Johns Hopkins University

Johns Hopkins University, Dept. of Biostatistics Working Papers

Year  Paper 

Bayesian Hierarchical Distributed Lag Models

for Summer Ozone Exposure and

Cardio-Respiratory Mortality

Yi Huang∗ Francesca Dominici†

Michelle L. Bell‡

∗

Johns Hopkins Bloomberg School of Public Health, Department of Biostatistics,

yhuang19@jhmi.edu

†The Johns Hopkins Bloomberg School of Public Health, fdominic@jhsph.edu

‡Yale University, School of Forestry & Environmental Studies, michelle.bell@yale.edu

This working paper is hosted by The Berkeley Electronic Press (bepress) and may not be commercially reproduced without the permission of the copyright holder.

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Copyright 
c 2004 by the authors.
Bayesian Hierarchical Distributed Lag Models

for Summer Ozone Exposure and

Cardio-Respiratory Mortality

Yi Huang, Francesca Dominici, and Michelle L. Bell

Abstract

In this paper, we develop Bayesian hierarchical distributed lag models for estimating associations between daily variations in summer ozone levels and daily

variations in cardiovascular and respiratory (CVDRESP) mortality counts for 19

U.S. large cities included in the National Morbidity Mortality Air Pollution Study

(NMMAPS) for the period 1987 - 1994.

At the first stage, we define a semi-parametric distributed lag Poisson regression model to estimate city-specific relative rates of CVDRESP associated with

short-term exposure to summer ozone. At the second stage, we specify a class of

distributions for the true city-specific relative rates to estimate an overall effect by

taking into account the variability within and across cities. We perform the calculations with respect to several random effects distributions (normal, t-student,

and mixture of normal), thus relaxing the common assumption of a two-stage

normal-normal hierarchical model. We assess the sensitivity of the results to: 1)

lag structure for ozone exposure; 2) degree of adjustment for long-term trends;

3) inclusion of other pollutants in the model;4) heat waves; 5) random effects

distributions; and 6) prior hyperparameters.

On average across cities, we found that a 10ppb increase in summer ozone

level for every day in the previous week is associated with 1.25 percent increase

in CVDRESP mortality (95% posterior regions: 0.47, 2.03). The relative rate estimates are also positive and statistically significant at lags 0, 1, and 2. We found

that associations between summer ozone and CVDRESP mortality are sensitive

to the confounding adjustment for PM 10, but are robust to: 1) the adjustment for

long-term trends, other gaseous pollutants (NO 2, SO 2, and CO); 2) the distributional assumptions at the second stage of the hierarchical model; and 3) the prior
distributions on all unknown parameters.

Bayesian hierarchical distributed lag models and their application to the NMMAPS

data allow us estimation of an acute health effect associated with exposure to ambient air pollution in the last few days on average across several locations. The

application of these methods and the systematic assessment of the sensitivity of

findings to model assumptions provide important epidemiological evidence for

future air quality regulations.
BAYESIAN HIERARCHICAL DISTRIBUTED LAG MODELS FOR

SUMMER OZONE EXPOSURE AND CARDIO-RESPIRATORY

MORTALITY

Yi Huang, Francesca Dominici, Michelle L. Bell

Abstract

In this paper we develop Bayesian hierarchical distributed lag models for estimating

associations between daily variations in summer ozone levels and daily variations in cardiovascular and respiratory (CVDRESP) mortality counts for 19 large U.S. cities included in

the National Morbidity Mortality Air Pollution Study (NMMAPS) for the period 1987 -

1994.

At the first stage, we define a semi-parametric distributed lag Poisson regression model

to estimate city-specific relative rates of CVDRESP associated with short-term exposure to

summer ozone. At the second stage, we specify a class of distributions for the true cityspecific relative rates to estimate an overall effect by taking into account the variability

within and across cities. We perform the calculations with respect to several random effects distributions (normal, t-student, and mixture of normal), thus relaxing the common

assumption of a two-stage normal-normal hierarchical model. We assess the sensitivity of the

results to: 1) lag structure for ozone exposure; 2) degree of adjustment for long-term trends;

3) inclusion of other pollutants in the model; 4) heat waves; 5) random effects distributions;

and 6) prior hyperparameters.

On average across cities, we found that a 10ppb increase in summer ozone level for every

day in the previous week is associated with 1.25 percent increase in CVDRESP mortality

(95% posterior regions: 0.47, 2.03). The relative rate estimates are also positive and statistically significant at lags 0, 1, and 2. We found that associations between summer ozone and

CVDRESP mortality are sensitive to the confounding adjustment for PM10, but are robust

to: 1) the adjustment for long-term trends, other gaseous pollutants (NO2, SO2, and CO);

2) the distributional assumptions at the second stage of the hierarchical model; and 3) the

prior distributions on all unknown parameters.

Bayesian hierarchical distributed lag models and their application to the NMMAPS data

allow us estimation of an acute health effect associated with exposure to ambient air pollution in the last few days on average across several locations. The application of these

methods and the systematic assessment of the sensitivity of findings to model assumptions

provide important epidemiological evidence for future air quality regulations.

