American Journal of Epidemiology Advance Access originally published online on April 15, 2008
American Journal of Epidemiology 2008 167(12):1511-1517; doi:10.1093/aje/kwn078
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PRACTICE OF EPIDEMIOLOGY |
A Regression Approach for Estimating Multiday Adverse Health Effects of PM10 When Daily PM10 Data Are Unavailable
From the School of Finance and Applied Statistics, College of Business and Economics, Australian National University, Canberra, Australia
Correspondence to Dr. Steven Roberts, School of Finance and Applied Statistics, College of Business and Economics, Australian National University, Canberra, ACT 0200, Australia (e-mail: steven.roberts{at}anu.edu.au).
Received for publication October 4, 2007. Accepted for publication March 11, 2008.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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The authors propose a regression-based approach for obtaining multiday estimates of the adverse health effects of ambient particulate matter less than 10 µm in diameter (PM10) when daily PM10 time-series data are unavailable. This situation is common in the United States, because most US cities take PM10 measurements every 6 days. Current evidence suggests that adverse effects of PM10 are not concentrated on a single day but rather are spread out over multiple days, so the unavailability of daily PM10 data presents a problem for the estimation of these effects. The proposed model estimates weights that are used to construct a linear combination of single-lag PM10 effect estimates obtained from the available PM10 data. It is shown that this new approach provides estimates of the effect of PM10 on mortality that have less bias and mean squared error than currently available methods. Application of this method to the US cities contained in the National Morbidity, Mortality, and Air Pollution Study database produces an estimated national average effect of PM10 on nonaccidental mortality in persons over age 65 years, corresponding to a 0.32% increase per 10-µg/m3 increment in PM10. The estimated effects for cardiorespiratory mortality and other mortality are 0.34% and 0.22%, respectively.
air pollution; epidemiologic methods; models, statistical; mortality; particulate matter
Abbreviations: NMMAPS, National Morbidity, Mortality and Air Pollution Study; PM10, particulate matter less than 10 µm in diameter
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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Time-series studies utilizing data from US cities have been extensively used to investigate the association between daily mortality and pollution from ambient particulate matter less than 10 µm in diameter (PM10) (1–3). Perhaps the most comprehensive such study is the National Morbidity, Mortality, and Air Pollution Study (NMMAPS), funded by the Health Effects Institute, which uses data from a large number of US cities (4, 5). A problem with studies such as this is the lack of daily observations of PM10 for the majority of US cities, as most monitors that measure PM10 operate only on an every-6-day collection schedule (6). This lack of daily PM10 measurements means that in the majority of US cities it is possible only to estimate the isolated effect of a single day's PM10 on mortality, not how the short-term effects of PM10 are distributed over time. Toxicologic evidence and the results of some previous time-series investigations suggest that the mortality effects of PM10 extend over multiple days (7, 8). For this reason, the development of methods for estimating multiday effects of PM10 when daily PM10 data are unavailable would allow a better understanding of the mortality effects of PM10 in the United States.
In three previous articles (9–11), we proposed methods for obtaining estimates of the effect of PM10 on mortality when daily PM10 data are unavailable. In 2005 (9) and 2006 (10), we proposed the use of a moving total mortality count (the moving total approach). The moving total approach was shown to produce estimates with enhanced statistical precision and typically smaller bias than that incurred using a single day's PM10 data. In a 2007 paper (11), we suggested that three separate models be fitted to estimate the effect of the current day's PM10 (lag 0), the previous day's PM10 (lag 1), and the two previous days' PM10 (lag 2) on mortality, and we further proposed that these three estimates be combined using weights (the weighted approach). In that paper (11), simple ad hoc weights of 0.5, 1, and 0.5 were assigned to the lag-0, lag-1, and lag-2 estimates, respectively. The weighted approach was shown to produce estimates with smaller bias than the moving total approach. We argued that the weighted approach was an improvement over the moving total approach because the increased use of large multicity studies, with enhanced statistical precision as a by-product, means that bias reduction should be targeted at the individual city level.
