Vol. 162, No. 1
Printed in U.S.A.
DOI: 10.1093/aje/kwi157
American Journal of Epidemiology
Copyright ª 2005 by the Johns Hopkins Bloomberg School of Public Health
All rights reserved
Are the Acute Effects of Particulate Matter on Mortality in the National Morbidity,
Mortality, and Air Pollution Study the Result of Inadequate Control for Weather
and Season? A Sensitivity Analysis using Flexible Distributed Lag Models
Leah J. Welty and Scott L. Zeger
From the Department of Biostatistics, Johns Hopkins Bloomberg School of Public Health, Baltimore, MD.
Received for publication April 28, 2004; accepted for publication February 16, 2005.
Time-series studies have linked daily variations in nonaccidental deaths with daily variations in ambient particulate matter air pollution, while controlling for qualitatively larger influences of weather and season. Although timeseries analyses typically include nonlinear terms for weather and season, questions remain as to whether models
to date have completely controlled for these important predictors. In this paper, the authors use two flexible
versions of distributed lag models to control extensively for the confounding effects of weather and season. One
version builds on the current approach to controlling for weather, while the other version offers a new approach.
The authors conduct a comprehensive sensitivity analysis of the particulate matter–mortality relation by applying
these methods to the recently updated National Morbidity, Mortality, and Air Pollution Study database that comprises air pollution, weather, and mortality time series from 1987 to 2000 for 100 US cities. They combine cityspecific estimates of the short-term effects of particulate matter on mortality using a Bayesian hierarchical model.
They conclude that, within the broad classes of models considered, national average estimates of particulate
matter relative risk are consistent with previous estimates from this study and are robust to model specification
for weather and seasonal confounding.
air pollution; longitudinal studies; mortality; regression analysis; seasons; temperature; weather
Abbreviations: APHEA, Air Pollution and Health: a European Approach; NMMAPS, National Morbidity, Mortality, and
Air Pollution Study; PM10, particulate matter of less than 10 lm in aerodynamic diameter.
A substantial literature of epidemiologic studies demonstrates an association between mortality and exposure to
particulate matter air pollution (1). Much of this evidence
comes from time-series studies (2) that compare daily fluctuations in mortality with daily fluctuations in particulate
matter while controlling for confounding factors that also
vary over time, such as weather and season (3). Initially,
time-series studies of air pollution and mortality estimated
the particulate matter effect for individual cities (4–8). More
recently, multicity studies, such as the Air Pollution and
Health: a European Approach (APHEA) project, the National Morbidity, Mortality, and Air Pollution Study
(NMMAPS), and a study of 11 Canadian cities, have provided consistent evidence of the widespread association between current particulate matter levels and mortality (9–16).
Despite the evidence, doubt about the findings remains, in
part because of the potential confounding effects of weather
and season (17). Time-series studies of temperature and
mortality have found temperature effects (18–22) that are
larger than particulate matter effects and that may be correlated with particulate matter levels (7, 23). Time-series studies of the effects of both air pollution and weather on
mortality have identified the importance of adequate control
for temperature and humidity when estimating air pollution
effects (24–28). Multicity time-series studies of particulate
matter and mortality have accounted for the potential confounding effects of weather by including nonlinear covariates for current and previous days’ weather and by
excluding extreme weather days. These studies have also
conducted sensitivity analyses on the degree of flexibility
Correspondence to Dr. Leah J. Welty, Department of Preventive Medicine, Northwestern University Feinberg School of Medicine,
680 North Lake Shore Drive, Suite 1120, Chicago, IL 60611 (e-mail: [email protected]).
80
Am J Epidemiol 2005;162:80–88
NMMAPS Distributed Lag Model Sensitivity Analysis 81
for weather and season covariates (12, 16). However, concern remains that ‘‘the possibility of subtler effects within
the normal climactic range continues’’ (17, p. 67). A recent
report on the results from multicity time-series studies states
that ‘‘further exploration of weather effects is merited (for
example, considering correlated cumulative effects of multiday temperature or humidity)’’ (17, p. 67).
