American Journal of Epidemiology
ª The Author 2011. Published by Oxford University Press on behalf of the Johns Hopkins Bloomberg School of
Public Health. All rights reserved. For permissions, please e-mail: [email protected].
Vol. 174, No. 1
DOI: 10.1093/aje/kwr038
Advance Access publication:
May 3, 2011
Practice of Epidemiology
Potential for Bias in Case-Crossover Studies With Shared Exposures Analyzed
Using SAS
Shirley V. Wang, Brent A. Coull, Joel Schwartz, Murray A. Mittleman, and Gregory A. Wellenius*
* Correspondence to Dr. Gregory A. Wellenius, Center for Environmental Health and Technology, Brown University, 121 South Main
Street, Box G-S121-2, Providence, RI 02912 (e-mail: [email protected]).
Initially submitted September 23, 2010; accepted for publication January 28, 2011.
The case-crossover method is an efficient study design for evaluating associations between transient exposures
and the onset of acute events. In one common implementation of this design, odds ratios are estimated using
conditional logistic or stratified Cox proportional hazards models, with data stratified on each individual event. In
environmental epidemiology, where aggregate time-series data are often used, combining strata with identical
exposure histories may be computationally convenient. However, when the SAS software package (SAS Institute
Inc., Cary, North Carolina) is used for analysis, users can obtain biased results if care is not taken to properly
account for multiple cases observed at the same time. The authors show that fitting a stratified Cox model with the
‘‘Breslow’’ option for handling tied failure times (i.e., ties ¼ Breslow) provides unbiased health-effects estimates in
case-crossover studies with shared exposures. The authors’ simulations showed that using conditional logistic
regression—or equivalently a stratified Cox model with the ‘‘ties ¼ discrete’’ option—in this setting leads to healtheffect estimates which can be biased away from the null hypothesis of no association by 22%–39%, even for small
simulated relative risks. All methods tested by the authors yielded unbiased results under a simulated scenario with
a relative risk of 1.0. This potential bias does not arise in R (R Foundation for Statistical Computing, Vienna,
Austria) or Stata (Stata Corporation, College Station, Texas).
air pollution; bias (epidemiology); environmental exposure; environmental health; epidemiologic methods
Abbreviation: PM2.5, particulate matter with an aerodynamic diameter 2.5 lm.
The case-crossover design is used to evaluate associations
between transient exposures and the onset of acute events
(1). In one common implementation of this design, each
subject’s exposure before a case-defining event (case period) is compared with his or her own exposure experience
during 1 or more control periods in which the subject did not
become a case. The use of matched, within-subject comparisons provides effective control of confounding by measured
or unmeasured subject characteristics that are stable over
time, although confounding by time-varying characteristics
or exposures is still possible (1, 2). This implementation of
the case-crossover design is analogous to a matched casecontrol study, and the odds ratio estimated with conditional
logistic regression—or equivalently the stratified Cox proportional hazards model—is typically used as the measure
of association.
Case-crossover methods have been applied to studies in
a number of different substantive areas, including pharmacoepidemiology (3–6), cardiovascular disease (7–10), injury
(11–14), infectious disease (15–17), and environmental epidemiology (18–20). The application of case-crossover
methods is a common approach in studies evaluating the
association between short-term changes in levels of ambient
air pollutants and the risk of acute morbid or fatal events (8,
21–24). In this setting, it is common to assign the same
exposure history to all persons experiencing the event of
interest on a given day. In theory, case-crossover analyses
should treat each event as a separate stratum in the analysis.
In settings where exposure is shared, it is often convenient to
condition the analysis on calendar day rather than the individual. Because conditioning on day allows data to be
aggregated into fewer strata, analytic data sets are smaller
118
Am J Epidemiol. 2011;174(1):118–124
Handling Ties in Case-Crossover Analyses
and computation time can be reduced substantially. However, we show that when the analyses are performed using
the SAS statistical software package, version 9.2 (SAS Institute Inc., Cary, North Carolina), failure to properly account for this aggregation can lead to health-effects
estimates that are biased away from the null hypothesis of
no association. Our goal in this paper is to characterize the
magnitude, direction, and conditions under which this potential bias occurs.
