American Journal of Epidemiology

American Journal of Epidemiology 2005 162(1):80-88; doi:10.1093/aje/kwi157

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American Journal of Epidemiology Copyright © 2005 by the Johns Hopkins Bloomberg School of Public Health All rights reserved

ORIGINAL CONTRIBUTIONS

Are the Acute Effects of Particulate Matter on Mortality in the National Morbidity, Mortality, and Air Pollution Study the Result of Inadequate Control for Weather and Season? A Sensitivity Analysis using Flexible Distributed Lag Models

Leah J. Welty and Scott L. Zeger

From the Department of Biostatistics, Johns Hopkins Bloomberg School of Public Health, Baltimore, MD

Correspondence to Dr. Leah J. Welty, Department of Preventive Medicine, Northwestern University Feinberg School of Medicine, 680 North Lake Shore Drive, Suite 1120, Chicago, IL 60611 (e-mail: lwelty{at}northwestern.edu).

Received for publication April 28, 2004. Accepted for publication February 16, 2005.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX References

 
Time-series studies have linked daily variations in nonaccidental deaths with daily variations in ambient particulate matter air pollution, while controlling for qualitatively larger influences of weather and season. Although time-series analyses typically include nonlinear terms for weather and season, questions remain as to whether models to date have completely controlled for these important predictors. In this paper, the authors use two flexible versions of distributed lag models to control extensively for the confounding effects of weather and season. One version builds on the current approach to controlling for weather, while the other version offers a new approach. The authors conduct a comprehensive sensitivity analysis of the particulate matter–mortality relation by applying these methods to the recently updated National Morbidity, Mortality, and Air Pollution Study database that comprises air pollution, weather, and mortality time series from 1987 to 2000 for 100 US cities. They combine city-specific estimates of the short-term effects of particulate matter on mortality using a Bayesian hierarchical model. They conclude that, within the broad classes of models considered, national average estimates of particulate matter relative risk are consistent with previous estimates from this study and are robust to model specification for weather and seasonal confounding.

air pollution; longitudinal studies; mortality; regression analysis; seasons; temperature; weather

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Abbreviations: APHEA, Air Pollution and Health: a European Approach; NMMAPS, National Morbidity, Mortality, and Air Pollution Study; PM₁₀, particulate matter of less than 10 µm in aerodynamic diameter

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX References

 
A substantial literature of epidemiologic studies demonstrates an association between mortality and exposure to particulate matter air pollution (1). Much of this evidence comes from time-series studies (2) that compare daily fluctuations in mortality with daily fluctuations in particulate matter while controlling for confounding factors that also vary over time, such as weather and season (3). Initially, time-series studies of air pollution and mortality estimated the particulate matter effect for individual cities (4–8). More recently, multicity studies, such as the Air Pollution and Health: a European Approach (APHEA) project, the National Morbidity, Mortality, and Air Pollution Study (NMMAPS), and a study of 11 Canadian cities, have provided consistent evidence of the widespread association between current particulate matter levels and mortality (9–16).

Despite the evidence, doubt about the findings remains, in part because of the potential confounding effects of weather and season (17). Time-series studies of temperature and mortality have found temperature effects (18–22) that are larger than particulate matter effects and that may be correlated with particulate matter levels (7, 23). Time-series studies of the effects of both air pollution and weather on mortality have identified the importance of adequate control for temperature and humidity when estimating air pollution effects (24–28). Multicity time-series studies of particulate matter and mortality have accounted for the potential confounding effects of weather by including nonlinear covariates for current and previous days' weather and by excluding extreme weather days. These studies have also conducted sensitivity analyses on the degree of flexibility for weather and season covariates (12, 16). However, concern remains that "the possibility of subtler effects within the normal climactic range continues" (17, p. 67). A recent report on the results from multicity time-series studies states that "further exploration of weather effects is merited (for example, considering correlated cumulative effects of multiday temperature or humidity)" (17, p. 67).

