Cox Models for Ecologic Time-Series Data?
Referencing: Survival Analysis to Estimate Association between Short-Term Mortality and Air Pollution
In a recent article, Lepeule et al. (2006) proposed
using Cox regression with time-dependent covariates to estimate the
acute health effects of air pollution. Their results were similar to
those they obtained in a previous case-crossover analysis (Filleul et
al. 2004), and they claimed that the Cox model approach is more precise.
Understanding their results and why the claim is misleading requires
considering how case-crossover and Cox model analyses work.
The case-crossover design (Maclure 1991) requires
a choice of referent strategy or a method for choosing control time
periods (referent windows). With a valid referent strategy—a localizable
design (Janes et al. 2005a; Janes et al. 2005b)—a conditional
likelihood is constructed by conditioning on the number of events experienced
by each person over the study period. Conveniently, there is no information
on the exposure effect from people who do not have an event, so no information
is lost by dropping them from the analysis. The information comes from
variations in exposure within person and within referent window. We
must assume that all variables that confound the variation in risk within
an individual across a referent window have been measured. The estimated
β is the value that equates the exposure on the index day to its
expected value over the referent window, averaged over all subjects.
The Cox model (Cox 1972) uses the same principle
of equating the observed and expected exposure, but across people rather
than within a person. Time points with no events do not contribute information
for estimating the exposure effect and may be discarded. The information
comes from comparisons between people at the same point in time. We
must assume that all variables that confound variation in risk between
individuals at the same point in time have been measured. The estimated
β is the value for which the exposure for the person with the event
equals its expected value over the at-risk cohort, averaged over all
time points.
If the same time scale is used for the case-crossover
and Cox analyses, the two sets of information do not overlap: the case-crossover
analysis is purely within person; the Cox model analysis is purely between
persons. When exposure measurements vary both over time and by individual,
the two analyses provide independent estimates of risk. In a data set
that includes only chronic exposure measurements, there is no temporal
exposure variation so the Cox model captures all the information. Conversely,
in an ecologic time-series data set, there is no variation in exposure
between people at a given time; therefore, the case-crossover analysis
uses all of the information.
In order to estimate acute effects with ecologic
exposure measurements using a Cox model, Lepeule et al. (2006) used
age as the time scale. That is, they chose β, so that the exposure
for an individual who died at a given age is equal to the average exposure
for at-risk individuals at exactly that age. Because all individuals
have the same exposure measurement on any given day, this is equivalent
to comparing exposure on the day of death with exposure on a selected
set of other days determined by the dates other members of the cohort
reach that age. That is, it is a case-crossover design, albeit one with
an unusual choice of referent strategy. Note also that the Cox regression
estimating equations are exactly the same as those used in conditional
logistic regression, making the case-crossover and Cox regression estimates
identical.
This Cox model approach is a case-crossover design.
Theoretical development is needed to determine whether it is a localizable
design. It is more effcient than a semisymmetric bidirectional case-crossover
design only because more referent time points are used.
We see at least two potential biases associated
with this design. First, it is not clear that the strong seasonality
and time trends in air pollution and mortality data are controlled with
this referent strategy; typically, referent windows are designed to
be small to control for time-dependent confounders by design. This referent
strategy necessitates controlling such factors by modeling, as these
authors have done. Second, there may be minor bias due to subjects who
die very young or very old being dropped from the analysis because they
have no referents (no one else is at risk at that age).
The authors declare they have no competing financial
interests.
Thomas Lumley
Department of Biostatistics
University of Washington
Seattle, Washington
Holly Janes
Department of Biostatistics
Johns Hopkins Bloomberg School
of Public Health
Baltimore , Maryland
Lianne Sheppard
Department of Occupational and
Environmental Health
Sciences
University of Washington
Seattle, Washington
References
Cox DR. 1972. Regression models and life tables.
JR Stat Soc B 34:187–220.
Filleul L, Rondeau V, Cantagrel A, Dartigues JF.
2004. Do subject characteristics modify the effects of particulate air
pollution on daily mortality among the elderly? J Occup Environ Med
46:1115–1122.
Janes H, Sheppard L, Lumley T. 2005a. Case-crossover
analyses of air pollution exposure data: referent selection strategies
and their implications for bias. Epidemiology 16:717–726.
Janes H, Sheppard L, Lumley T. 2005b. Overlap bias
in the case-crossover design, with application to air pollution exposures.
Stat Med 24:285–300.
Lepeule J, Rondeau V, Filleul L, Dartigues JF. 2006.
Survival analysis to estimate association between short-term mortality
and air pollution. Environ Health Perspect 114:242–247.
Maclure M. 1991. The case-crossover design: a method
for studying transient effects on the risk of acute events. Am J Epidemiol
133:144–153.
Cox Models: Lepeule et al. Respond
We read with interest the letter by Lumley et al.
regarding our article (Lepeule et al. 2006), and we appreciate their
comments and interesting suggestions.
Our results (Lepeule et al. 2006) showed that the
Cox model (Cox 1972) approach gave more precise results for cohort data
than the case-crossover design (Maclure 1991). As stated by Lumley et
al., the Cox model is more efficient than the semisymmetric bidirectional
case-crossover design because more referent time points are used. In
fact, because the case-crossover design is a within-people approach,
people who do not have the event are not included in the analysis, whereas
they are in the Cox model and so contribute to the information for estimating
the exposure effect.
