Estimating Particulate Matter-Mortality Dose-Response Curves and Threshold Levels: An Analysis of Daily Time-Series for the 20 Largest US Cities
Michael J. Daniels1,
Francesca Dominici2,
Jonathan M. Samet3 and
Scott L. Zeger2
1 Department of Statistics, Iowa State University, Ames, IA.
2 Department of Biostatistics, School of Hygiene and Public Health, The Johns Hopkins University, Baltimore, MD.
3 Department of Epidemiology, School of Hygiene and Public Health, The Johns Hopkins University, Baltimore, MD.
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ABSTRACT
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Numerous studies have shown a positive association between daily mortality and particulate air pollution, even at concentrations below regulatory limits. These findings have motivated interest in the shape of the exposure-response relation. The authors have developed flexible modeling strategies for time-series data that include spline and threshold exposure-response models; they apply these models to daily time-series data for the 20 largest US cities for 19871994, using the concentration of particulate matter <10 µm in aerodynamic diameter (PM10) as the exposure measure. The spline model showed a linear relation without indication of threshold for PM10 and relative risk of death for all causes and cardiorespiratory causes; by contrast, for other causes, the risk did not increase until approximately 50 µg/m3 PM10. For all-cause mortality, a linear model without threshold was preferred to the threshold model and to the spline model, using the Akaike information criterion (AIC). The findings were similar for cardiovascular and respiratory deaths combined. By contrast, for causes other than cardiovascular and respiratory, a threshold model was more competitive with a threshold value estimated at 65 µg/m3. These findings indicate that linear models without a threshold are appropriate for assessing the effect of particulate air pollution on daily mortality even at current levels. Am J Epidemiol 2000;152:397406.
air pollution; models; statistical; mortality
Abbreviations:
AIC, Akaike's information criterion; BIC, Bayesian information criterion; CI, confidence interval; PM10, particulate matter <10 µm in aerodynamic diameter.
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INTRODUCTION
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In spite of improvements in air quality in many developed countries, adverse health effects of particulate air pollution remain a regulatory and public health concern. This continued concern is motivated largely by recent epidemiologic studies that have examined both acute and longer-term effects of exposure to particulate air pollution in different cities in the United States and elsewhere in the world (1


5
). Many of these studies have shown positive associations between levels of particulate air pollution and daily mortality and morbidity rates, and some of these studies suggest that morbidity and mortality from respiratory and cardiovascular diseases are increased at levels of particulate air pollution below the current US National Ambient Air Quality Standard (150-µg/m3 24-hour average) (6
, 7
) for particulate matter <10 µm in aerodynamic diameter (PM10).
Together, the epidemiologic evidence and the legislative mandate have motivated substantial interest in the shape of the exposure-response relation of PM10 with risk for adverse health effects. In this paper we introduce flexible modeling strategies to describe the form of the relation of PM10 with mortality and then apply these models to daily time-series data for air pollution and mortality for the largest 20 US cities. We consider and compare three plausible models for the relation between log mortality and PM10 concentrations, each having potentially different regulatory implications: model 1, a linear model without a threshold; model 2, a spline dose-response model in which log mortality is a smooth function of PM10 concentration; and model 3, a threshold model that assumes no relation between PM10 and log mortality up to a threshold and a linear relation after the threshold. Under this spectrum of models our goals are the following: 1) to pool information from the 20 cities to more precisely determine the shape of the average dose-response curve; 2) to test the hypothesis that the air pollution-mortality association is approximately linear; and 3) to identify possible threshold pollution concentrations below which an effect of air pollution on daily mortality cannot be detected.
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MATERIALS AND METHODS
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These analyses use a database developed for a larger project, the National Morbidity, Mortality, and Air Pollution Study. Full details of database development are given elsewhere (8
).