Motivated by this lingering challenge to the particulate matter–mortality association, we investigate possible subtler effects of weather and season on estimates of mortality relative risk for particulate matter of less than 10 µm in aeordynamic diameter (PM₁₀). Using the recently updated NMMAPS database with time series from 1987 to 2000 for over 100 US cities (16), we conduct sensitivity analyses of city-specific and national average estimates of the acute effect of PM₁₀ on mortality to control for the effects of weather and season, including cumulative temperature effects. For comparison, our sensitivity analysis includes models similar to those used in previous analyses of the NMMAPS.

Many single-city time-series studies investigating more aggressive control for weather and season have shown that particulate matter effect estimates are generally robust (7, 25, 26, 28). Results from single-city studies do not necessarily generalize to other cities, and single-city particulate matter effect estimates are more variable than are national estimates, making it difficult to distinguish noise from confounding. A substantial portion of evidence for the acute effects of PM₁₀ on mortality comes from multicity studies, such as the NMMAPS and the APHEA project, so it is important to conduct extensive sensitivity analyses for these multicity studies.

We consider two strategies to control for the effects of temperature and season on mortality. Both are based on distributed lag models, time-series models that allow an exposure to affect response over an extended period of time (29). Using distributed lag models, researchers have shown that temperature affects mortality over several days (18, 19), so distributed lag models provide a natural framework for considering the effects of weather on mortality. Distributed lag models also provide a natural framework to quantify the health effects of multiple-day exposure to particulate matter, and they have been used throughout the air pollution epidemiology literature (11, 14, 15, 28, 30).

Our distributed lag models are formulated to capture the complexity of confounding by temperature and season and to account for documented aspects of the temperature–mortality relation. Temperatures up to 2 weeks prior are likely sufficient for capturing the lagged effects of temperature on mortality (18), so we use distributed lag models with 2 weeks of temperature lags. Numerous studies and our own exploratory analyses have shown that the relation between temperature and mortality is nonlinear and varies by season. The relation between temperature and mortality relative risk is convex (i.e., U shaped), implying that cold temperatures in winter or hot temperatures in summer are worse for health than are moderate temperatures (20, 22). The exact nature of the nonlinearity, or the degree to which temperature extremes affect mortality, varies by location (20). We formulate two versions of distributed lag models to control for the nonlinear effects of temperature on mortality: one that builds on the current approach to control for temperature and one that takes a new approach.

In early air pollution time-series studies, methods of control for temperature included categorizing days by weather regime, using sine and cosine terms or indicator variables for season and using indicator, linear, and polynomial terms for meteorologic covariates (4–6, 10, 21). In recent time-series studies, smooth, unspecified functions of temperature have been used to account for the nonlinearity between temperature and mortality. In multicity time-series studies, this flexible specification allows for different nonlinear temperature–mortality relations in different cities. The advantage of models with flexible functions of temperature is that the exact nature of the temperature–mortality relation need not be explicitly defined. However, these models do not account for the possibility that the nonlinear relation between temperature and mortality may change slowly over time. For example, if in a city air conditioner use had increased over time, warm summer temperatures in the later years of the study could be less harmful than warm summer temperatures in the earlier years of the study. A single U-shaped function of temperature in this situation might capture only the average effect of warm temperatures on mortality and possibly have some subtle effect on the particulate matter effect estimate.

We consider two versions of distributed lag models to control for these complex effects of temperature on mortality. One version builds on the current use of flexible functions of temperature, while the new version allows for the nonlinear temperature–mortality relation to vary over time. The first part of this paper describes these two types of distributed lag models. In Materials and Methods, we introduce a basic distributed lag model for temperature and show how it may be extended so that temperature coefficients trend and vary seasonally, thereby allowing the temperature–mortality relation to vary nonlinearly and in time. We also illustrate how models with nonlinear temperature covariates are derived from the basic distributed lag model, and we consider more extensive versions than those often used. We follow up with subsections on the inclusion of temperature interactions and other covariates. The last portion of Materials and Methods outlines a hierarchical model for combining city-specific PM₁₀ effect estimates into national estimates. The next part of this paper compares fitted-model results, both at the national and at the individual-city levels, to assess the sensitivity of the estimated particulate matter–mortality relation to control for weather and season.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX References