Key Words: Bayesian hierarchical model, distributed lag model, Ozone, Cardiovascular and respiratory mortality.

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Contact Information: Francesca Dominici, Department of Biostatistics, Bloomberg

School of Public Health, Johns Hopkins University, 615 N. Wolfe Street, Baltimore,MD

21205-3179, USA. ph: 410-614-5107, fax: 410-955-0958, email: fdominic@jhsph.edu.

Acknowledgments: Funding for Michelle Bell and Francesca Dominici was provided

by a grant from the Environmental Protection Agency (EPA 3D-6867-NAEX). Funding for

Francesca Dominici was also provided by NIEHS RO1 grant (ES012054-01) and by the

NIEHS Center in Urban Environmental Health (P30 ES 03819).

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1 Introduction

The health effects associated with short-term exposure to ambient ozone are well documented

in air pollution studies (Galizia and Kinney, 1999; Devlin et al., 1991; Folinbee et al., 1988;

Kinney et al., 1996; Stern et al., 1994; Levy et al., 2001; Kelsall et al., 1997; Moolgavkar, 2000;

Thurston and Ito, 2001; Glodberg et al., 2001). Exposure to ozone levels less than 100ppb

can irritate the respiratory tract, particularly for sensitive populations, concentrations above

300ppb can cause nose and throat irritation, and levels above 1000ppb (typical levels in very

smoggy air) can cause breathing problems and headaches. In addition, long-term exposure

to low ozone concentrations may damage sensitive areas of skin, causing premature aging of

lung tissue (Turco, 1997).

Although several time series studies in single cities have shown significant positive associations between mortality/morbidity and summer ozone levels (Kinney and Ozkaynak,

1991; Kinney et al., 1996; Burnett et al., 1997a, 1994, 1997b; Moolgavkar, 2000; Glodberg

et al., 2001; Kelsall et al., 1997), few multi-site time series studies have been used to estimate

an average short-term effect of ozone on mortality and morbidity by combining information

across several locations (Stern et al., 1994; Burnett et al., 1997a,b; Thurston and Ito, 2001).

Critics of single-site studies focus on: 1) inconsistencies of findings among studies; 2) arbitrariness in the selection of the cities to be studied; and 3) model choices that might lead to

relative rates estimates biased upward. Multi-site time series studies address these criticisms

because: 1) site-specific data are assembled under a common framework; 2) the same statistical model is fitted to all the cities; and 3) the average effect is estimated by taking into

account the within site and across sites variability by use of hierarchical models (Dominici,

2002).

In this paper we extend previous modelling approaches developed in NMMAPS for PM10

(Dominici et al., 2000) to take into account the following characteristics of the summer ozone

data (from June 1st to September 30th). Unlike PM10, summer ozone levels are measured

daily, thus allowing estimation of health effects associated with exposure to ozone in the

past few days instead of health effects associated with a one day impulse in air pollution

concentrations. In addition, the estimated city-specific relative rates for ozone show some

outliers, indicating that a normal random effects distribution might be inappropriate.

We develop Bayesian hierarchical distributed lag models for estimating the percent increase in cardiovascular and respiratory (CVDRESP) mortality associated with short-term

changes in summer ozone levels on average across 19 cities in the United States. The distributed lag models have been used in economics after Almon (Almon, 1965) popularized

them in 1965. Then, Pope, Schwartz and others (Pope and Schwartz, 1996; Zanobetti et al.,

2000; Schwartz, 2000) used these methods in environmental health studies. Building upon

these methods, at the first stage, we estimate city-specific relative rates of CVDRESP mortality associated with summer ozone levels using a distributed lag Poisson regression models

with natural cubic splines to take into account non-linear confounding by long-term trend,

and weather. At the second stage, we combine the information across the 19 cities by assuming that the true city specific effects are normally distributed with unknown mean and

variance (Gelman et al., 1995). In addition to the usual assumption of normality, at the

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second stage of the hierarchical model we also assume that the “true” city specific effects

are distributed as a t-student and as a mixture of two normal distributions. Third we assign

prior distributions to all the unknown parameters. Our model specification is then applied to

19 large cities of the National Mortality Morbidity Air Pollution Study (NMMAPS) (Samet

et al., 2000b,a; Dominici et al., 2003) for which ozone data were available.

We choose CVRESP mortality in summer months as the outcome to capture a population

subgroup at higher susceptibility than the general population (Burnett et al., 1997a, 1994,

1997b; Kinney and Ozkaynak, 1992; Kinney et al., 1996; Galizia and Kinney, 1999). We

choose ambient summer ozone levels (from June 1st to September 30th) as the exposure,

since summer ozone is highest among all seasons and measured more frequently.

Controlling for confounding bias and exploring effect modification in time series studies

of air pollution and mortality is challenging (Samet et al., 2000a,b; Dominici et al., 2004),

and requires a systematic assessment of the sensitivity of findings to model specification.