Motivated by this lingering challenge to the particulate
matter–mortality association, we investigate possible subtler
effects of weather and season on estimates of mortality
relative risk for particulate matter of less than 10 lm in
aeordynamic diameter (PM10). Using the recently updated
NMMAPS database with time series from 1987 to 2000 for
over 100 US cities (16), we conduct sensitivity analyses of
city-specific and national average estimates of the acute effect of PM10 on mortality to control for the effects of weather
and season, including cumulative temperature effects. For
comparison, our sensitivity analysis includes models similar
to those used in previous analyses of the NMMAPS.
Many single-city time-series studies investigating more
aggressive control for weather and season have shown that
particulate matter effect estimates are generally robust (7,
25, 26, 28). Results from single-city studies do not necessarily generalize to other cities, and single-city particulate
matter effect estimates are more variable than are national
estimates, making it difficult to distinguish noise from confounding. A substantial portion of evidence for the acute
effects of PM10 on mortality comes from multicity studies,
such as the NMMAPS and the APHEA project, so it is
important to conduct extensive sensitivity analyses for these
multicity studies.
We consider two strategies to control for the effects of
temperature and season on mortality. Both are based on
distributed lag models, time-series models that allow an
exposure to affect response over an extended period of time
(29). Using distributed lag models, researchers have shown
that temperature affects mortality over several days (18, 19),
so distributed lag models provide a natural framework for
considering the effects of weather on mortality. Distributed
lag models also provide a natural framework to quantify the
health effects of multiple-day exposure to particulate matter,
and they have been used throughout the air pollution epidemiology literature (11, 14, 15, 28, 30).
Our distributed lag models are formulated to capture
the complexity of confounding by temperature and season
and to account for documented aspects of the temperature–
mortality relation. Temperatures up to 2 weeks prior are
likely sufficient for capturing the lagged effects of temperature on mortality (18), so we use distributed lag models
with 2 weeks of temperature lags. Numerous studies and
our own exploratory analyses have shown that the relation
between temperature and mortality is nonlinear and varies
by season. The relation between temperature and mortality
relative risk is convex (i.e., U shaped), implying that cold
temperatures in winter or hot temperatures in summer are
worse for health than are moderate temperatures (20, 22).
The exact nature of the nonlinearity, or the degree to which
temperature extremes affect mortality, varies by location
(20). We formulate two versions of distributed lag models
to control for the nonlinear effects of temperature on
Am J Epidemiol 2005;162:80–88
mortality: one that builds on the current approach to control
for temperature and one that takes a new approach.
In early air pollution time-series studies, methods of control for temperature included categorizing days by weather
regime, using sine and cosine terms or indicator variables for
season and using indicator, linear, and polynomial terms
for meteorologic covariates (4–6, 10, 21). In recent timeseries studies, smooth, unspecified functions of temperature
have been used to account for the nonlinearity between
temperature and mortality. In multicity time-series studies,
this flexible specification allows for different nonlinear
temperature–mortality relations in different cities. The
advantage of models with flexible functions of temperature
is that the exact nature of the temperature–mortality relation
need not be explicitly defined. However, these models do not
account for the possibility that the nonlinear relation between
temperature and mortality may change slowly over time. For
example, if in a city air conditioner use had increased over
time, warm summer temperatures in the later years of the
study could be less harmful than warm summer temperatures
in the earlier years of the study. A single U-shaped function of
temperature in this situation might capture only the average
effect of warm temperatures on mortality and possibly have
some subtle effect on the particulate matter effect estimate.
We consider two versions of distributed lag models to
control for these complex effects of temperature on mortality. One version builds on the current use of flexible functions of temperature, while the new version allows for the
nonlinear temperature–mortality relation to vary over time.
The first part of this paper describes these two types of
distributed lag models. In Materials and Methods, we introduce a basic distributed lag model for temperature and show
how it may be extended so that temperature coefficients
trend and vary seasonally, thereby allowing the temperature–
mortality relation to vary nonlinearly and in time. We also
illustrate how models with nonlinear temperature covariates
are derived from the basic distributed lag model, and we
consider more extensive versions than those often used.
We follow up with subsections on the inclusion of temperature interactions and other covariates. The last portion of
Materials and Methods outlines a hierarchical model for
combining city-specific PM10 effect estimates into national
estimates. The next part of this paper compares fitted-model
results, both at the national and at the individual-city levels,
to assess the sensitivity of the estimated particulate matter–
mortality relation to control for weather and season.