Lumley et al. specify that the estimating equations
for the Cox regression are the same as those used in the conditional
logistic regression for the case-crossover design and that we applied
them to the same data. Despite that, we cannot say that the Cox model
is a case-crossover design with an unusual choice of referent strategy.
As we stated in our article (Lepeule et al. 2006), the results of both
approaches are very similar, and when a cohort is available, the Cox
model should be applied because survival analysis uses all available
information and increases the power of the study. The case-crossover
analysis is a within-person approach; the referent time points are chosen
by the operator, and the design is the same for all the subjects. The
Cox model is a between-people approach; the referent time points cannot
be chosen because they depend on the number of live subjects who will
be included in the risk set, which varies at each time of death. Moreover,
with age used as the basic time scale, the dispersion of the referent
time points included in the risk set around the time of death varies
at each age of death. Otherwise, the number of referent time points
is almost always higher in the Cox model than in the case-crossover
design (i.e., two referent time points in the bidirectional design).
They chose β; thus, the exposure for an individual
who died at a given age is equal to the average exposure for at-risk
individuals at exactly that age.
In fact, we assess β as the exposure for a person
who died at a given age compared with the exposures for at-risk people
at exactly that age: β is the mean effect for an increase in air
pollution concentration on the mortality, whatever the age. Thus, in
both cases, when the exposure is either a chronic measurement or an
ecologic time-series data set, the Cox model captures all of the information
available, whereas the case-crossover design cannot be used with chronic
exposure measurements. Therefore, the Cox model should prove particularly
useful in the future to simultaneously analyze both the chronic (long-term)
and the short-term effects of air pollution concentrations.
Concerning the first possible bias noted by Lumley
et al., the adjustment of the results for the seasonality effect and
for time trends in air pollution concentration is more of an advantage
than a disadvantage. These pieces of information are very easy to take
into account with truncated power basis splines (Heuer 1997) without
data collection. Moreover, this process allows for the assessment of
the magnitude of these effects, which is not possible with the case-crossover
design. The second bias noted by Lumley et al. on the extreme age of
death is a very minor bias that was not present in our study. This bias
appears only if there is no risk set for the first or the last subject
who has the event.
Furthermore, numerous results from time-series studies
have shown an association between mortality and particulate air pollution,
and the results observed were similar (Filleul et al. 2001; Goldberg
et al. 2001; Samet et al. 2000). Despite that, causality was discussed
(Filleul et al. 2003) and statistical methods have sometimes been criticized.
For example, generalized additive models using nonparametric smoothing,
which could lead to biased estimates and to underestimation of the true
variance (Dominici et al. 2002, Ramsay et al. 2003). Thus, using the
Cox model could be an alternative approach if data are available.
Our study (Lepeule et al. 2006) is the first in
which a Cox model has been used to study the short-term effect of air
pollution. We found that the Cox method and case-crossover design gave
the same results as times series. This information supports the hypothesis
of a causal relationship between mortality and air pollution.
The authors declare they have no competing financial
interests.
Johanna Lepeule
Virginie Rondeau
INSERM, EMI 0338 (Biostatistique)
Université Victor Segalen
Bordeaux, France
Jean-François Dartigues
INSERM, U 593
Université Victor Segalen
Bordeaux, France
Laurent Filleul
Institut de Veille Sanitaire - CIRE Aquitaine
Bordeaux, France
References
Cox DR. 1972. Regression models and life tables.
JR Stat Soc B 34:187–220.
Dominici F, McDermott A, Zeger SL, Samet JM. 2002.
On the use of generalized additive models in time-series studies of
air pollution and health. Am J Epidemiol 156:193–203.
Filleul L, Medina S, Cassadou S. 2003. Urban particulate
air pollution: from epidemiology to health impact in public health [in
French]. Rev Epidemiol Santé Publique 51(5):527–542.
Filleul L, Zeghnoun A, Declercq C, Le Goaster C,
Le Tertre A, Eilstein D, et al. 2001. Short-term relationships between
urban atmospheric pollution and respiratory mortality: time series studies
[in French]. Rev Mal Respir 18:387–395.
Goldberg MS, Burnett RT, Bailar JC, Brook J, Bonvalot
Y, Tamblyn R, et al. 2001. The association between daily mortality and
ambient air particle pollution in Montreal, Quebec. 2. Cause specific
mortality. Environ Res 86(1): 26–36.
Heuer C. 1997. Modeling of time trends and interactions
in vital rates using restricted regression splines. Biometrics 53:161–177.
Lepeule J, Rondeau V, Filleul L, Dartigues JF. 2006.
Survival analysis to estimate association between short-term mortality
and air pollution. Environl Health Perspect 114:242–247.
Maclure M. 1991. The case-crossover design: a method
for studying transient effects on the risk of acute events. Am J Epidemiol
133:144–153.
Ramsay TO, Burnett RT, Krewski D. 2003. The effect
of concurvity in generalized additive models linking mortality to ambient
particulate matter. Epidemiology 14(1): 18–23.
Samet JM, Dominici F, Curriero FC, Coursac I, Zeger
SL. 2000. Fine particulate air pollution and mortality in 20 US cities.
N Engl J Med 343:1742–1749. |