Data
The analysis database included mortality, weather, and air pollution data for the 20 largest metropolitan areas in the United States for the 7-year period 19871994 (figure 1; table 1) (8
). The air pollution data were obtained from the Aerometric Information Retrieval System database maintained by the US Environmental Protection Agency. The Aerometric Information Retrieval System is a computer-based repository of information about air pollution in the United States and various other countries (http://www.epa.gov/airs/airs.html). In some locations, a high percentage of days had missing values for PM10 because measurements have been required only every 6 days since 1987 by the Environmental Protection Agency. Daily cause-specific mortality data, aggregated at the level of county, were obtained from the National Center for Health Statistics. After excluding deaths from external causes and in nonresidents of the county, we classified the deaths (9
) by age group (<65, 6574, and
75 years) and by cause according to the International Classification of Diseases, Ninth Revision: cardiac (codes 390448); respiratory, including chronic obstructive pulmonary disease and related disorders (codes 490496), influenza (code 487), and pneumonia (codes 480486, 507); and other remaining diseases. The hourly temperature and dew point data for each site were obtained from the EarthInfo CD-ROM database (http://www.sni.net/earthinfo). After extensive preliminary analyses that considered various daily summaries of temperature and dew point as predictors, such as daily average, maximum, and 8-hour maximum, we used the 24-hour mean for each day (8
, 10
, 11
). If a city had more than one weather station, we took the average of the measurements from all available stations. The PM10 data were also averaged over all monitors in each county. To protect against outliers, a 10 percent trimmed mean was used to average across monitors, after correction for yearly averages for each monitor. This yearly correction is appropriate since long-term trends in mortality are also controlled for in the log-linear regressions. See the reports by Kelsall et al. (12
), Dominici et al. (13
), and Samet et al. (14
) for further details.

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FIGURE 1. Maximum likelihood estimates and 95% confidence intervals of the city-specific mean lag particulate matter <10 µm in aerodynamic diameter (PM10) effects on total mortality (TOTAL), cardiovascular and respiratory mortality (CVDRESP), and other-causes mortality (OTHERS), 20 largest US cities, 19871994. At the far right are plotted posterior means and 95% credible regions of the overall effect. la, Los Angeles; ny, New York; chic, Chicago; dlft, Dallas-Ft. Worth; hous, Houston; sand, San Diego; staa, Santa Ana-Anaheim; phoe, Phoenix; det, Detroit; miam, Miami; phil, Philadelphia; minn, Minneapolis; seat, Seattle; sanj, San Jose; clev, Cleveland; sanb, San Bernadino; pitt, Pittsburgh; oakl, Oakland; atla, Atlanta; and sana, San Antonio.
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TABLE 1. Selected counties, populations, average total deaths, average cardiorespiratory deaths, average other deaths, number of days recording PM10* (DPM10), and PM10 daily mean with 10th and 90th percnetiles, 20 largest US cities, 19871994
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Models
In this section, we summarize the log-linear models used to estimate the air pollution/mortality relative rate separately for each location, while accounting for age-specific longer-term trends, weather, and day of the week. The core analysis for each city was a generalized additive model with log link and Poisson error that accounts for smooth fluctuations in mortality that may potentially confound estimates of the pollution effect and/or introduce autocorrelation in mortality series. These models had been developed in earlier work using data from Philadelphia (12
) and for the national-level analyses (9
, 15
).
To specify the approach, let yat be the observed mortality for each age group a = (<65, 6574, and
75 years), PM10t be the level of PM10 at day t, and uat = E(Yat) be the expected number of deaths at day t. We consider a log-linear model of the form log(uat) = ßPM10t + confounders, where ß represents the log-relative rate of mortality associated with a unit increase in PM10, and PM10t is the mean of the current day and the previous day's concentration of PM10. If the previous (current) day's concentration of PM10 is missing, then we use only the current (previous) day's concentration of PM10. As there were missing values for some variables on some days, we restricted analysis to days with no missing values for all covariates. To assess the sensibility of findings, we also fit the models using the current day's pollution data and also the 1-day lag data.
To control for potential confounding of the pollution relative rates ß by longer-term trends due, for example, to changes in health status, seasonality, or influenza epidemics, and to account for any additional temporal correlation in the count time-series, we estimated the pollution effect using only shorter-term variations in mortality and air pollution. To do so, we partialled out the smooth fluctuations in mortality over time by including smooth functions of calendar time S(time,df) for each city. Here, df is a smoothness parameter that we prespecified, on the basis of epidemiologic knowledge of the time scale of the major possible confounders. Taking into account the literature review and our prior work, we selected 7 df per year so that little information from time scales longer than approximately 2 months is included when estimating ß. This choice largely eliminates confounding from seasonal influenza epidemics and from longer-term trends possibly due to changing medical practices and health behaviors while retaining as much unconfounded information as possible on the time scale of interest. We also controlled for age-specific longer-term and seasonal variations in mortality, adding a separate smooth function of time with a total of 8 df over all years for each age group. To control for weather, we also fit smooth functions of the same day's temperature (temp0), average temperature for the 3 previous days (temp13), each with 6 df, and the analogous functions for dew point (dew0, dew13), each with 3 df.