MATERIALS AND METHODS
Data
The data were assembled from publicly available sources
as part of the NMMAPS, funded by the Health Effects Institute. The data consist of daily time series of mortality,
temperature, dew point, and PM10 for 109 US cities for the
period 1987–2000. Daily death counts were obtained from
the National Center for Health Statistics and classified by age
(<65, 65–74, 75 years); accidental deaths and homicides
were excluded. Twenty-four hour averages of temperature
and dew point were computed from hourly observations
82
Welty and Zeger
assembled by the National Climactic Data Center in the
EarthInfo database. Daily measures of PM10 were obtained
from the US Environmental Protection Agency’s Aerometric
Information Retrieval and AirData Systems. Several cities
had no PM10 measurements, and many cities had PM10 nonmissing values every sixth day. The 100 cities with PM10
measured at least every sixth day for more than 4 years were
used in the following analysis. Additional details regarding
data assembly are available at http://www.ihapss.jhsph.edu/
and are discussed in previous NMMAPS analyses (31).
Methods
Distributed lag models for temperature. Poisson regression (32) is frequently used to estimate the relation between
fluctuations in daily mortality counts and fluctuations in air
pollution, while taking into account fluctuations in weather
and other time-varying confounders (3). We assume that Ytc
is an overdispersed random variable with E½Ytc ¼ lct and
Var½Ytc ¼ /c lct . The overdispersion parameter /c represents the variation in mortality not captured by the regression model. We let xct and pmct be the daily temperature and
PM10 time series for city c. Our analysis considers the effects of previous days’ temperature and particulate matter
on current day mortality, so let xctn and pmctl refer to the
temperature and PM10 time series lagged by n and l days,
respectively. A model for lag l PM10 on daily mortality with
distributed lags of temperature is
c
c c
c c
c c
logðlt Þ ¼ k0 xt þ k1 xt1 þ þ kn xtn
c
c
þ bl pmtl þ Sðt;a 3 yearsÞ þ covariates
ð1Þ
where covariates include other confounding variables, such
as the day of the week. The term S(t, a 3 years), a smooth
function of time with a degrees of freedom per year, controls
for the seasonality in mortality that is not directly related to
temperature. We restrict our attention to models with single
lags of PM10 because PM10 is sampled every sixth day for
the majority of the 100 cities in our analysis.
Time lags of exposure variables are often correlated,
making the distributed lag effects kcj difficult to estimate.
A common solution is to constrain the kcj to a functional
form, such as a polynomial (29) or a spline (33). Here, we let
n ¼ 14, allowing for temperatures up to 2 weeks prior to
affect mortality, and we constrain the kcj to lie on a step
function. This formulation minimizes the number of distributed lag parameters, allows for effect differences among
more and less recent temperatures, and facilitates interpretation of coefficients and of the total effect of temperature.
We set steps at lag 0, lag 2, and lag 7, thereby constraining
the kcj so that lag 1 and lag 2 temperatures have the same
effect on mortality, lag 3–7 temperatures have the same
effect on mortality, etc. Defining xc1t as the average of the
past 2 days’ temperatures ði:e:, xc1t ¼ ½xct1 þ xct2 =2Þ, xc2t as
the average of the past 7 days’ temperatures, and xc3t as the
average of the past 14 days’ temperatures, we have
c
c c
c c
c c
c c
logðlt Þ ¼ h0 xt þ h1 x1t þ h2 x2t þ h3 x3t
c
c
þ bl pmtl þ Sðt;a 3 yearsÞ þ covariates:
ð2Þ
A 10F (5.6C) increase in average temperature on the current and on each of the past 14 days corresponds with
a 1,000ðhc0 þ hc1 þ hc2 þ hc3 Þ percent increase in current day
mortality, which we refer to as the ‘‘total temperature effect.’’
To illustrate this step distributed lag function as well as
the seasonal variability in temperature effects, figure 1
shows estimates of the step distributed lags (equation 2)
fitted separately for each of the 14 summers (May–August)
and each of the 13 winters (November–February) of the
NMMAPS data for New York City. The covariates day of
week (as a factor) and day of month (as a linear term) were
also included. The smooth function of time S(t, a 3 years)
was not included since we fitted the model separately over
the 4-month periods. For New York City, a 10F increase in
current day temperature in May–August results in greater
increase in mortality than does a 10F increase in current
day temperature in November–February (figure 1). Increases in lag 1 and lag 2 day temperatures generally result
in increases in mortality in May–August and decreases in
November–February (figure 1).