 
Data
The data were assembled from publicly available sources as part of the NMMAPS, funded by the Health Effects Institute. The data consist of daily time series of mortality, temperature, dew point, and PM₁₀ for 109 US cities for the period 1987–2000. Daily death counts were obtained from the National Center for Health Statistics and classified by age (<65, 65–74, 75 years); accidental deaths and homicides were excluded. Twenty-four hour averages of temperature and dew point were computed from hourly observations assembled by the National Climactic Data Center in the EarthInfo database. Daily measures of PM₁₀ were obtained from the US Environmental Protection Agency's Aerometric Information Retrieval and AirData Systems. Several cities had no PM₁₀ measurements, and many cities had PM₁₀ nonmissing values every sixth day. The 100 cities with PM₁₀ measured at least every sixth day for more than 4 years were used in the following analysis. Additional details regarding data assembly are available at http://www.ihapss.jhsph.edu/ and are discussed in previous NMMAPS analyses (31).

Methods
Distributed lag models for temperature.
Poisson regression (32) is frequently used to estimate the relation between fluctuations in daily mortality counts and fluctuations in air pollution, while taking into account fluctuations in weather and other time-varying confounders (3). We assume that is an overdispersed random variable with and . The overdispersion parameter ^c represents the variation in mortality not captured by the regression model. We let and be the daily temperature and PM₁₀ time series for city c. Our analysis considers the effects of previous days' temperature and particulate matter on current day mortality, so let and refer to the temperature and PM₁₀ time series lagged by n and l days, respectively. A model for lag l PM₁₀ on daily mortality with distributed lags of temperature is

(1)

where covariates include other confounding variables, such as the day of the week. The term S(t, x years), a smooth function of time with degrees of freedom per year, controls for the seasonality in mortality that is not directly related to temperature. We restrict our attention to models with single lags of PM₁₀ because PM₁₀ is sampled every sixth day for the majority of the 100 cities in our analysis.

Time lags of exposure variables are often correlated, making the distributed lag effects difficult to estimate. A common solution is to constrain the to a functional form, such as a polynomial (29) or a spline (33). Here, we let n = 14, allowing for temperatures up to 2 weeks prior to affect mortality, and we constrain the to lie on a step function. This formulation minimizes the number of distributed lag parameters, allows for effect differences among more and less recent temperatures, and facilitates interpretation of coefficients and of the total effect of temperature. We set steps at lag 0, lag 2, and lag 7, thereby constraining the so that lag 1 and lag 2 temperatures have the same effect on mortality, lag 3–7 temperatures have the same effect on mortality, etc. Defining as the average of the past 2 days' temperatures as the average of the past 7 days' temperatures, and as the average of the past 14 days' temperatures, we have

(2)

A 10°F (5.6°C) increase in average temperature on the current and on each of the past 14 days corresponds with a 1,000 percent increase in current day mortality, which we refer to as the "total temperature effect."

To illustrate this step distributed lag function as well as the seasonal variability in temperature effects, figure 1 shows estimates of the step distributed lags (equation 2) fitted separately for each of the 14 summers (May–August) and each of the 13 winters (November–February) of the NMMAPS data for New York City. The covariates day of week (as a factor) and day of month (as a linear term) were also included. The smooth function of time S(t, x years) was not included since we fitted the model separately over the 4-month periods. For New York City, a 10°F increase in current day temperature in May–August results in greater increase in mortality than does a 10°F increase in current day temperature in November–February (figure 1). Increases in lag 1 and lag 2 day temperatures generally result in increases in mortality in May–August and decreases in November–February (figure 1).