In summary, we fit the following log-linear generalized additive model (16
) to obtain the estimated pollution log-relative rate
and the sample variance V(
) at each location:
 | (1) |
where DOW are indicator variables for day of week. Samet et al. (14
, 17
), Kelsall et al. (12
), and more recently Dominici et al. (15
) give additional details about the rationale for the functions used to control for longer-term trends and weather.
To critically examine the log-linear assumption for particulate levels in model 1, we allowed for greater flexibility in the air pollution/mortality association by assuming that the logarithm of the expected value of the mortality counts is a smooth function of air pollution, specifically, a spline dose-response model; we modeled mortality as a smooth function of S(PM10,
), with
denoting the degrees of freedom for the smoothness of the dose-response curve. A straightforward and flexible approach is to restrict the choice of the smooth functions to the natural cubic splines with a fixed number of knots. These are cubic polynomials within each pair of knots with smooth connections between adjoining segments (continuous first and second derivatives). Our spline dose-response model assumed that:
 | (2) |
We used a natural cubic spline with knots at 30 and 60 µg/m3, which is sufficiently flexible to capture the PM10-mortality association in the range of our air pollution data. We located the two knots at 30 and 60 µg/m3 because these values are approximately the 25 percent and 75 percent quantiles of the PM10 distributions for many of the 20 cities.
To examine the question of whether the mortality effects of particulate pollution are negligible below some level, we also fit a threshold model that we compare with the linear and spline models detailed above.
In the threshold model, we replaced the PM10t term in model 1 with a term of the form (PM10t h)+, where (x+ = x if x
0 and x+ = 0 if x < 0) and h is an unknown change point that is estimated from the data. We then assumed:
 | (3) |
Note that the parameters ß in model 1 and
in model 3 have different interpretations; ß measures the percentage increase in mortality per 10-µg/m3 increase in PM10 at any level of the pollutant;
measures the percentage increase in mortality per 10-µg/m3 increase in PM10 when PM10t is higher than h. For PM10t values smaller than h, the log-relative rate of mortality is set to zero.
Estimation
Using the gam() function in Splus (MathSoft, Inc., Cambridge, Massachusetts), we fit the generalized additive linear and spline models 1 and 2, respectively, within each city to obtain city-specific estimates and standard errors of the linear effect and of the parameter vector corresponding to the splines. Let
c denote the city-specific parameters measuring linear and nonlinear effects of PM10 on mortality for models 12 for city c: c = 1, ..., 20. For the linear model,
c = ßc, and for the dose-response model,
c equals the vector of coefficients corresponding to the splines. To combine the coefficients across cities, we first test for heterogeneity using a generalization of the statistic proposed by DerSimonian and Laird (18
) in the context of meta-analysis:
 | (1) |
where
= (
cV-1c)-1
cV-1c
c. Under the null hypothesis of no heterogeneity, X2 follows a chi-squared distribution with 20x dim(
c) - dim(
c) degrees of freedom. When we fail to reject the null, we combine the city-specific estimates using a fixed-effects model with weights Wc = V-1c and an estimator of the form
= (
cWc)-1
cWc
c with variance, V(
) = (
cWc)-1.
If we reject the null, we fit a two-level Bayesian normal hierarchical model (see, e.g., Daniels and Kass (19
)),
 | (2) |
 | (3) |
with flat priors on
, the overall coefficient vector, and D, the between-city covariance matrix (i.e., p(
)
1 and p(D)
1). This model is fit using the Gibbs sampler and gives similar results to random effects models (18
) with weights of the form Wc = (D + Vc)-1 for some estimate of the between-city covariance matrix D.
Because the threshold model 3 assumes that the threshold h is unknown, we implemented the following procedure to jointly estimate
and h within each city. First, we created a grid of possible thresholds, extending from 5 µg/m3 to 200 µg/m3 with a spacing of 5 µg/m3. Then, conditional on each possible threshold in the grid, we fit model 3 in each city using the gam function, thus obtaining a set of coefficients {
(h), h = 5, 10, ... 200} of maximum likelihood estimates of
, conditional on h. For each of the 20 cities, we then found the maximum likelihood estimate of h by searching over the grid for the point
that maximizes the likelihood l(
(h), h), and we set
=
(
). This method is equivalent to choosing the pair (
(h),h) that maximizes the joint log likelihood, l(
,h). We do not pool the coefficients
c across cities for the threshold model because coefficients have different interpretations for different h.