Distributed lag models for temperature with seasonally and
temporally varying coefficients. In these models, distrib-
uted lag temperature coefficients vary by season and time.
This implicitly allows the relation between temperature and
mortality to be nonlinear (since the temperature coefficient
for a summer day may differ from that for a winter day) and
the nonlinearity to change over time (since the temperature
coefficient for a summer day may differ from the coefficient
for the same summer day a year later). We replace the temperature log relative risks hck in equation 2 by time-varying
temperature log relative risks hck ðtÞ, which we model as
parametric functions of time. We use sine and cosine functions to specify seasonal change and smooth functions to
describe temporal change in the hck ðtÞs (details in Appendix). The distributed lag model then becomes
c
c
c
c
c
c
c
c
c
x1t þ h2 ðtÞ
x2t þ h3 ðtÞ
x3t
logðlt Þ ¼ h0 ðtÞxt þ h1 ðtÞ
c
ð3Þ
c
þ bl pmtl þ Sðt;a 3 yearsÞ þ covariates:
Replacing the fixed temperature coefficients (equation 2)
with time-varying coefficients allows for temperatures up
to 2 weeks prior to affect mortality nonlinearly and for this
nonlinearity to change over time.
Distributed lag models for temperature with nonlinear
temperature covariates. These models include those cur-
rently used in multicity studies and use smooth functions of
distributed lags of temperature to explicitly account for nonlinearity in the temperature–mortality relation. We let S(, q)
denote a smooth function with q degrees of freedom for a
city c, and we consider distributed lag models of the form
c
c
logðlt Þ ¼ Sðxt ;qÞ þ
K
X
c
c
c
Sð
xkt ;qÞ þ bl pmtl
ð4Þ
k¼1
þ Sðt;a 3 yearsÞ þ covariates:
The value for q determines the nonlinearity of distributed
lag temperature covariates, and K determines how many
lags of temperature to include. By K ¼ 0, we denote the
model that contains only current day temperature. Previous
Am J Epidemiol 2005;162:80–88
NMMAPS Distributed Lag Model Sensitivity Analysis 83
FIGURE 1. Percent increase in mortality associated with temperature increases at lag days 0–14, New York City, New York, 1987–2000. Percent
increases were estimated separately for the winter and summer months and separately for each year, with the use of a distributed lag step function
and temperature and mortality data from the National Morbidity, Mortality, and Air Pollution Study. Lines indicate the distributed lag effcts estimated
for each year; shaded regions indicate the distributed lag effects averaged over years.
multicity studies have used smooth functions of the current
day’s temperature and the average of the past 2 or 3 days’
temperatures (9, 12–16), that is, K ¼ 1, so we consider K ¼
0, 1, 2, 3. Previous multicity studies have generally set q ¼ 3,
and corresponding sensitivity analyses have considered the
effects of varying q on particulate matter log relative risk
(13, 16). We accordingly investigate how both the smoothness of the distributed lag temperature variables (q) and the
number of distributed lag variables (K) may influence particulate matter log relative risk.
In models with nonlinear temperature terms (equation 4),
the exact nature of the seasonal relation between temperature and mortality is not explicitly defined. However, these
models do not allow for the nonlinearity between temperature and mortality to change over time. The model with
seasonally–temporally varying temperature coefficients
(equation 3) allows implicitly for a nonlinear relation between temperature and mortality and additionally allows for
this nonlinearity to change over time. We compare PM10 log
Am J Epidemiol 2005;162:80–88
relative risk estimates bcl from both formulations to determine if either formulation alters PM10 estimates.
Interactions among lagged temperatures. Models that include interactions of distributed lag temperature variables
allow for synergy between current and previous days’
temperatures when these affect mortality. We accordingly
estimate our distributed lag models with and without interactions of temperature distributed lags. The interaction
terms for seasonally–temporally varying distributed lag
models (equation 3) take the form hc ðtÞxct xc1t , etc. For models
with smooth functions of temperature distributed lags
(equation 4), interactions take the form Sðxct xc1t ; qÞ, etc. To
keep the number of regression parameters from growing too
large, we excluded all interactions with xc3t .
