Pollution on Daily Mortality Leah J. Welty, PhD Department of Preventive Medicine Northwestern University Joint work with Scott Zeger, Roger Peng, Francesca Dominici Johns Hopkins Department of Biostatistics

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Extreme Air Pollution Events London Fog, December 1952 When Smoke Ran Like Water, D. Davis, Perseus Books © 2002

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Time Series: Daily Deaths & Daily Air Pollution

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Current Air Quality Standards: Adverse Health Effects? Energy Information Administration United States Dept. of Energy United States EPA

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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

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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.

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Distributed Lag Models Allow response to depend on exposure over several days Effect of yesterdays pollution on todays mortality Total effect = % increase in daily mortality associated w/ 1 unit increase in PM on previous days

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Distributed Lag Functions Effect of unit increase in pollution 7 days ago on todays mortality lag i

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Estimating Distributed Lag Models Allow response to depend on exposure over several days Difficult to estimate since x s are correlated

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Example distributed lag functions Yesterday and day before constrained to have same effect

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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?

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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

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Estimation Problems (Chicago, PM10)

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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

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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 reflect i

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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

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Fast Variances 0 Slow Less Correlation More Simulated Dist Lag Functions from Prior

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Bayesian Model If the likelihood for the distributed lag coefficients is normal, then

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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)

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Simulation Study BDLM vs. unconstrained, maximum likelihood polynomial of degree 4 p-spline estimated via GCV p-spline estimated via REML 25 different true distributed lag functions some consistent with prior knowledge some not -- e.g. the dist lag fun a non-zero constant 500 outcome series for each function

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Comparing BDLM to Common Methods black = true DL function white = estimated DL function gray = 95% posterior/confidence bands

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Comparing BDLM to Common Methods black = truth DL function white = estimated DL function gray = 95% posterior/confidence bands

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Comparing BDLM to Common Methods: MSEs* Distributed Lag Function Bayes Poly GCV REML * Expressed as a percent of the MSE for the MLE (unconstrained) Distributed Lag Model TOTAL EFFECT LAG 14

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Comparing BDLM to Common Methods BDLM performs consistently well Captures features of DL function Narrower confidence bands When the goal is estimating the total effect BDLM 10-15% better Better estimation at longer lags

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Application: Chicago Mortality & PM10 log (expected mortality) ~ long term trends + season + weather + pollution + day of week everything else: trends, season, weather, dow

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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

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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:

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Posterior Mean & 95% Posterior Region PM 10 on Mortality Chicago 1987-2000 Using last 4000 of 5000 iterations of Gibbs sampler

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Joint Posterior for Hyperparameters η PM 10 on Mortality Chicago 1987-2000 Using last 4000 of 5000 iterations of Gibbs sampler Fast Variances 0 Slow More Correlation Less

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Estimation Method Comparison: Normal Approx vs Gibbs Sampler Using last 4000 of 5000 iterations of Gibbs sampler

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Sensitivity of BDLM for Chicago to value of σ and prior on η Altering σAltering η

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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

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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 3 rd derivative Difficult to relate to biological or prior knowledge BDLMs transparent method for eliciting P-spline penalties consistent w/ objective function

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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

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Conclusions: PM 10 and Mortality Chicago 1987-2000 Largest effect lag day 3 10 μg/m 3 increase in PM 10 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

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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

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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: lwelty@northwestern.edu

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Constructing Prior: More Details 3.Lag effects tend to zero smoothly X has desired correlation structure - - -

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Comparing BDLM to Common Methods

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