MATERIALS AND METHODS
Suppose that one wants to evaluate the relation between
levels of ambient fine particles (particulate matter with an
aerodynamic diameter 2.5 lm (PM2.5)) and the risk of
hospitalization for a specific cause. The left side of Table 1
shows how one might structure the data set for a timestratified case-crossover study. In this design, PM2.5
levels at the time of each event are compared with
PM2.5 levels during referent periods sampled from other
times in the same calendar month (25, 26). In this example, we assume that data are available on the date (but not
the time) of each event, and we match case periods and
referent periods on day of the week, but neither is necessary to the method or to our demonstration. Each event
has a unique identifier. As is typical in this type of study,
the same value of PM2.5 is assigned to all events that
occur at the same time.
We use conditional logistic regression, stratifying on each
event, to estimate the association between PM2.5 and the risk
of hospitalization. In SAS, we can implement this analysis
using the PHREG procedure as follows.
PROC PHREG data ¼ expanded;
MODEL time 3 case(0) ¼ pm25 covariate1. . .
covariatep/ties ¼ Breslow;
STRATA eventid;
run;
The MODEL statement describes the regression model to be
fitted and the STRATA statement denotes that the analysis is
conditioned on each individual event. Note that as part of the
MODEL statement, SAS enables the user to specify how tied
failure times will be handled in the analysis (e.g., ties ¼
Breslow). However, since we are conditioning on each event
(i.e., there is exactly 1 case in each stratum), there are no ties and
all of the available ties-handling options yield exactly the same
answer. Equivalently, one could use SAS’s LOGISTIC procedure to fit the conditional logistic regression model as follows:
PROC LOGISTIC data ¼ expanded;
MODEL case ¼ pm25 covariate1. . .covariatep;
STRATA eventid;
run;
Since all events that occur at the same time are assigned
the same exposure value, an attractive alternative approach
is to create a data set with 1 stratum per calendar day and to
weight each observation by the number of cases observed on
that day (see right side of Table 1). We can implement this
weighted analysis in SAS as follows:
Am J Epidemiol. 2011;174(1):118–124
119
PROC PHREG data ¼ weighted;
MODEL time*case(0) ¼ pm25 covariate1. . . covariatep/
ties ¼ Breslow;
STRATA date;
FREQ count;
run;
In this case, the PHREG procedure treats each observation
as if it appears n times, where n is the value of the FREQ
variable for the observation. Since in this example we are
conditioning on strata defined by calendar day, each stratum
contains nd events, where nd may be more than 1. This
approach is attractive because the analysis is computationally much faster, since there are many fewer strata. For large
data sets, the difference in computation times can be
substantial.
Now the approach used to handle tied failure times matters, since each stratum can contain multiple events. The
default ties-handling option uses the approach described
by Breslow (27), which maximizes a partial likelihood function that is identical to the partial likelihood function for the
case-crossover design, as previously described (25, 26, 28)
and as shown in the Appendix. However, the SAS manuals
advise that when the PHREG procedure is being used to
implement conditional logistic regression models (such as
when analyzing data from a matched case-control study),
the ‘‘discrete’’ ties-handling option should be used (29). As
is shown in the Appendix, this construction differs from that
which assumes independence across all matched sets in the
case-crossover design. Therefore, the partial likelihood
function generated from the ‘‘discrete’’ ties option differs
from the correct partial likelihood function for case-crossover
analyses, suggesting that use of the ‘‘discrete’’ option with
the PHREG procedure may lead to biased effect estimates.
However, the direction and magnitude of this potential bias
are not clear.
Alternatively, we could use SAS’s LOGISTIC procedure
to fit the conditional logistic model incorporating the
weights as follows:
PROC LOGISTIC data ¼ weighted;
MODEL case ¼ pm25 covariate1. . .covariatep;
STRATA date;
FREQ count;
run;
This yields effect estimates identical to those of the PHREG
procedure with the ‘‘ties ¼ discrete’’ option but is computationally less efficient. The partial likelihood maximized by
the LOGISTIC procedure with the frequency weights again
differs from the correct partial likelihood function for casecrossover analyses.