Other covariates. Measures of humidity, such as dew
point, are important weather components in mortality–air
pollution models (34). However, directly including dew point
in models with many temperature covariates may result in
variable parameter estimates due to collinearity. Residualized
84
Welty and Zeger
values of dew point variables regressed on temperature
covariates are orthogonal to the temperature covariates but
retain variation in dew point not explained by temperature.
We regress the current-day dew point and the average of the
past 2 days’ dew points on temperature covariates and include
the respective residuals in our models.
For convenience, we denote the distributed lag models
with seasonally–temporally varying temperature coefficients by SVIa and those with nonlinear smooth functions
of temperature by NLIa ðK þ 1; qÞ, where a indicates the
degrees of freedom per year in the smooth time trend, and
the presence of superscript I indicates the inclusion of temperature interactions. For the nonlinear models, K þ 1 refers
to the number of distributed lag variables in the model (current day plus K additional lag averages), and q is the degree
of smoothness of the distributed lag variables. For both
model formulations, we include the covariates day of week
and age category (as factors), day of month (as a linear
term), and for each age category a natural spline of time
with 14 df. The smooth of time by age category accounts for
mortality trends related to an aging population or differing
migration rates within age categories. Although not included in prior NMMAPS analyses, a few previous timeseries studies of air pollution and health have adjusted for
day of month (35). We conservatively included a linear dayof-month term in our models.
The PM10 log relative risk was estimated for all distributed
lag models separately for each of 100 cities, using the
‘‘glm’’ function in R, version 1.8.0, software (see below).
The models included PM10 exposure from the current day,
the previous day, or 2 days previously (l ¼ 0, 1, or 2). For
each of these exposures in the seasonally–temporally varying model, the degree of adjustment for seasonal effects on
mortality was varied, with a ¼ 1, 2, 4, or 8. Based on the
results for the seasonally–temporally varying model, a was
set to 4 for the nonlinear model. All R programs are available in an R-script tdlm.R at http://www.ihapss.jhsph.edu/
data/NMMAPS/R/, and the data are available as part of the
NMMAPSdata Package in R (36).
Multicity estimates. The comparison of city-specific log
relative risks across models informs the sensitivity of particulate matter estimates to control for weather and season
for a single city but does not quantify the sensitivity of
particulate matter estimates to control for weather and season generally. With a Bayesian hierarchical model (37),
city-specific PM10 log relative risks bcl for a particular model
may be used to estimate a pooled PM10 log relative risk b*l
for that model. Then, b*l is a national average PM10 log
relative risk. The hierarchical model supposes that
b̂cl j bcl ; rcl ~ Nðbcl ; rcl Þ and bcl are independent Nðb*l ; r*l Þ
and that rcl s are uniformly distributed. TLNise statistical
software (37) was used to estimate b*l and r*l .
RESULTS
National average estimates of the percent increase in
mortality associated with a 10-lg/m3 increase in PM10 at
lags 0, 1, and 2 are robust to model specification for weather
and season. Figure 2 shows national average PM10 posterior
means for lags l ¼ 0, 1, 2 for all distributed lag models
considered. Effect estimates are consistent within the lag
of PM10; overlapping 95 percent pointwise posterior intervals demonstrate no consistent differences in estimates
across models. Previous day PM10 has a statistically significant association with daily mortality for all models, with an
increase of 10 lg/m3 in PM10 corresponding generally to
a 0.2 percent increase in mortality. Table 1 shows national
average PM10 posterior means and standard errors for all
models with lag 1 PM10. Estimates are consistent with previous NMMAPS analyses (16).
Estimated PM10 log relative risks for individual cities are
qualitatively similar across models, but model choice may
determine effect significance. Figure 3 displays unpooled,
city-specific lag 1 PM10 log relative risks for eight of the 24
models and for 10 of the largest US cities. Within-city effect
estimates from one model to the next are not different in
a material way. For the eight models shown, the 95 percent
level of significance for the PM10 effect changes across
models for Los Angeles, New York, and Chicago, while it
remains the same for the other seven cities.