For example, temperature is both a potential confounder and effect modifier of the ozonemortality association. Ozone and mortality tend to be higher at elevated temperatures,

and the health effects of ozone could be exacerbated at extreme temperatures. In addition,

ozone levels can be highly correlated with other pollutants, making it difficult to disentangle

its effect from those of other environmental agents. Other long-term varying factors such

as influenza epidemics and weather patterns are also determinants of mortality and ozone

levels, and their incorrect assessment could lead to spurious associations. We systematically

explore the sensitivity of the results to the adjustment for confounding bias due to long-term

trends, other pollutants, and heat waves.

2 Data

We used data from NMMAPS including mortality counts, weather, and air pollutant concentrations for the 19 large metropolitan areas in the U.S. from 1987 to 1994. Data are

available on the internet at http://ihapss.biostat.jhsph.edu/data/data.htm. See also

Samet et al. (2000a,b); Dominici et al. (2003) for additional details.

Cause-specific mortality data, excluding non-residents and accidental deaths, were obtained from the National Center for Health Statistics. Because mortality information contained residence information at county level only to protect confidentiality, all predictor variables were aggregated at county level as well. Mortality data were classified by age (< 65, 65

to 74, and ≥ 75 years), and by cause according to the International Classification of Disease

Ninth Revision. The CVDRESP mortality used in this analyses include the following ICD-9

codes: cardiovascular (codes 390-448), respiratory, including chronic obstructive pulmonary

disease and related disorders (codes 490-496), influenza (code 487), and pneumonia (codes

480-486,507). The study outcome is the daily counts of CVDRESP mortality from June 1st

to September 30th for 1987 - 1994 (eight years of summer months).

Hourly temperature and dew point data for each site were obtained from the EarthInfo CD database (http://www.sni.net/earthinfo/cdroms/). After extensive preliminary

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analyses that considered various daily summaries of temperature and dew point, we used

24-hour averages. If a county has more than one weather station, we used the average of the

measurements from all available stations.

Daily time series of ambient air pollution levels for each county were obtained from the

Aerometric Information Retrieval Service (AIRS) database maintained by the U.S. Environmental Protection Agency (http://www.wpa.gov/airs/airs.html). The daily ozone levels

used in analysis for each county are obtained by: 1) for each day and each monitor in a

county, calculating the mean ozone concentration with moving window of 365 days centering

around that day, called annual mean; 2) calculating daily residuals from the annual mean

for each monitor; 3) to protect against outliers, calculating the 10% trimmed mean of the

daily residuals across all monitors in this county. Other daily pollutant data are obtained in

the similar fashion. Summer daily ozone levels from June 1st to September 30th for 1987 -

1994 (eight years of summer months) are the exposure measures of interest.

3 Methodology

In this section we describe the two-stage hierarchical model used for estimating an overall

relative rate of CVDRESP mortality associated with short-term summer ozone exposure.

Let Y

c

t be the daily count of CVDRESP deaths, and x

c

t−` be the ambient ozone levels at lag

` in city c. We assume that Y

c

t has a over-dispersed Poisson distribution with mean μ

c

t

and

over-dispersion parameter φ

c and that:

log μ

c

t = age-specific intercept + P6

l=0 β

c

`x

c

t−`+

+ long-term trend + long-term trend specific to each age group + day of week+

+ temperature and dew point + copollutants, or heat wave.

(1)

We first provide the interpretation of the parameters of interest, that is the summer ozone

health effects. At the end of the section, we describe all the potential confounders and how

we adjust for confounding in our model specification. Let βb

c

= [βbc

0

, . . . , βbc

6

] be the vector of

the estimated city-specific estimates of the lag coefficients, and let V

c be the corresponding

statistical covariance matrix. Let θbc =

P6

`=0 βbc

` be the estimated city-specific cumulative

effect and let ˆv

c be the corresponding statistical variance calculated as 1

0

Vbc1, where 1 is a

vector of ones. We assume:

θbc| θ

c

, vc ∼ N(θ

c

, vc)

θ

c

| μ, τ 2 ∼ g(μ, τ 2),

(2)

The parameter θ

c =

P6

`=0 β

c

` denotes the true city-specific cumulative log-relative rate of

CVDRESP mortality associated with a 10ppb increase in summer ozone level every day of

the previous week. The detailed explanation of this interpretation is listed in Appendix1. As an alternative, we could specify a multivariate normal distribution for the vector of

parameters β

c = (β

c

0

, . . . , βc

6

) at the second stage then calcuate θ

c =

P6

`=0 β

c

`

, rather then

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modelling θ

c directly. However, the high collinearity of the ozone levels at different lags

creates numerical problems in the lag-specific effect estimations.

The parameters μ and τ

2 denote the true average cumulative log-relative rate across the

19 cities and the between city variance of the θ

c

s. Our goal is to approximate the marginal

posterior distributions of μ and τ

2

.

