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Bloomberg School BIO 751 - Daily Mortality

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Biometrics DOI: 10.1111/j.1541-0420.2007.01039.xBayesian Distributed Lag Models: Estimating Effects of ParticulateMatter Air Pollution on Daily MortalityL. J. Welty,1, ∗R. D. Peng,2S. L. Zeger,2and F. Dominici21Department of Preventive Medicine, Northwestern University, Feinberg School of Medicine,680 North Lake Shore Drive, Suite 1102, Chicago, Illinois 60611, U.S.A.2Department of Biostatistics, Johns Hopkins Bloomberg School of Public Health,615 North Wolfe Street, Baltimore, Maryland 21205, U.S.A.∗email: [email protected]. A distributed lag model (DLagM) is a regression model that includes lagged exposure vari-ables as covariates; its corresponding distributed lag (DL) function describes the relationship between thelag and the coefficient of the lagged exposure variable. DLagMs have recently been used in environmentalepidemiology for quantifying the cumulative effects of weather and air pollution on mortality and morbid-ity. Standard methods for formulating DLagMs include unconstrained, polynomial, and penalized splineDLagMs. These methods may fail to take full advantage of prior information about the shape of the DLfunction for environmental exposures, or for any other exposure with effects that are believed to smoothlyapproach zero as lag increases, and are therefore at risk of producing suboptimal estimates. In this article,we propose a Bayesian DLagM (BDLagM) that incorporates prior knowledge about the shape of the DLfunction and also allows the degree of smoothness of the DL function to be estimated from the data. Weapply our BDLagM to its motivating data from the National Morbidity, Mortality, and Air Pollution Studyto estimate the short-term health effects of particulate matter air pollution on mortality from 1987 to 2000for Chicago, Illinois. In a simulation study, we compare our Bayesian approach with alternative methodsthat use unconstrained, polynomial, and penalized spline DLagMs. We also illustrate the connection be-tween BDLagMs and penalized spline DLagMs. Software for fitting BDLagM models and the data used inthis article are available online.Key words: Air pollution; Bayes; Distributed lag; Mortality; NMMAPS; Penalized splines; Smoothing;Time series.1. IntroductionDistributed lag models (DLagMs; Almon, 1965) are regressionmodels that include lagged exposure variables, or distributedlags (DLs), as covariates. They have recently been employedin environmental epidemiology for estimating short-term cu-mulative effects of environmental exposures on daily mortal-ity or morbidity (e.g., Pope et al., 1991; Pope and Schwartz,1996; Braga et al., 2001; Zanobetti et al., 2002; Kim, Kim,and Hong, 2003; Bell McDermott, Zeger, Samet, and Do-minici, 2004; Goodman, Dockery, and Clancy, 2004; Weltyand Zeger, 2005). DLagMs are specialized types of varying-coefficient models (Hastie and Tibshirani, 1993) and dynamiclinear models (Ravines, Schmidt, and Migon, 2006).For Poisson log-linear DLagMs that estimate the effectsof lagged air pollution levels on daily mortality counts, thesum of the DL coefficients is interpreted as the percentageincrease in daily mortality associated with a one unit in-crease in air pollution on each of the previous days. Becausethe time from exposure to event will almost certainly vary ina population, this sum is a more appropriate measure of theeffect of short-term exposure than a single day’s coefficient.Results from previous time series studies suggest that com-pared to DLagMs, models with single day pollution exposuresmight underestimate the risk of mortality associated with airpollution (Schwartz, 2000; Zanobetti et al., 2003; Goodmanet al., 2004; Roberts, 2005).Exposure variables, such as ambient air pollution levels,may be highly correlated over time, making DL coefficientsdifficult to estimate. A general solution is to constrain the co-efficients as a function of lag. Common constraints include apolynomial (Almon, 1965) or a spline (Corradi, 1977). Esti-mating DLagMs as varying-coefficient models constrains thecoefficients to follow a natural cubic spline (Hastie and Tib-shirani, 1993). The DL function for air pollution and mor-tality has been estimated with polynomial constraints (e.g.,Schwartz, 2000, Braga et al., 2001; Kim et al., 2003; Bell,Samet, and Dominici, 2004; Goodman et al., 2004), splineconstraints (Zanobetti et al., 2000), and without constraints(Zanobetti et al., 2003).Each type of constraint on the DL coefficients is an appli-cation of prior knowledge to model specification. In the con-text of air pollution and mortality, prior knowledge suggeststhat short-term risk of mortality varies smoothly as a func-tion of lag and decreases to zero. Prior knowledge about theeffects of air pollution on mortality at early lags is limited.There may be short delays in health effects after exposure,C2008, The International Biometric Society 12 Biometricsas suggested by studies of single day pollution exposures thatfind the largest effect on mortality at lag day 1 (Zmirou et al.,1988; Katsouyanni et al., 2001; Dominici et al., 2003). In thescenario of mortality displacement (Schimmel and Murawsky,1978), in which high air pollution levels may advance by sev-eral days the deaths of frail individuals, the DL function maybe zero or positive at early lags, then decrease and becomenegative (Zanobetti et al., 2000, 2002). If there were both adelay in health effect and mortality displacement, hypothesesconcerning the sign or smoothness of the DL function at earlylags would be tenuous at best.For more appropriate model specification and improved es-timation, it may be advisable to formulate DLagMs so that(i) coefficients are constrained to approach zero smoothlywith increasing lag and (ii) early coefficients are relativelyunconstrained. Neither polynomial nor spline constraints, themost common methods for specifying DLagMs, include thisprior information in estimation. In this article, we developBayesian DLagMs (BDLagMs) that incorporate our under-standing of the relationship between short-term fluctuationsof particulate matter (PM) air pollution and daily fluctuationsin mortality counts. Our prior distribution specifies that aslag increases, the DL function will have increasing smooth-ness and approach zero. An advantage of our approach isthat the degree of smoothness of the DL function is estimatedfrom the data. We note that BDLagMs have been explored ineconomics (e.g., Leamer, 1972; Schiller, 1973; Ravines et


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