A difficulty with the weighted approach is the ad hoc nature of the weighting scheme; while it has the advantage of simplicity, it has a limited theoretical basis. In this article, we propose a data-driven method of estimating appropriate weights for use with the weighted approach. The weighted approach implemented with this new method produces improved estimates in the sense of having both less bias and less mean squared error than are incurred with the ad hoc weighted approach.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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Materials
The data used in this paper were obtained from the publicly available NMMAPS database (4, 5). The NMMAPS database contains daily mortality, weather, and air pollution data for over 100 US cities for the 5,114-day period 1987–2000, inclusive. We extracted daily measurements of the number of nonaccidental deaths among persons over age 65 years (hereafter called total mortality >65), numbers of deaths by cause (cardiorespiratory mortality and other nonaccidental mortality), 24-hour averages of temperature and dew-point temperature, and 24-hour trimmed averages of PM10. Further details on the data can be obtained from the Internet-based Health and Air Pollution Surveillance System (http://www.ihapss.jhsph.edu/).
The extent of "missingness" in the available PM10 data is evidenced by the fact that across the 108 cities in the NMMAPS database, the 10th, 25th, 50th, 75th, and 90th percentiles of the number of days (expressed as a percentage) with recorded PM10 concentrations are 9 percent, 15 percent, 18 percent, 38 percent, and 75 percent, respectively. The small number of days with recorded PM10 concentrations is largely a result of PM10 collection's only being required every 6 days for regulatory purposes.
Methods
A commonly used method in time-series investigations of the association between mortality and PM10 is to model the daily mortality counts using an additive log-linear model (4, 8, 12). Specifically, the mortality counts are modeled as independent Poisson random variables with mean µt on day t, given by
 | (1) |
Ideally, when daily PM10 data are available, g(PM10)t is approximated using a distributed lag model (8). Under an unconstrained distributed lag model, g(PM10)t=β0PM10,0t+...+βjPM10,jt (where PM10,kt is the PM10 concentration on day t–k) and the effect of PM10 is given by 
However, in a situation where daily PM10 data are unavailable, a distributed lag model cannot be used and the PM10 exposure measure is restricted to a single day's PM10—that is, g(PM10)t=
kPM10,kt. Note that βk represents the lag-k PM10 effect obtained from an unconstrained distributed lag model, while k represents the lag-k PM10 effect obtained from a model that includes only a single day's PM10 data. In the analyses that follow, we restrict attention to the lag-0, lag-1, and lag-2 PM10 effects, because previous research has shown that daily mortality has little association with more than the previous few days' PM10 and because these are the lags considered in the multicity NMMAPS analyses (5).
In the NMMAPS analyses, the lag-0, lag-1, and lag-2 PM10 effects were estimated for each city by fitting models similar to model 1 with g(PM10)t=kPM10,kt and k = 0, 1, 2, respectively (5). The estimates corresponding to a particular value of k were then pooled to obtain regional and/or national average effects of PM10 on mortality at lag k. If daily PM10 data were available for each city, model 1 could be fitted individually to each city with g(PM10)t=β0PM10,0t+β1PM10,1t+β2PM10,2t and then the values of 
could be pooled. In the material that follows, model 1 with g(PM10)t=β0PM10,0t+β1PM10,1t+β2PM10,2t is referred to as the distributed lag model and model 1 with g(PM10)t=kPM10,kt is referred to as the lag-k model.
The weighted approach we proposed in 2007 (11) relies on the fact that even without daily PM10 data, it remains possible to obtain estimates for the effect of PM10 on mortality at different lag times. We suggested that the k estimates obtained by fitting the lag-k model for each of k = 0, 1, and 2 be combined using the weighted combination 0.50+1+0.52. In the ideal situation of daily PM10 data, a distributed lag model could be fitted and the effect of PM10 on mortality could be given by β0+β1+β2. In our 2007 paper (11), we used the linear combination 0.50+1+0.52 in an attempt to mimic this ideal situation, noting that simply adding up the k estimates obtained from three separately fitted models would result in positive bias because the three estimates would be dependent, so to an extent they would result in "double-counting."