We did not compute the covariance matrix for the coefficients and threshold jointly, or a standard error for the threshold as for most cities, estimation of the threshold was quite unstable and standard information-based approaches failed. Consequently, to find an estimate of the overall threshold, we chose the value,
, which minimizes the deviance (or equivalently, maximizes the log likelihood):
where Dev(h) =
cDevc(h) = -2
cl(h) with Devc(h) and lc(h) are the city-specific deviance and log-likelihood evaluated at h, respectively. We consider h ranging in {0, 5, 10, ..., 75 µg/m3}, where 75 µg/m3 is the largest PM10 level for which the parameters of the threshold model were identified for all 20 cities. We consider the linear case (h = 0) as a special case of the threshold model when estimating the overall h. To find a measure of uncertainty for the threshold, we first calculate the following weights:
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where wh measures the extent to which the data support the hypothesis that h is the true threshold H = {0, 5, ..., 75}. We then define a 95 percent credible set as {(hL,hU):
h=hUh =hL wh
0.95}. The approach of computing the credible set for the overall threshold using the weights, wh, is similar in concept to inverting the likelihood ratio (see, e.g., (20
)), and it can be further justified by showing its connection to the Bayesian information criterion (BIC) and posterior mode probabilities. The BIC for a model M corresponding to a threshold h, BIC(Mh), is given by Dev(h) + (number of parameters) x log(sample size). The approximate posterior probability of Mh, when also considering the models with the other thresholds, is
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The first equality holds if we assume that each threshold is equally likely a priori. The second equality holds since all models being considered here have the same number of parameters. The weight wh can then be interpreted as an approximate posterior probability of the model Mh.
In addition to obtaining an overall curve that draws information from all cities, we also compared the models within each city and over all cities to determine which best fits the data. We used the Akaike information criterion (AIC) for this purpose (21
). The AIC for each model is a combination of the deviance and the number of parameters in the model: AIC = deviance + 2(number of parameters). The second term provides a penalty for models with more parameters. The model with a lower value of AIC is preferred. To compare the threshold and the linear models using AIC, we exclude h = 0 (linear) when we fit the threshold model. However, to compute our overall estimate, we include h = 0 as a possible value.
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RESULTS
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Figure 1 gives city-specific and overall estimates and 95 percent confidence intervals for the PM10 effects under the linear model. The results are reported using the mean of concurrent day (lag 0) and previous day (lag 1) pollution values. For all causes, there was an increase in mortality with increasing level of PM10. Overall, the largest effect was estimated for respiratory and cardiovascular mortality, with an increase in mortality of 0.69 percent (95 percent confidence interval (CI): 0.40, 0.98) for a 10-µg/m3 increase of PM10. The estimates for total and other mortality were 0.54 percent (95 percent CI: 0.33, 0.76) and 0.34 percent (95 percent CI: 0.17, 0.51), respectively. Table 2 reports point estimates and 95 percent credible regions of the overall effects of PM10 using concurrent day (lag 0) and previous day (lag 1) pollution levels and mean lag. The overall effects of PM10 are robust with respect to lag specification.
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TABLE 2. Posterior means and 95% credible regions (numbers in parantheses) of the overall effects (percentage increase in mortality per 10-unit increase in PM10*) at mean lag, lag 0, and lag 1 on total, cardiovascular and respiratory, and other-causes mortality, 20 largest US cities, 19871994
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Figure 2 shows the spline dose-response curves for all cause, cardiorespiratory, and other cause mortality for the mean lag and lags 0 and 1. For total and cardiorespiratory mortality, the spline curves are roughly linear, consistent with the absence of a threshold. For mortality from other causes, there appears to be little increase in risk until the PM10 concentration exceeds 50 µg/m3. Thus, this curve suggests that the threshold model (model 3) may be reasonable for other-cause mortality. The shapes of the dose-response curves do not change substantially for mean lag, lag 0, and lag 1.

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FIGURE 2. Mortality-particulate matter <10 µm in aerodynamic diameter (PM10) dose-response curves for total (TOTAL) mortality, cardiovascular and respiratory (CVDRESP) mortality, and other causes (OTHERS) mortality, 20 largest US cities, 19871994. The dose-response curves for the mean lag, current day, and previous day PM10 are denoted by solid lines, squared points, and triangle points, respectively.