View larger version (37K): [in this window] [in a new window]   FIGURE 1. Percent increase in mortality associated with temperature increases at lag days 0–14, New York City, New York, 1987–2000. Percent increases were estimated separately for the winter and summer months and separately for each year, with the use of a distributed lag step function and temperature and mortality data from the National Morbidity, Mortality, and Air Pollution Study. Lines indicate the distributed lag effcts estimated for each year; shaded regions indicate the distributed lag effects averaged over years.

 
Distributed lag models for temperature with seasonally and temporally varying coefficients.
In these models, distributed lag temperature coefficients vary by season and time. This implicitly allows the relation between temperature and mortality to be nonlinear (since the temperature coefficient for a summer day may differ from that for a winter day) and the nonlinearity to change over time (since the temperature coefficient for a summer day may differ from the coefficient for the same summer day a year later). We replace the temperature log relative risks in equation 2 by time-varying temperature log relative risks , which we model as parametric functions of time. We use sine and cosine functions to specify seasonal change and smooth functions to describe temporal change in the (details in Appendix). The distributed lag model then becomes

(3)

Replacing the fixed temperature coefficients (equation 2) with time-varying coefficients allows for temperatures up to 2 weeks prior to affect mortality nonlinearly and for this nonlinearity to change over time.

Distributed lag models for temperature with nonlinear temperature covariates.
These models include those currently used in multicity studies and use smooth functions of distributed lags of temperature to explicitly account for nonlinearity in the temperature–mortality relation. We let S(·, ) denote a smooth function with degrees of freedom for a city c, and we consider distributed lag models of the form

(4)

The value for determines the nonlinearity of distributed lag temperature covariates, and K determines how many lags of temperature to include. By K = 0, we denote the model that contains only current day temperature. Previous multicity studies have used smooth functions of the current day's temperature and the average of the past 2 or 3 days' temperatures (9, 12–16), that is, K = 1, so we consider K = 0, 1, 2, 3. Previous multicity studies have generally set = 3, and corresponding sensitivity analyses have considered the effects of varying on particulate matter log relative risk (13, 16). We accordingly investigate how both the smoothness of the distributed lag temperature variables () and the number of distributed lag variables (K) may influence particulate matter log relative risk.

In models with nonlinear temperature terms (equation 4), the exact nature of the seasonal relation between temperature and mortality is not explicitly defined. However, these models do not allow for the nonlinearity between temperature and mortality to change over time. The model with seasonally–temporally varying temperature coefficients (equation 3) allows implicitly for a nonlinear relation between temperature and mortality and additionally allows for this nonlinearity to change over time. We compare PM₁₀ log relative risk estimates from both formulations to determine if either formulation alters PM₁₀ estimates.

Interactions among lagged temperatures.
Models that include interactions of distributed lag temperature variables allow for synergy between current and previous days' temperatures when these affect mortality. We accordingly estimate our distributed lag models with and without interactions of temperature distributed lags. The interaction terms for seasonally–temporally varying distributed lag models (equation 3) take the form , etc. For models with smooth functions of temperature distributed lags (equation 4), interactions take the form , etc. To keep the number of regression parameters from growing too large, we excluded all interactions with .

Other covariates.
Measures of humidity, such as dew point, are important weather components in mortality–air pollution models (34). However, directly including dew point in models with many temperature covariates may result in variable parameter estimates due to collinearity. Residualized values of dew point variables regressed on temperature covariates are orthogonal to the temperature covariates but retain variation in dew point not explained by temperature. We regress the current-day dew point and the average of the past 2 days' dew points on temperature covariates and include the respective residuals in our models.