We perform the calculations under the following assumptions for the random effects

distribution g:

1. g is normal with mean μ and variance τ

2

;

2. g is t-student with 3 degrees of freedom and with mean μ and variance τ

2

;

3. g is a mixture of two normals with unknown weights, that is: g(μ, τ 2) = (1−p)N(λ1, τ 2)+

pN(λ2, τ 2). Then μ = (1 − p)λ1 + pλ2.

We fit model (2) by use of Monte Carlo Markov Chain Methods (MCMC) (Gelman

et al., 1995) implemented by use of Winbugs (Spiegelhalter et al., 2000). As prior distributions we use the “default non-informative priors”: that is μ ∼ N(0, 10000) and 1/

τ

2 ∼ Gamma(0.01, 0.01). In the next section, we report the sensitivity of the posterior inference on μ to the specification of the prior distribution for τ

2 and to the choice of the prior

hyperparameters. Winbugs software is detailed in the Appendix-2.

Day-to-day associations between air pollution levels and mortality counts can be confounded by time-varying factors that vary in a similar manner as pollution and mortality.

Below are listed the potential confounding factors for the associations between summer ozone

and CVDRESP, and how they are included into the regression model to adjust relative rates

estimates for confounding bias.

Age group: We have adjusted for age by adding age-specific intercepts which allow for

age-specific baseline mortality rate. The variable age is a categorical variable with 3 levels:

< 65, 65 to 74, and ≥ 75 years.

Yearly trend: The number of deaths for cardiovascular and respiratory mortality and

ozone levels have both declined in the past few years because of improvements in medical

practice and air quality regulations. We have employed an 8 degree of freedom smoothing

spline - piecewise-cubic splines over 8 years with 4-summer-month period in each year and

knots at the separation of each summer - to remove the yearly trend. This form of smoothing

spline adjustment will allow for increases or decreases over time within each summer, which

is a more flexible way of time adjustment than using yearly dummies. In addition, to

further remove trends in the summer mortality time series specific to each age group, we add

interaction terms between smooth functions (natural cubic spline) of time with 2.4 degree

of freedom, and the second and third of the three age groups.

Day of the week: Day of the week is a surrogate for traffic patterns and daily activity.

In addition there is evidence that the daily number of deaths and pollution levels are different

in weekday versus weekend. We adjust for the potential confounding effect of day of the week

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by including into the model a categorical variable that allows for different baseline mortality

counts on the different days of the week.

Temperature and dew point temperature: Since the chemistry of ozone formation

is temperature and sunlight driven (Seinfeld and Pandis, 1998), ozone levels are correlated

with temperature. In addition, epidemiological studies (Curriero et al., 2002) have reported

associations between summer temperatures and mortality. To account for potential confounding by weather, we included in the model natural cubic splines of daily temperature,

dew point temperature, and adjusted average temperatures and dew point temperatures over

the three previous days (both adjusted for current-day temperature and dew point) with six

degrees of freedom.

Heat waves: Correlations between ozone and temperature and between mortality and

temperature can be even higher for a subset of days with extreme temperatures (heat waves).

We further explored the sensitivity of the short-term effects of summer ozone on CVDRESP

mortality to the potential confounding of heat-waves. We modelled a heat-wave as a natural

cubic spline of the interaction between current temperature and average temperature in the

three previous days with 2 degrees of freedom.

Other pollutants: Because previous studies have found evidence of a short-term association between PM10 and mortality (Samet et al., 2000a,b; Dominici et al., 2003), and

because both PM10 and O3 levels are correlated with ultra-fine particles, PM10 might be a

confounder of the ozone-mortality association. Similar arguments apply to other pollutants,

including carbon monoxide (CO), sulfur dioxide (SO2), and nitrogen dioxide (NO2). Further

exploration of the sensitivity of the short-term effects of summer ozone on CVDRESP mortality to inclusion of copollutants were performed. To control for their potential confounding

effect, we fit several multi-pollutant models within each city having as predictors the current day co-pollutant concentrations. Notice that, since PM10 measurements are available

only every 1 to 6 days, a distributed lag model is not appropriate to examine the potential

confounding effect of PM10. Therefore we explore the sensitivity of the ozone health effects

to the inclusion of PM10 by use of a single-lag model (we use ambient ozone exposure at lag

2). Remark that only the days with available PM10 measures are included in this analysis.

In summary, this paper introduces the modelling approach with a focus on summer data

and explores the sensitivity of the results to model choices (adjustment for confounding, random effects distributions, and prior hyperparameters). The design of the sensitivity of results

to random effects distributions and prior hyperparameters is listed at the beginning of this

section. Other sensitivity studies include: 1) lag structure for ozone exposure: distributed

lag model versus single lag models where we included only one air pollution predictor at lags

0, 1, and 2 separately. 2) degree of adjustment for long-term trends: monitor the changes of

model inferences to various degrees of freedoms in the natural cubit splines of time, ranging

from 4 to 16 with increasing step=2. 3) inclusion of other pollutants in the model; 4) heat

waves.