Varying a, the degrees of freedom per year in the smooth
time trend, shifted the portion of mortality attributed to
temperature instead of season, but it did not substantially
alter their combined predictive effect or the PM10 log relative risk. In seasonally–temporally varying models with small
values of a, the distributed lag variable x3t ¼ ðxt1 þ þ
xt14 Þ=14 acted as a surrogate for season. Its estimated effects
were that of a seasonal covariate: large and positive in cooler
months, when influenza epidemics increase mortality, and
negative or near zero in summer. For all values of a, the total
temperature effect was strongly negatively correlated with
the smooth time trend S(t, a 3 years). The combined predictor for the temperature effect and the smooth time trend
did not change substantially when a ranged over 1, 2, 4, and 8
and correlations of the combined predictors for different
values of a were all near 0.9. The effects of season and temperature on mortality, therefore, may not be fully separated,
although distinction is unnecessary for robust estimation of
the PM10–mortality relation.
DISCUSSION
The results above address concerns that the PM10–
mortality association from multicity time-series studies (9,
12, 16) may be biased by residual temperature and seasonal
confounding. The comparison of PM10 effect estimates
across distributed lag models for temperature, some of
which are substantially more flexible than previously considered models, shows that, across 100 US cities in the
NMMAPS, the short-term effects of PM10 on mortality
are not an artifact of inadequately modeled weather and
season. Consistent with previous NMMAPS analyses (16),
a 10-lg/m3 increase in the previous day’s PM10 is associated
with an approximate 0.2 percent increase in daily mortality.
Distributed lag models with seasonally–temporally varying
temperature coefficients and with nonlinear temperature
covariates produce similar estimates of PM10 log relative
risk. There is no clear division between daily mortality
Am J Epidemiol 2005;162:80–88
NMMAPS Distributed Lag Model Sensitivity Analysis 85
FIGURE 2. Posterior means and 95% posterior intervals of the national average short-term effects of particulate matter of less than 10 lm in
aerodynamic diameter (PM10) on mortality, by PM10 at lags 0, 1, and 2, National Morbidity, Mortality, and Air Pollution Study, 1987–2000. The
national average estimates were obtained by use of different versions of seasonally–temporally varying and nonlinear distributed lag models, both
with and without temperature interaction terms, to account for the confounding effects of temperature. SVa denotes a seasonally–temporally
varying model with a degrees of freedom per year in the smooth time trend; NLa(K þ 1, q) denotes a nonlinear model with a degrees of freedom per
year in the smooth time trend and K þ 1 temperature distributed lag variables (current-day plus K additional lag averages), and q denotes degrees
of freedom for the smooth functions of the distributed lags of temperature. Addition of superscript I denotes a model with distributed lag temperature
interactions included as well.
attributable to temperature and daily mortality attributable
to seasonality; however, this division does not substantively
alter PM10 effect estimates. We have not investigated if
barometric pressure or wind speed may account for mortality currently attributed to PM10, but these factors have yet to
be consistently identified as potential confounders.
The variability of evidence from single-city analyses investigating confounding by weather and season highlights
the advantages of a multicity approach. Samet et al. (26)
found that estimates in Philadelphia, Pennsylvania, were
robust to control for weather, while Smith et al. (34) found
that PM10 coefficients for Birmingham, Alabama, and
Chicago, Illinois, were sensitive to control for humidity. Our
findings show that, though different weather and season
models may alter the significance of estimates for specific
Am J Epidemiol 2005;162:80–88
cities, they do not significantly or substantially alter national
PM10 log relative risk estimates.
Our findings support the use of smooth functions of
current-day and average temperature and dew point from
the past few days to control for weather effects. The nonlinear
distributed lag model designated NL4(2, 4) is similar to the
models used in multicity studies, such as the NMMAPS and
the APHEA project (10, 12–14, 16), and we found that the
PM10 log relative risk from NL4(2, 4) is similar to estimates
from our other nonlinear models with more distributed lags of
temperature. Time-series studies of temperature and mortality have found the strongest temperature effects on the current day and the past few days (18–20), perhaps explaining
why including distributed lags of temperature for more than
a few days does not alter particulate matter effect estimates.