For convenience, we denote the distributed lag models with seasonally–temporally varying temperature coefficients by and those with nonlinear smooth functions of temperature by , where indicates the degrees of freedom per year in the smooth time trend, and the presence of superscript I indicates the inclusion of temperature interactions. For the nonlinear models, K + 1 refers to the number of distributed lag variables in the model (current day plus K additional lag averages), and is the degree of smoothness of the distributed lag variables. For both model formulations, we include the covariates day of week and age category (as factors), day of month (as a linear term), and for each age category a natural spline of time with 14 df. The smooth of time by age category accounts for mortality trends related to an aging population or differing migration rates within age categories. Although not included in prior NMMAPS analyses, a few previous time-series studies of air pollution and health have adjusted for day of month (35). We conservatively included a linear day-of-month term in our models.

The PM₁₀ log relative risk was estimated for all distributedlag models separately for each of 100 cities, using the "glm" function in R, version 1.8.0, software (see below). The models included PM₁₀ exposure from the current day, the previous day, or 2 days previously (l = 0, 1, or 2). For each of these exposures in the seasonally–temporally varying model, the degree of adjustment for seasonal effects on mortality was varied, with = 1, 2, 4, or 8. Based on the results for the seasonally–temporally varying model, was set to 4 for the nonlinear model. All R programs are available in an R-script tdlm.R at http://www.ihapss.jhsph.edu/data/NMMAPS/R/, and the data are available as part of the NMMAPSdata Package in R (36).

Multicity estimates.
The comparison of city-specific log relative risks across models informs the sensitivity of particulate matter estimates to control for weather and season for a single city but does not quantify the sensitivity of particulate matter estimates to control for weather and season generally. With a Bayesian hierarchical model (37), city-specific PM₁₀ log relative risks for a particular model may be used to estimate a pooled PM₁₀ log relative risk for that model. Then, is a national average PM₁₀ log relative risk. The hierarchical model supposes that , and are independent and that s are uniformly distributed. TLNise statistical software (37) was used to estimate and .

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX References

 
National average estimates of the percent increase in mortality associated with a 10-µg/m³ increase in PM₁₀ at lags 0, 1, and 2 are robust to model specification for weather and season. Figure 2 shows national average PM₁₀ posterior means for lags l = 0, 1, 2 for all distributed lag models considered. Effect estimates are consistent within the lag of PM₁₀; overlapping 95 percent pointwise posterior intervals demonstrate no consistent differences in estimates across models. Previous day PM₁₀ has a statistically significant association with daily mortality for all models, with an increase of 10 µg/m³ in PM₁₀ corresponding generally to a 0.2 percent increase in mortality. Table 1 shows national average PM₁₀ posterior means and standard errors for all models with lag 1 PM₁₀. Estimates are consistent with previous NMMAPS analyses (16).

View larger version (34K): [in this window] [in a new window]   FIGURE 2. Posterior means and 95% posterior intervals of the national average short-term effects of particulate matter of less than 10 µm in aerodynamic diameter (PM10) on mortality, by PM10 at lags 0, 1, and 2, National Morbidity, Mortality, and Air Pollution Study, 1987–2000. The national average estimates were obtained by use of different versions of seasonally–temporally varying and nonlinear distributed lag models, both with and without temperature interaction terms, to account for the confounding effects of temperature. SV denotes a seasonally–temporally varying model with degrees of freedom per year in the smooth time trend; NL(K + 1, ) denotes a nonlinear model with degrees of freedom per year in the smooth time trend and K + 1 temperature distributed lag variables (current-day plus K additional lag averages), and denotes degrees of freedom for the smooth functions of the distributed lags of temperature. Addition of superscript I denotes a model with distributed lag temperature interactions included as well.

 

View this table: [in this window] [in a new window]   TABLE 1. Posterior means and standard errors of national average short-term effects of particulate matter of less than 10 µm in aerodynamic diameter on mortality using seasonally–temporally varying and nonlinear distributed lag models to control for the confounding effects of weather and season, National Morbidity, Mortality, and Air Pollution Study, 1987–2000*

 
Estimated PM₁₀ log relative risks for individual cities are qualitatively similar across models, but model choice may determine effect significance. Figure 3 displays unpooled, city-specific lag 1 PM₁₀ log relative risks for eight of the 24 models and for 10 of the largest US cities. Within-city effect estimates from one model to the next are not different in a material way. For the eight models shown, the 95 percent level of significance for the PM₁₀ effect changes across models for Los Angeles, New York, and Chicago, while it remains the same for the other seven cities.