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4 Results

Figure 1 shows the box-plot of summer ozone levels in the 19 large U.S. cities. Daily levels

of ambient ozone vary from 0 to 100ppb. Figure 2 shows the box-plot of the logarithm of

the daily counts of CVDRESP mortality in the 19 large cities. The CVDRESP mortality

counts in Los Angeles, New York, and Chicago, are much larger than others. Figure 3 shows

city-specific estimates and 95% posterior regions of the percentage increases CVDRESP

mortality associated with a 10ppb increase in summer ozone levels at lags 0, 1, 2, and under

a distributed lag model (cumulative effect in the last week). At the far right are shown

the posterior means and 95% posterior intervals of the corresponding overall relative rates.

City-specific and overall relative rates estimates under a distributed lag model and their

95% posterior regions are also reported in Table 1. Figure 4 (a and b) shows the posterior

distributions of the average relative rate (μ) and of the between city standard deviation (τ ).

On average across the 19 cities, we found strong evidence of an association between

summer ozone levels and CVDRESP mortality under a distributed lag model, and at lags 0,

1 and 2. For example, we found that, on average across the 19 large cities, a 10ppb increase

in summer ozone levels for the same lag and each of the previous 6 days is associated with

a 1.25 percent increase in CVDRESP mortality (95% posterior regions: 0.47%, 2.02%). The

average effects for ozone exposure at lags 0, 1, and 2 are 0.73 (95% posterior regions: 0.27,

1.19), 0.70 (95% posterior regions 0.26, 1.12), and 0.64 (95% posterior regions 0.17,1.07),

respectively.

Notice that in several cities the estimated effects show monotonicity, with the smallest

effect at lag 0 and the largest effect corresponding to the cumulative relative rate under

the distributed lag model. This monotonicity suggests a consistency of findings across lag

specifications with associations between ozone and mortality that persist for lags of at least

6 days. Posterior means of the heterogeneity parameter τ are similar across lag specifications

with wider posterior intervals for the distributed lag model.

Table 2 summarizes the posterior means and 95% posterior regions of the overall relative

rates μ with respect to: 1) distributional assumptions for the random effects distribution

(normal, t-student, and mixture of two normal); 2) prior distributions on the heterogeneity

variance τ

2

(Inverse Gamma, Jeffrey prior, uniform on the shrinkage factor); and 3) choice of

the prior hyperparameters. Posterior inference on μ are very robust to these model choices.

Figure 5 shows the robustness of the city-specific estimates to the inclusion of other

pollutants in the model. At each panel, two sets of city-specific estimates for ozone at lag 2

are plotted against each other. On the x-axes, the estimates do not include any adjustment

for other pollutants, whereas the estimates on the y-axes are adjusted for one additional

pollutant. The cross mark corresponds to the overall estimates and their 95 % posterior

regions. Notice that, except when PM10 is included in the model, the two sets of estimates

and their averages are very close to the diagonal line, indicating very little sensitivity of

the ozone effect to the inclusion of gaseous pollutants into the model. City-specific ozone

estimates vary with the adjustment for PM10; however, on average the ozone effect estimate

remains positive and significant. Since PM10 measurements are available only every 1 to

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6 days and only the days with available PM10 measures are included in the analysis, the

sensitivity of the ozone health effects to the inclusion of PM10 in the model might indicate

a true confounding effect or might be attributed to higher statistical uncertainty due to the

lack of daily PM10 measurements. In summary, the average relative rates under the several

bi-pollutant models are summarized in Table 3.

To assess the confounding effect of heat waves, we fit the distributed lag model within

each city by including a smooth function of interaction between current day temperature

and average temperature in the last three days. The average relative rate is robust to the

inclusion of this interaction term and moved from 1.25, 95% posterior regions (0.47, 2.03) to

1.11, 95% posterior regions (0.38, 1.86), as listed in Table 4.

Figure 6 shows the robustness of ozone effects at lag 2 to the degrees of freedom in the

smooth functions of time. More specifically, the midline of each box denotes the relative

rate estimates as we vary the number of degrees of freedom in the smooth function of time

from 4 to 16 with increasing step=2, and the polygon captures the corresponding posterior

regions. The polygon at the far right shows the posterior means of the overall relative rates

and their 95% posterior regions. The picture shows that the average relative rate estimates

are somewhat sensitive to the selection of the number of degrees of freedom in the smooth

function of time.

In summary, we found strong evidence of associations between short-term summer exposure to ozone and CVDRESP mortality on average across 19 large cities in the U.S. We

found that this association persists at different lags and it is robust to the adjustment for

long-term trend, temperature, heat waves, and other pollutants.

5 Discussion

In this paper, we extend modelling approaches developed as part of the NMMAPS by introducing Bayesian hierarchical distributed lag models with non-normal random effects distributions for estimating associations between short-term variations in ozone and CVDRESP

mortality. On average across the 19 large cities in the U.S., we found that a 10ppb increase in

summer ozone levels every day for the last week is associated with a 1.25 percent increase in

CVDRESP mortality (95% CI: 0.47, 2.03). Results were robust to: 1) distributional asusmptions on the random effects distribution; 2) inclusion of other gaseous pollutants (CO, SO2,

and NO2) in the model; 3) potential confounding effects of long-term trend, temperatures,

and heat waves; 4) lag specification; and 5) prior distributions.