Simulation studies
We performed simulations to quantify the direction and
magnitude of the bias that results when the ties-handling
procedure is incorrectly specified for a typical time-series
study of air pollution health effects. First, we simulated the
number of cause-specific hospital admissions on day i (Yi)
over a 10-year period as a Poisson random variable:
120 Wang et al.
Table 1. Sample Data Set Structure for a Case-Crossover Analysis Conditioning on Each Individual (Left) or Conditioning on Each Calendar Day
(Right) (n ¼ 4 cases)a
Conditioning on Event ID
Row
Event IDb
Datec
Conditioning on Day
Daily PM2.5
Cased
Timee
Countf
Datec
Daily PM2.5
Cased
Timee
1
1
January 1, 2000
26.3
1
1
2
January 1, 2000
26.3
1
1
2
1
January 1, 2000
16.0
0
2
2
January 1, 2000
16.0
0
2
3
1
January 1, 2000
19.3
0
2
2
January 1, 2000
19.3
0
2
4
1
January 1, 2000
26.6
0
2
2
January 1, 2000
26.6
0
2
5
1
January 1, 2000
28.5
0
2
2
January 1, 2000
28.5
0
2
6
2
January 1, 2000
26.3
1
1
3
January 2, 2000
27.4
1
1
7
2
January 1, 2000
16.0
0
2
3
January 2, 2000
19.7
0
2
8
2
January 1, 2000
19.3
0
2
3
January 2, 2000
28.3
0
2
9
2
January 1, 2000
26.6
0
2
3
January 2, 2000
24.0
0
2
10
2
January 1, 2000
28.5
0
2
3
January 2, 2000
22.5
0
2
11
3
January 2, 2000
27.4
1
1
12
3
January 2, 2000
19.7
0
2
13
3
January 2, 2000
28.3
0
2
14
3
January 2, 2000
24.0
0
2
15
3
January 2, 2000
22.5
0
2
16
4
January 2, 2000
27.4
1
1
17
4
January 2, 2000
19.7
0
2
18
4
January 2, 2000
28.3
0
2
19
4
January 2, 2000
24.0
0
2
20
4
January 2, 2000
22.5
0
2
21
5
January 2, 2000
27.4
1
1
22
5
January 2, 2000
19.7
0
2
23
5
January 2, 2000
28.3
0
2
24
5
January 2, 2000
24.0
0
2
25
5
January 2, 2000
22.5
0
2
Abbreviations: ID, identifier; PM2.5, particulate matter with an aerodynamic diameter 2.5 lm.
All cases that occur on the same calendar day are assumed to involve exposure to the same levels of PM2.5. Controls are chosen according to
the time-stratified approach, such that all days in the same month and on the same day of the week as the case day are used as control days.
b
Unique individual identifier.
c
Calendar day during follow-up.
d
1 if case day, 0 if control days.
e
1 if case day, 2 if control days.
f
Number of cases that occurred on that day.
a
Yi ~ Poissonðlnðki Þ ¼ b0 þ b1 3 PMi Þ;
where PMi represents PM2.5 levels on day i and b1 represents
the hypothesized log rate ratio associated with a 1-lg/m3 increase in PM2.5 on the same day. For PMi, we simulated a time
series of daily measures of PM2.5 where the natural log of the
exposure was normally distributed with a mean of 2.58 and
a standard deviation of 0.54. These values were those observed
in Chicago, Illinois, between 2000 and 2006 and were used in
the applied example below. b0 was chosen such that the mean
number of events at the average PM2.5 level was 5. Second, for
each simulated data set, we evaluated the association between
same-day PM2.5 and risk of hospitalization using the timestratified case-crossover design (25, 26), where referents were
all days in the same year, month, and day of the week as the
simulated case day, excluding the case day.
We simulated data sets for true rate ratios ranging from
0.80 to 2.00 per 10-lg/m3 increase in PM2.5, including the
null hypothesis of no association (rate ratio ¼ 1.0). We used
SAS’s PHREG procedure with either the ‘‘Breslow’’ or the
‘‘discrete’’ ties-handling option to estimate odds ratios and
95% confidence intervals. We stratified on calendar day in all
analyses. For each effect size, we simulated results for 250 data
sets, and we report the average b̂1 and the relative bias, defined
as ðb1 b̂1 Þ=b1 3 100: The simulated data sets were created
in R, version 2.7.2 (30), and exported into SAS for analysis.
Applied example
As an applied example, we evaluated the association between daily PM2.5 and the risk of hospital admission for
congestive heart failure among Medicare beneficiaries aged
Am J Epidemiol. 2011;174(1):118–124
Handling Ties in Case-Crossover Analyses
121
Figure 1. Results of analyses performed in SAS for simulated relative risks of hospitalization for heart failure ranging from 0.8 to 2.0 per 10-lg/m3
increase in particulate matter with an aerodynamic diameter 2.5 lm. Box plots show the distribution of effect estimates from 250 simulated data
sets. Each box denotes the median value and interquartile range (25th–75th percentiles), and whiskers denote the upper and lower adjacent
values. The vertical dashed lines denote the simulated relative risk.