View larger version (31K): [in this window] [in a new window]   FIGURE 3. Maximum likelihood estimates and 95% confidence intervals of the short-term effects of particulate matter of less than 10 µm in aerodynamic diameter (PM10) on mortality for the 10 largest cities in the National Morbidity, Mortality, and Air Pollution Study, 1987–2000, by use of eight different distributed lag models to control for confounding by weather and season. The 10 cities are Los Angeles, California (La); New York, New York (Ny); Chicago, Illinois (Chic); Dallas/Fort Worth, Texas (Dlft); Houston, Texas (Hous); Phoenix, Arizona (Phoe); San Diego, California (Sand); Santa Ana/Anaheim, California (Staa); Miami, Florida (Miam); and Detroit, Michigan (Det). "All" denotes national average posterior means and 95% posterior intervals for the short-term effects of PM10 on mortality. Triangles indicate seasonally–temporally varying (SV) distributedlag models, and dots indicate nonlinear (NL) distributed lag models. From left to right, the specific models shown are SV2, SV8, , and and ,,, and .

 
Varying , the degrees of freedom per year in the smooth time trend, shifted the portion of mortality attributed to temperature instead of season, but it did not substantially alter their combined predictive effect or the PM₁₀ log relative risk. In seasonally–temporally varying models with small values of , the distributed lag variable acted as a surrogate for season. Its estimated effects were that of a seasonal covariate: large and positive in cooler months, when influenza epidemics increase mortality, and negative or near zero in summer. For all values of , the total temperature effect was strongly negatively correlated with the smooth time trend S(t, x years). The combined predictor for the temperature effect and the smooth time trend did not change substantially when ranged over 1, 2, 4, and 8 and correlations of the combined predictors for different values of were all near 0.9. The effects of season and temperature on mortality, therefore, may not be fully separated, although distinction is unnecessary for robust estimation of the PM₁₀–mortality relation.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX References

 
The results above address concerns that the PM₁₀–mortality association from multicity time-series studies (9, 12, 16) may be biased by residual temperature and seasonal confounding. The comparison of PM₁₀ effect estimates across distributed lag models for temperature, some of which are substantially more flexible than previously considered models, shows that, across 100 US cities in the NMMAPS, the short-term effects of PM₁₀ on mortality are not an artifact of inadequately modeled weather and season. Consistent with previous NMMAPS analyses (16), a 10-µg/m³ increase in the previous day's PM₁₀ is associated with an approximate 0.2 percent increase in daily mortality. Distributed lag models with seasonally–temporally varying temperature coefficients and with nonlinear temperature covariates produce similar estimates of PM₁₀ log relative risk. There is no clear division between daily mortality attributable to temperature and daily mortality attributable to seasonality; however, this division does not substantively alter PM₁₀ effect estimates. We have not investigated if barometric pressure or wind speed may account for mortality currently attributed to PM₁₀, but these factors have yet to be consistently identified as potential confounders.

The variability of evidence from single-city analyses investigating confounding by weather and season highlights the advantages of a multicity approach. Samet et al. (26) found that estimates in Philadelphia, Pennsylvania, were robust to control for weather, while Smith et al. (34) found that PM₁₀ coefficients for Birmingham, Alabama, and Chicago, Illinois, were sensitive to control for humidity. Our findings show that, though different weather and season models may alter the significance of estimates for specific cities, they do not significantly or substantially alter national PM₁₀ log relative risk estimates.