Our methods have recently been applied to a more extensive NMMAPS analysis on ozone

and mortality, which includes the 95 large cities in the U.S. for the period 1987-2000 (Bell

et al., 2004). While this paper focuses on summer mortality, Bell et al. (2004) focuses on

yearly mortality. These more extensive ozone analyses found a strong association between

short-term variations in ozone and mortality. In fact, Bell et al. (2004) report a national

average equal to 0.64% (95% posterior regions: 0.31, 0.98) in CVDRESP mortality for a 10ppb

increase in the preceding week’s ozone levels, using a constrained distributed lag model.

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Our results provide additional evidence in support of an association between ozone and

mortality and are consistent with the literature. Epidemiological studies have found associations between low ozone levels and: 1) decline in pulmonary functions (Galizia and Kinney,

1999; Folinbee et al., 1988; Kinney et al., 1996; Devlin et al., 1991); 2) increase in hospital

admissions for asthma, especially among children (Stern et al., 1994; Galizia and Kinney,

1999); and 3) congestive heart failure among the elderly (Burnett et al., 1997b). Positive

associations between short-term 1-hour max ozone levels and mortality, adjusted for the potential confounding effects of other pollutants and weather, have been found in Los Angeles

and New York (Kinney and Ozkaynak, 1991; Kinney et al., 1996). In addition a multi-site

time-series study of 16 Canadian cities showed a significant positive association between

respiratory admissions and summer daily 1-hour maximum ozone levels at lag 1 (Burnett

et al., 1997a).

Strengths of our approach are: 1) the use of routinely collected data on air pollution,

mortality and weather; 2) the implementation of Bayesian hierarchical models for combining information across locations and for estimating the average relative rate accounting for

within city statistical uncertainty and heterogeneity of the effects across locations; and 3)

the systematic assessment of the sensitivity of findings to adjustment for confounding bias

and model choice.

This analysis has the following limitations. First, the 24-hour daily average of hourly

ozone measurements is used as the exposure variable. Because of the volatile nature of

ozone, with the maximum generally appearing in the early afternoon and lowest levels at

night, the 24-hour average might not be an adequate summary measure of the ozone exposure. Therefore additional work is needed to explore the sensitivity of our results with

respect to other summary measures, such as the daily 1-hour and 8-hour maximums. Bell

et al. (2004) investigated multiple metrics of ozone concentration, and found all to be significantly associated with mortality. Second, we use ambient levels of ozone as a surrogate

for exposure without taking into account potential exposure measurement error. Third, our

model-based methods for adjusting for confounders, although extensive, cannot completely

rule out confounding bias in the estimated associations. In this paper we used 1 degree of

freedom per summer to adjust for trend. Therefore the lack of a more severe adjustment

for long-term trend and might have biased some of our relative rate estimates. However

when we repeat the analyses by using 2 degrees of freedom per summer, we still found a

positive association between ozone and mortality with a national average estimate equal

to 0.93% with 95% posterior regions equal to (-0.06,1.99). Unfortunately, no data-driven

method exists to estimate an optimal number of degree of freedom for removing confounding

due to long-term trends. This issue has been discussed extensively in the paper by Dominici

et al (Dominici et al., 2004). Fourth, the association between ozone and mortality might

also be confounded by other pollutants which are not currently measured in NMMAPS. For

example, ozone contributes to the secondary formation of organic aerosols, which are a large

source of ultra-fine particles (Andreae and Crutzen, 1997; Seinfeld and Flagan, 1997).

These results provide evidence to policy-makers on the harmful effects of ozone on human health. Such epidemiological evidence is central to the regulatory process, including the

setting of the National Ambient Air Quality Standards (NAAQS) for ozone. Regulations are

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designed to protect human health with an adequate margin of safety for susceptible populations, such as those with cardiovascular or respiratory disease. This research extends earlier

time-series analysis and indicates that summer ozone levels adversely impact cardiovascular/respiratory mortality.

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Figure Legend

1. Box-plots of summer ozone levels for 19 large U.S. cities in NMMAPS for the period

1987 - 1994.

2. Box-plots of log(Daily CVDRESP Death Count) for 19 large U.S. cities in NMMAPS

in summer months of 1987 - 1994.

3. City-specific relative rates estimates (percent increases in CVDRESP mortality associated with 10ppb increase in ozone) and their 95% confidence intervals, with overall

ozone effect and its 95% posterior regions at the end. These effects are estimated by

use of a distributed lag model for one week and at lags 0, 1, and 2 days.

4. (a) Posterior distribution of average relative rate (μ) and (b) of the heterogeneity

parameter (τ ) obtained by combining city-specific relative rates estimates under a

distributed lag model and single-lag models.