65 years residing in the Chicago metropolitan area between January 1, 2000, and December 31, 2006. We obtained hospital admission records from the Centers for
Medicare and Medicaid Services and defined cases as
patients admitted from the emergency department with a primary discharge diagnosis of heart failure (International
Classification of Diseases, Ninth Revision, code 428). We
obtained daily measures of PM2.5 from the Environmental
Protection Agency and computed daily mean concentrations, as previously described (19). We obtained meteorologic data from the National Weather Service (National
Climatic Data Center, Asheville, North Carolina). Finally,
we evaluated the association between PM2.5 and same-day
risk of hospital admission using the time-stratified casecrossover design as above. We used SAS’s PHREG
procedure with either the ‘‘Breslow’’ or the ‘‘discrete’’
ties-handling option to estimate odds ratios and 95% confidence intervals. In all models, we controlled for confounding by temperature and dew point. This analysis was
approved by the institutional review boards of the Harvard
School of Public Health (Boston, Massachusetts) and Beth
Israel Deaconess Medical Center (Boston, Massachusetts).
RESULTS
Simulation studies
Figure 1 shows the results of using SAS to analyze simulated data sets. Use of the ‘‘Breslow’’ ties-handling option
when stratifying on day resulted in unbiased estimates of the
simulated associations between PM2.5 and same-day hospital
admissions. As expected, when the ‘‘Breslow’’ ties option
was used, results were identical regardless of whether we
conditioned on each individual event or conditioned on
Am J Epidemiol. 2011;174(1):118–124
calendar day weighted by the number of cases per day. However, using the ‘‘discrete’’ ties option, conditioning on calendar day, and weighting by the number of cases per day
resulted in estimates that were potentially biased away from
the null hypothesis of no association. Specifically, when the
simulated relative risk differed from 1.0, effect estimates
were biased by 22%–39% away from the null hypothesis.
Using SAS’s LOGISTIC procedure conditioned on calendar
day and weighted by the number of cases per day yielded
identical (biased) results. Under the simulated null hypothesis
of no association, all methods yielded unbiased estimates.
Applied example
We evaluated the association between ambient PM2.5 and
the risk of hospitalization for congestive heart failure among
Medicare beneficiaries residing in Chicago using the timestratified case-crossover design (Table 2). Using SAS’s
PHREG procedure with the ‘‘Breslow’’ ties option and conditioning on each event, we observed a 1.2% (95% confidence interval: 0.1, 2.4) increase in risk of hospitalization
for heart failure per 10-lg/m3 increase in PM2.5. Conditioning on calendar day but still using the ‘‘Breslow’’ ties option
yielded identical results. However, conditioning on calendar
day and using the ‘‘discrete’’ ties option, we found a 1.6%
(95% confidence interval: 0.3, 2.9) increase in risk of hospitalization per 10-lg/m3 increase in PM2.5, representing
a 30% relative bias away from the null on the log odds scale.
Analysis of the weighted data set using the LOGISTIC
procedure yielded (biased) estimates identical to those observed with the PHREG procedure using the ‘‘discrete’’ ties
option, but took almost 3 times longer (25.5 seconds vs. 9.3
seconds). The PHREG procedure using the ‘‘Breslow’’ tieshandling option took 0.8 seconds to execute.
122 Wang et al.
Table 2. Association Between Ambient PM2.5 and Risk of Hospitalization for Congestive Heart Failure, Estimated With Different SASa
Procedures, Ties-Handling Options, and Frequency Weights, Among Medicare Beneficiaries Residing in Chicago, Illinois, 2000–2006
SAS Procedure
Ties Option
c
Conditioning On:
Weighted By:
Estimate, %b
95% Confidence Interval
Computational Time, seconds
1.2
0.1, 2.4
18.3
PHREG
Breslow
Each event
PHREG
Breslow
Each day
Events/day
1.2
0.1, 2.4
0.8
PHREG
Discrete
Each day
Events/day
1.6
0.3, 2.9
9.3
1.2
0.1, 2.4
6.2
1.6
0.3, 2.9
25.5
LOGISTIC
Each event
LOGISTIC
Each day
Events/day
Abbreviation: PM2.5, particulate matter with an aerodynamic diameter 2.5 lm.