Our findings support the use of smooth functions of current-day and average temperature and dew point from the past few days to control for weather effects. The nonlinear distributed lag model designated NL₄(2, 4) is similar to the models used in multicity studies, such as the NMMAPS and the APHEA project (10, 12–14, 16), and we found that the PM₁₀ log relative risk from NL₄(2, 4) is similar to estimates from our other nonlinear models with more distributed lags of temperature. Time-series studies of temperature and mortality have found the strongest temperature effects on the current day and the past few days (18–20), perhaps explaining why including distributed lags of temperature for more than a few days does not alter particulate matter effect estimates. Models that control extensively for the effects of weather may be helpful when estimating the effects of pollutants that have more temperature dependence than does PM₁₀, such as ozone. We note that our models and the associated software implementation (36) can be used in other multisite time-series studies of particulate matter or other pollutants.

Our analysis demonstrates that the short-term effects of PM₁₀ on mortality estimated by Poisson regression are not artifacts of inadequate control for weather and season. Similar conclusions have been made for multicity time-series studies by use of alternative methods to Poisson regression, such as case-crossover analysis (38). Combined, the results suggest that the short-term effects of particulate matter on mortality are viable; they are the result of neither the lack of control for weather and season within the Poisson regression framework nor the use of Poisson regression itself.

Although the PM₁₀ effect estimates for the NMMAPS are robust to control for weather and season, they cannot capture all of the short- or long-term effects of particulate matter on mortality. Our sensitivity analysis is limited to single-day PM₁₀ exposures since the majority of cities in the NMMAPS lack daily PM₁₀ measurements, but other studies have reported larger PM₁₀ log relative risks for models that include distributed lags of PM₁₀ (14, 15, 28, 30). These studies that find significant air pollution log relative risks using models that include multiple-day exposures of temperature and air pollution provide additional evidence that residual confounding by weather is not responsible for the observed air pollution–mortality relation (28). Our estimated 0.2 percent increase in mortality due to a 10-µg/m³ increase in previous-day PM₁₀ reflects only a portion of the short-term health effects of PM₁₀ and, furthermore, does not estimate the larger magnitude of chronic health effects identifiable only through cohort studies. Our methods control for unmeasured differences across city populations that may confound cohort studies however, and therefore they provide important evidence that particulate matter (and not other factors) is responsible for adverse health effects.

	APPENDIX

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX References

 
As in equation 3, let be a seasonally–temporally varying log relative risk for our temperature-distributed lags. We define The D_m(t)s are seasonal and time-trend basis functions comprising harmonic functions to account for the seasonality in temperature effects, a smooth function of time to account for gradual changes in the level of temperature effects from 1987 to 2000, and the appropriate interactions between the two to account for possible changes in the seasonal effects of temperature over time. To specify the D_m(t), for day t of our time series, we let d_t represent the corresponding day number in year. We let D₁(t), D₂(t), D₃(t), and D₄(t) be harmonic basis vectors for seasonal variation in the temperature–mortality relation defined by the following:

In a leap year, we accordingly divide d_t by 366. We let D₅(t), D₆(t), and D₇(t) be basis vectors for slow temporal change in the temperature–mortality relation, designated by a natural spline over t = 1, 2, ..., 5,114 (i.e., years 1987–2000) with 3 df. To allow the seasonal effects of temperature to vary over time, we set D₈(t) = D₁(t) x D₅(t), D₉(t) = D₁(t) x D₆(t), ..., and D₁₉(t) = D₄(t) x D₇(t). Although understanding the sensitivity of PM₁₀ log relative risk is our primary interest, we note that the time-varying temperature log relative risks may be computed directly from the estimates of the

	ACKNOWLEDGMENTS

 
This research at the Johns Hopkins Bloomberg School of Public Health was supported by grant U50CCU322417 from the Center for Excellence in Environmental Public Health Tracking, Centers for Disease Control and Prevention; by National Institutes of Health grant R01ES012054; and by Health Effects Institute award HEI025.

The authors also wish to thank Drs. Francesca Dominici, Roger Peng, and Thomas Louis for their helpful comments.

The work herein does not necessarily reflect the views of the funding agencies nor was it subject to their review.

Conflict of interest: none declared.

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TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX References

 

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