We propose a regression approach for estimating weights for combining the k estimates obtained from the lag-k model. To facilitate this approach, we initially assume that we are in the ideal situation of having daily PM10 data for each city in the NMMAPS database. Under this assumption, let 
be the "gold standard" PM10 effect estimate for city i obtained from fitting a distributed lag model to the assumed daily PM10 data. Similarly, let 
be the estimates obtained for city i from fitting the lag-k model for each of k=0, 1, 2 to an every-6-day subset of the assumed daily PM10 data. Under this scenario, appropriate weights for combining are obtained by fitting a linear regression model relating the gold standard estimates to the estimates obtained from the lag-k models, that is, by fitting the following linear regression model:
 | (2) |
i are the regression coefficients and n is the number of cities. The estimated regression coefficients (
) from model 2 are appropriate in the sense that they minimize the squared error 
An estimate for the effect of PM10 on mortality in city i is given by the linear combination 
To make the above approach tractable, a method of estimating the regression weights in the situation where daily PM10 data are unavailable is required. A simple approach is to use information available from the US cities for which (roughly) daily PM10 data are available. For this purpose, we selected the nine US cities that had fewer than 1,000 days of missing PM10 data. Table 1 provides information on the distribution of PM10 concentrations in the nine cities that satisfied this criterion. For these cities, it is possible to fit both a distributed lag model to the daily PM10 data and the three relevant lag-k models to an every-6-day subset of the daily PM10 data and hence obtain values of 
and The regression weights () are obtained from model 2, fitted using the values of and from the nine cities with daily PM10 data. An estimate of the effect of PM10 on mortality for city i with daily PM10 data unavailable is obtained by fitting the lag-k model for each of k = 0, 1, and 2 to the available data and computing where are the regression coefficients obtained from fitting model 2 to the nine cities with daily PM10 data.
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In the discussion that follows, the weighted approach using weights of 0.5, 1, and 0.5 is referred to as the fixed-weight approach and the weighted approach using the regression-derived values
is referred to as the regression-weight approach. |
RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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Simulation study
We generate realistic mortality time series with known PM10 effects using the model
 | (3) |
is the explicitly specified effect of PM10 on mortality, and 
i, i=0, 1, 2 specify the distribution of over time. To generate a mortality time series in a given city, model 3 is fitted to the actual meteorologic data and total mortality >65 data for that city. The mean mortality counts from the fitted model are used as the means in a Poisson model to generate mortality counts. This process results in a simulated mortality time series with a PM10 effect of that is distributed over multiple days according to the weights i, i=0, 1, 2. The simulations are limited to the nine cities with daily PM10 data, since daily data are required for the simulated mortality time series to have a PM10 effect distributed over multiple days. In the simulations, days with missing values across any of the variables are removed.
To generate mortality across the nine cities, we assume that there is an overall average effect of PM10 on mortality 
and that the effect for a given city, , is a realization from a normal distribution with mean . Assumptions similar to those made here have been made in a number of investigations reporting national average effects of PM10 (1, 10, 14, 15). To evaluate the performance of the fixed-weight, regression-weight, and respective lag-k models, every-6-day subsets of PM10 are extracted from each city and the models are applied to the simulated mortality time series. We implement the regression-weight model within a particular city by fitting model 2 to the remaining eight cities to obtain the regression weights (). The estimates obtained for a particular model across the nine cities are combined using weights that are inversely proportional to the variances of the individual estimates. The combined estimates from each model are compared with the underlying overall average effect of PM10, . Assessing the performance of the various models relative to is appropriate, as regional and/or national average air pollution effects have been the quantities of interest in most recent studies (10, 14–16).
Table 2 shows the results of simulations in which the effect of PM10 for each city is N(, 0.5) and the distribution of this effect over time, (i, i=0, 1, 2), is constant across cities. The simulation results shown in table 3 relax the assumption of a constant distribution of the PM10 effect over time across cities and allow for positively correlated PM10 effects between cities. The results shown in both tables were based on 2,000 simulations.
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The results obtained under the various simulation scenarios are reasonably consistent in that the bias and root mean squared error of the estimates obtained from the regression-weight model are almost always smaller than the bias and root mean squared error of the estimates obtained from the fixed-weight model. This outcome suggests that the regression-weight model is producing estimates that are, on average, substantially closer to the true values than the fixed-weight model, the root mean squared error providing a measure of the increased closeness to the true values
afforded by the regression-weight model. The single-lag PM10 effect estimates obtained in large multicity studies such as NMMAPS are not intended to be used as estimates of multiday PM10 effects, so it is unsurprising that when considered in a multiday context, they do not perform as well as the methods introduced here. Indeed, note the relatively large bias and root mean squared error values for the lag-k models. These values suggest that single-lag PM10 effect estimates cannot serve as suitable proxies for the overall effect of PM10 on mortality.
Application
The weighted and lag-k models are applied to the cities in the NMMAPS database to produce pooled national average PM10 effect estimates. In the application, days with missing values across any of the variables are removed. The pooled estimates are obtained by combining the single-city estimates using weights inversely proportional to the variances of the individual estimates.