86
Welty and Zeger
TABLE 1. Posterior means and standard errors of national average short-term effects of particulate matter
of less than 10 mm in aerodynamic diameter on mortality using seasonally–temporally varying and
nonlinear distributed lag models to control for the confounding effects of weather and season, National
Morbidity, Mortality, and Air Pollution Study, 1987–2000*
Time trendz
Mean % increase
in mortality
Standard
error
S(t, 1 3 years)
0.229
0.053
S(t, 2 3 years)
0.220
0.053
S(t, 4 3 years)
0.187
0.050
S(t, 8 3 years)
0.178
0.049
S(t, 1 3 years)
0.195
0.048
S(t, 2 3 years)
0.200
0.051
S(t, 4 3 years)
0.176
0.050
S(t, 8 3 years)
0.149
0.050
0, 1–2, 1–7, 1–14
S(t, 4 3 years)
0.239
0.053
þ 0 3 1–2, 0 3 1–7, 1–2 3 1–7
S(t, 4 3 years)
0.172
0.045
S(0, 2), S(1–2, 2), S(1–7, 2),
S(1–14, 2)
S(t, 4 3 years)
0.186
0.046
þ S(0 3 1–2, 2), S(0 3 1–7, 2),
S(1–2 3 1–7, 2)
S(t, 4 3 years)
0.189
0.047
S(0, 4), S(1–2, 4), S(1–7, 4),
S(1–14, 4)
S(t, 4 3 years)
0.175
0.046
þ S(0 3 1–2, 4), S(0 3 1–7, 4),
S(1–2 3 1–7, 4)
S(t, 4 3 years)
0.190
0.048
0, 1–2, 1–7
S(t, 4 3 years)
0.252
0.053
þ 0 3 1–2, 0 3 1–7, 1–2 3 1–7
S(t, 4 3 years)
0.186
0.044
S(0, 2), S(1–2, 2), S(1–7, 2)
S(t, 4 3 years)
0.198
0.046
þ S(0 3 1–2, 2), S(0 3 1–7, 2),
S(1–2 3 1–7, 2)
S(t, 4 3 years)
0.201
0.047
S(0, 4), S(1–2, 4), S(1–7, 4)
S(t, 4 3 years)
0.189
0.045
þ S(0 3 1–2, 2), S(0 3 1–7, 4),
S(1–2 3 1–7, 2)
S(t, 4 3 years)
0.205
0.047
S(0, 4), S(1–2, 4)
S(t, 4 3 years)
0.250
0.045
þ S(0 3 1–2, 4)
S(t, 4 3 years)
0.253
0.044
S(0, 4)
S(t, 4 3 years)
0.220
0.045
Distributed lag
model
Seasonally–
temporally varying
Seasonally–
temporally varying
Nonlinear
Nonlinear
Nonlinear
Nonlinear
Temperature
variablesy
0, 1–2, 1–7, 1–14
0, 1–2, 1–7, 1–14, 0 3 1–2, 0 3
1–7, 1–2 3 1–7
* Means and standard errors are in units of percent increase in daily mortality associated with a 10-lg/m3
increase in previous-day particulate matter of less than 10-lm aerodynamic diameter.
y 0 indicates current-day temperature; 1–r indicates the average of lag 1 through lag r temperature; S(, q)
indicates a natural spline smooth with q degrees of freedom.
z S(t, a 3 years) indicates the natural spline smooth of time with degrees of freedom equal to a 3 (number of
years of data).
Models that control extensively for the effects of weather
may be helpful when estimating the effects of pollutants that
have more temperature dependence than does PM10, such as
ozone. We note that our models and the associated software
implementation (36) can be used in other multisite timeseries studies of particulate matter or other pollutants.
Our analysis demonstrates that the short-term effects of
PM10 on mortality estimated by Poisson regression are not
artifacts of inadequate control for weather and season. Similar conclusions have been made for multicity time-series
studies by use of alternative methods to Poisson regression,
such as case-crossover analysis (38). Combined, the results
suggest that the short-term effects of particulate matter on
mortality are viable; they are the result of neither the lack of
control for weather and season within the Poisson regression
framework nor the use of Poisson regression itself.