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The tests for heterogeneity for the dose-response curves across cities indicated heterogeneity for total and cardiovascular and respiratory mortality for mean lag. The heterogeneity of the curves for mean lag and total mortality appears in figure 3, which displays the overall curve and the city-specific curves estimated from the Bayesian hierarchical model. Although there appears to be considerable heterogeneity across cities, the curves have similar shapes and no one or two cities appear to dominate the estimate of the overall curve.

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FIGURE 3. Particulate matter <10 µm in aerodynamic diameter (PM10)-total mortality dose-response curve for the mean lag PM10 and 95% credible regions (solid lines), 20 largest US cities, 19871994. Dashed lines denote the Bayesian estimates of the city-specific dose-response curves.
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The histograms in figure 4 show the posterior probabilities (wh) of the thresholds (h) for each category of cause-specific mortality and for the mean of lag 0 and lag 1 PM10. The posterior distributions of the thresholds are skewed to the right for total mortality and cardiovascular and respiratory mortality and skewed to the left for other causes of mortality. The posterior mode and 95 percent credible regions for the threshold at lag 0 and lag 1 are summarized in table 3. The posterior mode does not change substantially across different lags. Likelihood functions are more informative for the mean lag than for lag 0 and lag 1 and, therefore, lead to more reliable threshold estimates with narrower confidence intervals. The threshold estimates ranged between about 15 and 65 µg/m3 PM10, corresponding approximately to the 10th and 80th interquartile ranges of the PM10 distributions for most of the cities. The category of other-cause mortality had the highest, most probable threshold, at 65 µg/m3 (95 percent credible regions: 55, 75). However, for cardiovascular and respiratory mortality, the data give more support to low values of h (as 0, 5, 10 µg/m3) than for total mortality, indicating that the threshold, if any, for cardiovascular and respiratory mortality may be lower than the threshold for total mortality. The posterior modes for the thresholds for cardiovascular and respiratory mortality and for total mortality were, respectively, 15 µg/m3 (95 percent credible region: 0, 20) and 15 µg/m3 (95 percent credible region: 10, 20).

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FIGURE 4. Posterior probabilities (wh) of the thresholds for each cause-specific mortality and for the mean lag particulate matter <10 µm in aerodynamic diameter (PM10), 20 largest US cities, 19871994. TOTAL, total mortality; CVDRESP, cardiovascular and respiratory mortality; OTHERS, other mortality.
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TABLE 3. Posterior mode and 95% credible regions (numbers in parantheses) of the threshold for mean lag 0 and lag 1, lag 0, and lag 1 PM10* values (expressed as µg/m3), 20 largest US cities, 19871994
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We more formally examined the hypothesis of linearity in the PM10-mortality relation by comparing the AIC values obtained under the linear, threshold, and spline dose-response models. Table 4 shows the difference in the AIC values between the threshold and linear models and between the spline dose-response and linear models for lags 0 and 1 and for the mean lag for total, cardiovascular and respiratory, and other-cause mortality. Positive values of these differences indicate that the linear model is preferred over the threshold and spline models, respectively. To compare the spline and the threshold models, the model with the smaller AIC difference is preferred.
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TABLE 4. AIC* for the threshold and dose-response models calibrated such that AIC for the linear model is 0, 20 largest US cities, 19871994 ,
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We can see that the spline model is never preferred at any lag or type of mortality. In comparing the linear and threshold models, we find that the linear model is preferred for all cases except for the mean lag in the case of "other" mortality. This preference is consistent with visual assessment of the spline dose-response curves for other causes of mortality.
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DISCUSSION
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The shape of the exposure-response relation between levels of an environmental agent and risk for adverse effects may have profound implications for developing regulations to protect public health (22
24
). Linear models are often assumed on a biologic basis or for computational simplicity, but nonlinear models may have plausibility as well, depending on the agent. For some environmental agents, for example, ionizing radiation, there has been extensive consideration for cancer risk of the shape of the dose-response or exposure-response relation (25
, 26
).
For air pollution, the shape of the exposure-response relation for various adverse health effects, including mortality, has also long been of interest as researchers have attempted to supply useful evidence for policymakers seeking to minimize risk to public health. For the "criteria air pollutants," regulated under section 109 of the Clean Air Act, the administrator of the Environmental Protection Agency is required to set standards that protect health with an "adequate margin of safety." This language provides a strong rationale for exploring the shape of the exposure-response relation. Approaches for doing so have included the fitting of threshold linear models and various nonlinear models to observational data and the restriction of analyses to lower concentrations. In fact, the continued demonstration of adverse effects of air pollution over recent decades, even as concentrations of pollutants have declined, suggests that exposures have not yet gone below no-effect thresholds, if such exist.