Table 4 shows the pooled estimates for total mortality >65, cardiorespiratory mortality, and other mortality. This table also contains the estimates obtained from fitting a distributed lag model to the nine cities with daily PM10 data along with the estimates from the fixed-weight, regression-weight, and lag-k models for these nine cities based on an every-6-day PM10 subset. The estimates obtained from the weighted models are larger than the estimates obtained from the lag-k models, as would be expected since the weighted models are estimating a 3-day effect of PM10 while the lag-k models are estimating a single-day effect. The larger estimates obtained from the fixed-weight model as compared with the regression-weight model are probably a result, as evidenced by the simulations, of the fixed-weight model's producing estimates with upward bias and the regression-weight model's producing estimates with downward bias. Comparing the estimates from the distributed lag model with those from the weighted models based on every-6-day subsets of PM10 further suggests that the regression-weight model is preferable to the fixed-weight model in the sense that it is producing estimates closer to those obtained from the "gold standard" distributed lag model fitted to daily PM10 data.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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A potential criticism of the regression-weight approach is that the weights applied to the estimates obtained in each city are the same across cities. For PM10, there are numerous reasons why weights peculiar to individual cities may be appropriate. The chemical composition of PM10 can differ by location, meaning that exposure to a given level of PM10 may have differential mortality effects across cities (14, 17). Additionally, within a given city, the mortality effects of PM10 may vary by season because of changes in the sources of air pollution and local meteorologic factors (14). If this is the case, then weights that also differ by season may be appropriate. Currently, with only nine US cities collecting daily PM10 measurements, the use of city-specific weights, or even weights that are the same for cities within particular regions of the United States, is not possible. If more cities had daily PM10 data available, the regression-weight model could be improved by expanding model 2 to include extra covariates that accounted for region within the United States as well as season. The inclusion of variables such as these would allow the weights used in the regression-weight model to vary by both region and season.
Another issue is how representative the nine cities with daily PM10 data are of the remaining cities in the NMMAPS database. If cities with daily PM10 data differ in some respect from the other cities, this may mean that the weights calculated using data from these nine cities are inappropriate for use in other cities. Thus, for example, collection of daily PM10 data in larger cities, cities with higher levels of PM10, or cities located in particular regions of the United States are all potential causes of such misrepresentation. Overall, the nine cities with daily PM10 data used in our analyses appear to have higher levels of PM10 than the remaining cities in the NMMAPS database. This problem needs to be kept in mind when inferring the potential benefits of applying the regression-weight model to situations outside our simulation studies, such as the application of the regression-weight model to other cities in the NMMAPS database.
We also investigated the use of a covariance-weighted model for finding appropriate weights. For this covariance-based model, the single-lag PM10 effect estimates are transformed so that their covariance structure mimics that of the distributed lag model estimates obtained from daily PM10 data. The transformed single-lag estimates are then summed to obtain an estimate of the effect of PM10 on mortality. The rationale behind this approach is that simple summing of unweighted lag-k estimates is clearly inappropriate, since they are generally correlated; thus, taking this correlation into account through a weighting scheme could mitigate bias incurred by simply summing them. Simulations showed that the regression-weight model produced estimates with smaller bias than the covariance-weighted method. Because of the advent of large multicity studies such as NMMAPS, which allow variance to be controlled through pooling of information, reduction in bias has become a more fundamental concern than reducing variance at the individual city level. For this reason, the larger bias reduction properties offered by the regression-weight model make it preferable to the covariance-based model.
Investigation into the adverse health effects of PM10 using time-series data is an extremely active area of current research (18–21). Unfortunately, the evidence available from US data is limited because of the small number of US cities that collect daily PM10 measurements. This lack of data has resulted in large US multicity studies' only being able to determine the effect of PM10 at a single lag time, resulting in the actual multiday mortality effect of PM10 going largely unexplored. This gap has important consequences for public health authorities and regulators, because without knowing the multiday influence of PM10, its adverse health effects and the potential benefits of regulatory measures cannot be properly gauged. The regression-weight approach offers a way to obtain estimates that more closely reflect the multiday effect of PM10 than is possible with existing models.
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ACKNOWLEDGMENTS |
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The authors thank Professor Paul Switzer for his helpful suggestions and comments on an earlier draft of this paper.
Conflict of interest: none declared.
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