Although the PM10 effect estimates for the NMMAPS are
robust to control for weather and season, they cannot capture
all of the short- or long-term effects of particulate matter on
mortality. Our sensitivity analysis is limited to single-day
PM10 exposures since the majority of cities in the NMMAPS
lack daily PM10 measurements, but other studies have
Am J Epidemiol 2005;162:80–88
NMMAPS Distributed Lag Model Sensitivity Analysis 87
FIGURE 3. Maximum likelihood estimates and 95% confidence intervals of the short-term effects of particulate matter of less than 10 lm in
aerodynamic diameter (PM10) on mortality for the 10 largest cities in the National Morbidity, Mortality, and Air Pollution Study, 1987–2000, by use of
eight different distributed lag models to control for confounding by weather and season. The 10 cities are Los Angeles, California (La); New York,
New York (Ny); Chicago, Illinois (Chic); Dallas/Fort Worth, Texas (Dlft); Houston, Texas (Hous); Phoenix, Arizona (Phoe); San Diego, California
(Sand); Santa Ana/Anaheim, California (Staa); Miami, Florida (Miam); and Detroit, Michigan (Det). ‘‘All’’ denotes national average posterior means
and 95% posterior intervals for the short-term effects of PM10 on mortality. Triangles indicate seasonally–temporally varying (SV) distributed
lag models, and dots indicate nonlinear (NL) distributed lag models. From left to right, the specific models shown are SV2, SV8, SVI4 , and SVI8 and
NLI4 ð1, 4Þ, NLI4 ð2, 4Þ, NLI4 ð3, 2Þ, and NLI4 ð4, 2Þ.
reported larger PM10 log relative risks for models that
include distributed lags of PM10 (14, 15, 28, 30). These
studies that find significant air pollution log relative risks
using models that include multiple-day exposures of temperature and air pollution provide additional evidence that
residual confounding by weather is not responsible for the
observed air pollution–mortality relation (28). Our estimated 0.2 percent increase in mortality due to a 10-lg/m3
increase in previous-day PM10 reflects only a portion of the
short-term health effects of PM10 and, furthermore, does not
estimate the larger magnitude of chronic health effects identifiable only through cohort studies. Our methods control for
unmeasured differences across city populations that may
confound cohort studies however, and therefore they provide
important evidence that particulate matter (and not other
factors) is responsible for adverse health effects.
ACKNOWLEDGMENTS
This research at the Johns Hopkins Bloomberg School of
Public Health was supported by grant U50CCU322417 from
the Center for Excellence in Environmental Public Health
Tracking, Centers for Disease Control and Prevention; by
National Institutes of Health grant R01ES012054; and by
Health Effects Institute award HEI025.
Am J Epidemiol 2005;162:80–88
The authors also wish to thank Drs. Francesca Dominici,
Roger Peng, and Thomas Louis for their helpful comments.
The work herein does not necessarily reflect the views of
the funding agencies nor was it subject to their review.
Conflict of interest: none declared.
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APPENDIX
As in equation 3, let hck ðtÞ be a seasonally–temporally
varying log relative riskP
for our temperature-distributed
c
lags. We define hck ðtÞ ¼ 19
m¼1 cmk Dm ðtÞ: The Dm(t)s are
seasonal and time-trend basis functions comprising harmonic functions to account for the seasonality in temperature effects, a smooth function of time to account for gradual
changes in the level of temperature effects from 1987 to
2000, and the appropriate interactions between the two to
account for possible changes in the seasonal effects of temperature over time. To specify the Dm(t), for day t of our
time series, we let dt represent the corresponding day number in year. We let D1(t), D2(t), D3(t), and D4(t) be harmonic
basis vectors for seasonal variation in the temperature–
mortality relation defined by the following:
D1 ðtÞ ¼ sinð2pdt =365Þ
D2 ðtÞ ¼ cosð2pdt =365Þ
D3 ðtÞ ¼ sinð2pdt =ð365=2ÞÞ
D4 ðtÞ ¼ cosð2pdt =ð365=2ÞÞ:
In a leap year, we accordingly divide dt by 366. We let D5(t),
D6(t), and D7(t) be basis vectors for slow temporal change in
the temperature–mortality relation, designated by a natural
spline over t ¼ 1, 2, . . ., 5,114 (i.e., years 1987–2000) with 3
df. To allow the seasonal effects of temperature to vary over
time, we set D8(t) ¼ D1(t) 3 D5(t), D9(t) ¼ D1(t) 3
D6(t), . . ., and D19(t) ¼ D4(t) 3 D7(t). Although understanding the sensitivity of PM10 log relative risk is our primary
interest, we note that the time-varying temperature log
relative risks hck ðtÞ may be computed directly from the
estimates of the ccmk s:
Am J Epidemiol 2005;162:80–88