5. City-specific ozone relative rates estimates at lag 2 adjusted by another pollutant

(PM10,NO2, SO2 or CO), plotted ozone relative rates estimates at lag 2. The cross

is plotted corresponding to the two overall relative rates (adjusted versus unadjusted)

and their 95 % posterior regions.

6. Sensitivity analysis of the city-specific effects at lag 2 with respect to the degrees of

freedom in the smooth functions of time. The number of degrees of freedom vary from

4 to 16 with increasing step = 2. Each polygon captures the change in the city-specific

relative rates estimates with their corresponding 95% confidence intervals. The last

polygon captures the change of overall ozone effect estimates with its 95% posterior

regions.

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Appendix - 1

To illustrate why the log-relative rate θ

c

can be interpreted as the percent increase in

CVDRESP associated with 10ppb increase in summer ozone levels every day for the previous

week, we consider the following:

log μ

c

t =

X

6

l=0

β

c

`x

c

t−` + all other covariates

log μ

c∗

t =

X

6

l=0

β

c

`

(x

c

t−` + 1) + all other covariates

log μ

c∗

t − log μ

c

t

log μ

c∗

t

μ

c

t

=

X

6

l=0

β

c

` = θ

c

μ

c∗

t

μ

c

t

= e

θ

c

, 1

st order approximation using Taylor expansion

≈ 1 + θ

c

μ

c∗

t − μ

c

t

μ

c

t

≈ θ

c

100 ×

μ

c∗

t − μ

c

t

μ

c

t

≈ (1000 × θ

c

)/10

In summary, 1000 × θ

c

, which is consistently reported in the paper, indicates the percent

increase in CVDRESP associated with 10ppb increasing in summer ozone levels every day

for the previous week. This interpretation used in this paper is consistent with other time

series studies published in the literature (Moolgavkar, 2000; Zeghnoun et al., 1990; Goldberg

et al., 2001). So, the interpretation of the parameters have been clarified.

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http://www.bepress.com/jhubiostat/paper56
Appendix - 2

model {

for (i in 1:N) {

ozone[i] <- y[i,1]

sigma[i] <- y[i,2]

prec[i] <- 1/(sigma[i]*sigma[i])

ozone[i] ~ dnorm(beta[i],prec[i])

beta[i] ~ dnorm(mu,tau)

}

mu ~ dnorm(0,0.0001)

tau ~ dgamma(0.01,0.01)

sig.O <- 1/sqrt(tau)

}

##mixture of two normal distributions

model { for (i in 1:N) {

ozone[i] <- y[i,1]

sigma[i] <- y[i,2]

prec[i] <- 1/(sigma[i]*sigma[i])

ozone[i] ~ dnorm(beta[i],prec[i])

beta[i] ~ dnorm(mu[i], tau)

mu[i] <- lambda[T[i]]

T[i] ~ dcat(P[])

}

P[1:2] ~ ddirch(alpha[])

theta ~ dnorm(4, 1.0E-6)I(0.0, )

lambda[2] <- lambda[1] + theta

lambda[1] ~ dnorm(-2, 1.0E-6)

tau ~ dgamma(0.001, 0.001)

sig.O <- 1 / sqrt(tau)

tot <- P[1]*lambda[1] + P[2]*lambda[2]

}

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Table 1: City-specific estimates of the percentage increase in CVDRESP mortality for a

10ppb increase in summer ozone levels for each of the 6 previous days obtained under a

distributed lag model (95% confidence intervals). Last row denotes the posterior mean and

the 95% posterior regions of the overall effect.

Los Angeles 0.79 (-0.69,2.28)

New York 2.33 (0.93,3.73)

Chicago -0.39 (-2.38,1.60)

Dallas/Fort Worth 2.35 (0.05,4.66)

Houston 1.11 (-1.62,3.84)

San Diego 1.58 (-1.50,4.67)

Santa Ana/Anaheim -3.03 (-6.40,0.35)

Phoenix 0.05 (-3.71,3.81)

Detroit 1.88 (-1.52,5.27)

Miami 0.68 (-3.51,4.86)

Philadelphia 2.42 (-0.22,5.07)

Seattle 8.08 (2.94,13.21)

San Jose 8.18 (0.61,15.75)

Cleveland 2.49 (-0.62,5.60)

San Bernardino -0.98 (-3.87,1.91)

Pittsburgh -0.05 (-3.17,3.07)

Oakland 7.97 (0.58,15.36)

Atlanta 0.77 (-3.45,4.99)

San Antonio 0.22 (-3.84,4.28)

overall 1.25 (0.47,2.03)

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Table 2: Posterior mean and 95% posterior region of the overall effect μ under alternative

specifications for the random effects distributions

Random Effect Distribution National average estimate

Normal & 1/τ 2 ∼ G(0.01, 0.01) 1.25 (0.47,2.03)

Normal & 1/τ 2 ∼ G(0.001, 0.001) 1.20 (0.47,1.93)

Normal & 1/τ 2 ∼ G(0.0001, 0.0001) 1.20 (0.51,1.89)

t-student with 3 df & 1/τ 2 ∼ G(0.001, 0.001) 1.26 (0.50,2.02)

Mixture of two normals & 1/τ 2 ∼ G(0.001, 0.001) 1.33(0.51,2.15)

Normal & 1/τ 2 ∝ constant 1.29 (0.35,2.23)

Normal & 1/τ 2/(1/τ 2 + 1/vc) ∼ U[0, 1] 1.27 (0.37,2.17)

Table 3: Percent increase in CVDRESP mortality associated with a 10ppb increase in ozone,

adjusting for co-pollutants (95% posterior regions).