For air pollution and daily mortality, the topic of this report, a number of investigators have explored the shape of the exposure-response relation (27
, 28
). These analyses have been exclusively carried out within single locations and consequently have limited statistical power to provide evidence in support of a particular model versus alternatives. Additionally, many of the early studies used methods that would no longer be considered optimal. In this paper, we introduce and apply two statistical models, a spline dose-response model and a threshold model, to daily time-series data for the 20 largest US cities. By use of multiple locations, power is gained and generalizability is enhanced. We also offer a method for characterizing the uncertainty of estimated thresholds. Smith et al. (29
) have used piecewise polynomials and cubic splines to estimate nonlinear relations of particulate air pollution with mortality, but their approach was limited to data for single locations.
For total and cardiovascular and respiratory mortality, our findings suggest that there is no threshold, but for other causes of mortality there is evidence for a threshold level below which an effect is small for PM10. As expected, the estimated threshold PM10 levels for cardiovascular and respiratory mortality and for total mortality are lower than the threshold PM10 level for other-cause mortality (15 µg/m3 and 65 µg/m3, respectively). The model comparisons based on the AIC always choose the log-linear dose-response model except for "other" causes of mortality, with mean lag. The linear model was found to be adequate for total and for respiratory and cardiovascular mortality. Results were not sensitive to the lag specification for PM10.
The findings are consistent with analyses of daily time-series data that have shown significant, positive associations between particle levels and mortality counts at current concentrations (3
, 27
). While analytical details of the studies are different, essentially all analyses have modeled log mortality as having a linear relation with particulate matter concentration. The general finding of a positive association implies that any threshold or no-effect level lies at the lower end of the concentration range. In fact, there is evidence of increasing effect at lower concentrations (9
, 15
, 27
). Descriptive analyses have been consistent in showing approximately linear relations as well (30
, 31
).
We note several limitations of our methodology that are amenable to solution with further development. One limitation of our estimation method for the threshold is that some cities offered only vague information and, therefore, we could not estimate city-specific thresholds very well or evaluate heterogeneity across cities in the threshold values. With regard to the spline dose-response model, the number and locations of the knots were fixed in advance. A small improvement might be to jointly estimate the spline dose-response curves and the number and location of the knots. Methods need to be developed for this purpose. On the other hand, we fixed the knots at reasonable locations in the span of PM10 levels and had sufficient flexibility at concentrations of public health concern.
The first limitation might be addressed using a two-level hierarchical model. For example, the following second level might be added to the current model 2:
where hc denotes the city-specific threshold, h the overall threshold, and
2 the between-location variability of the thresholds. Model fitting could be performed using Monte Carlo Markov chain techniques. If the threshold hc varies substantially across cities, the 20-city average curve would not have a threshold form but would tend to be smoother. Hence, our analysis, which indicates that the linear dose-response is preferred for total and cardiovascular and respiratory mortality, does not exclude the possibility that thresholds vary across the country. The second problem can be addressed by extending the work in Denison et al. (32
) on Bayesian curve fitting and applying it within the context of the two-level model.
The present results give an indication that the risk-free levels of PM10 are likely lower than the National Ambient Air Quality Standard for PM10. These findings indicate that linear models without a threshold are appropriate for assessing the effect of particulate air pollution on daily mortality, even at current levels.
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ACKNOWLEDGMENTS
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Research described in this article was conducted under contract to the Health Effects Institute, an organization jointly funded by the Environmental Protection Agency (EPA R824835) and automotive manufacturers. Funding was also provided by a grant from the National Science Foundation (DMS9816630), by a grant from the Health Effects Institute (Walter A. Rosenblith New Investigator Award), and by the Johns Hopkins Center in Urban Environmental Health (5P30ES03819-12).
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NOTES
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Reprint requests to Dr. Francesca Dominici, Department of Biostatistics, School of Hygiene and Public Health, The Johns Hopkins University, 615 N. Wolfe Street, Baltimore, MD 21205-3179 (e-mail: fdominic{at}jhsph.edu).
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Received for publication February 17, 2000.
Accepted for publication May 15, 2000.