O3 effect O3 O3 + PM10 O3 + NO2 O3 + SO2 O3 + CO

O3 lag 0 0.73 0.74 0.60 0.51 0.69

(0.27, 1.19) (-0.33, 1.72) (0.11, 1.10 ) (0.01, 1.03) (0.20, 1.18)

O3 lag 1 0.70 0.69 0.63 0.49 0.65

(0.26, 1.12) (-0.01, 1.38) (0.15, 1.12 ) (0.06, 0.95) (0.21, 1.08)

O3 lag 2 0.64 0.46 0.50 0.42 0.65

(0.17, 1.07) (-0.32, 1.17) (0.01, 0.99 ) (-0.01, 0.86) (0.20, 1.07)

Note: 1, All pollutants and ozone are at the same lag structure.

2,The availability of PM10 measurements is only every 1 to 6 days; Seattle and San Antonio

do not have any NO2 measurements; Miami, Seattle, San Jose, Oakland, and San Antonio

do not have any SO2 measurements.

Table 4: Percent increase in CVDRESP mortality associated with a 10ppb increase in ozone,

adjusting for heat wave (95% posterior regions).

O3 effect O3 O3 + Heat Wave

distributed 1.25 1.11

lag (DL) (0.47, 2.03) ( 0.38, 1.86)

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0 20 40 60 80

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Ozone <ppb>

Figure 1: Box plots of summer ozone levels for 19 large U.S. cities in NMMAPS for the

period 1987 - 1994.

http://www.bepress.com/jhubiostat/paper56
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log(Daily CVDRESP Death Count)

Figure 2: Box-plots of log(Daily CVDRESP Death Count) for 19 large U.S. cities in

NMMAPS in summer months of 1987 - 1994.

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% increase in mortality for 10ppb increase in summer ozone

-5 0 5 10 15

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overall

lag 0

lag 1

lag 2

distributed lag

Figure 3: City-specific relative rates estimates (percent increases in CVDRESP mortality

associated with 10ppb increase in ozone) and their 95% confidence intervals, with overall

ozone effect and its 95% posterior regions at the end. These effects are estimated by use of

a distributed lag model for one week and at lags 0, 1, and 2 days.

http://www.bepress.com/jhubiostat/paper56
(a)

0 1 2 3

0.0 0.1 0.2 0.3 0.4

Ozone at lag 0

Ozone at lag 1

Ozone at lag 2

Distributed lag model

 (b) 

0.0 0.5 1.0 1.5 2.0

0.0 0.1 0.2 0.3 0.4 0.5

SD at lag 0

SD at lag 1

SD at lag 2

Distributed lag model

Figure 4: (a) Posterior distributions of average relative rate (μ) and (b) of the heterogeneity

parameter (τ ) obtained by combining city-specific relative rates estimates under a distributed

lag model and single-lag models.

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Ozone+PM10 vs. ozone

O3 effect before adjusting PM10

O3 effect after adjusting PM10

-5 0 5 10

-5 0 5 10

**

*

*

*

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*

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*

ozone+NO2 vs. ozone

O3 effect before adjusting NO2

O3 effect after adjusting NO2

-2 -1 0 1 2 3

-2 -1 0 1 2 3

*

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Ozone+SO2 vs. ozone

O3 effect before adjusting SO2

O3 effect after adjusting SO2

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-2 -1 0 1 2 3

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Ozone+CO vs. ozone

O3 effect before adjusting CO

O3 effect after adjusting CO

-2 -1 0 1 2 3

-2 -1 0 1 2 3

*

*

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*

*

*

*

*

*

*

*

*

*

*

*

* *

*

*

Figure 5: City-specific ozone relative rates estimates at lag 2 adjusted by another pollutant

PM10,NO2, SO2 or CO, plotted versus ozone relative rates estimates at lag 2. The cross

is plotted in correspondence of the two average relative rates (adjusted versus unadjusted)

and their 95 % posterior regions.

http://www.bepress.com/jhubiostat/paper56
% increase in mortality for 10ppb increase in summer ozone

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Figure 6: Sensitivity analysis of the city-specific effects at lag 2 with respect to the degrees

of freedom in the smooth functions of time. The number of degrees of freedom vary from

4 to 16 with increasing step = 2. Each polygon captures the change in the city-specific

relative rates estimates with their corresponding 95% confidence intervals. The last polygon

captures the change of overall ozone effect estimates with its 95% posterior regions.

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