SAS Institute Inc., Cary, North Carolina.
b
Percentage increase in risk of hospitalization for heart failure per 10-lg/m3 increase in PM2.5.
c
Any of the available ties-handling options would yield identical results in this case.
a
DISCUSSION
The case-crossover study design has been applied in
a large number of epidemiologic studies, and its use is common in studies of the health effects of environmental exposures. It has been shown that when an appropriate control
selection strategy is used, conditional logistic regression
stratified on the individual subject yields unbiased estimates
of the relative risk (25, 26). When analyses are stratified on
the individual, there are no tied failure times, and all of the
commonly available partial likelihoods (i.e., ties options)
reduce to the same valid partial likelihood function. However, when multiple subjects are assigned a common exposure history, it is computationally advantageous to group
subjects with identical exposure histories and condition on
each group rather than each individual event. Because of the
analogy between case-crossover and matched case-control
studies—and by extension, Cox proportional hazards analysis of time-to-event data—we suspect that some investigators may use the discrete partial likelihood, as suggested by
the software manuals, to mimic the likelihood function used
in conditional logistic regression (29, 31). Other investigators may be using SAS’s LOGISTIC procedure, conditioning on each calendar day and weighting the analysis by the
number of cases observed per day. Our results show that use
of these approaches in SAS will result in estimates that are
biased away from the null hypothesis of no association.
Results from our simulations demonstrate that the relative
magnitude of this bias can be substantial even for small
effect sizes.
Notably, these results do not imply that there is a problem
with SAS’s implementation of the PHREG or LOGISTIC
procedures, the handling of tied failure times, or the handling of frequency weights. Rather, as the Appendix shows,
because of how these SAS procedures incorporate frequency weights into the analysis, the partial likelihood function fitted by SAS is not the correct one for case-crossover
analyses with shared exposures. Stata (Stata Corporation,
College Station, Texas) and R (R Foundation for Statistical
Computing, Vienna, Austria) are not prone to this bias because of the way in which each of these software packages
incorporates frequency weights into the analysis. Recall that
SAS’s FREQ statement treats each observation as if it appeared in the data set n times. This is equivalent to manually
replicating each row in the data set n times. In contrast,
Stata’s clogit function incorporates frequency weights directly into the partial likelihood function and therefore
yields unbiased estimates. In R, the clogit function of the
‘‘survival’’ package does not allow weights to be used. Neither the coxph function in R nor the stcox function in Stata
allows the use of weights when the exact partial likelihood
function is specified for handling tied failure times (the
‘‘exact’’ option in R and the ‘‘exactp’’ option in Stata).
Therefore, this potential bias arises in SAS only because
of the particular manner in which SAS implements the
FREQ statement.
Thus, we recommend that SAS users wishing to collapse
strata with identical exposure histories use the PHREG procedure with the ‘‘Breslow’’ ties-handling option. A weighted
analysis using SAS’s LOGISTIC procedure or using the
PHREG procedure with the ‘‘discrete’’ ties option will lead
to biased results. Of course, it must be emphasized that
using an expanded data set (as in Table 1) and conditioning
on each individual event yields unbiased estimates with any
of these procedures. Even so, for large data sets the PHREG
procedure may be preferable to the LOGISTIC procedure
because of computational efficiency. For very large data
sets, such as those typically used in recent studies (32–
34), the computational time needed to use the LOGISTIC
procedure may be problematic. Investigators in recent studies have performed similar analyses in more than 200 cities,
such that even with today’s high-performance computers,
the difference in computational time may be important.
Since few published articles describe the statistical implementation of the conditional logistic regression model, it
is difficult to know how prevalent this error is in the published literature. However, examples can be found in both
published research articles and publicly available presentations, suggesting that at least some researchers and educators are confused as to the appropriate approach.
The potential problem identified here is not unique to
studies of air pollution health effects. Shared exposures
are common in other areas of environmental epidemiology,
including studies of the health effects of weather and climate (15, 19) and water quality (18), and in areas outside of
environmental health, such as the health effects of alcohol
sales (35), hospital nurse staffing levels (36), and air travel
(37). As the popularity of the case-crossover method grows,
it is important that investigators be aware of potential
Am J Epidemiol. 2011;174(1):118–124
Handling Ties in Case-Crossover Analyses
analytic pitfalls. Moreover, this potential problem may also
arise in case-specular or case-control studies with shared or
ecologic exposure information among cases.
11.
12.
ACKNOWLEDGMENTS
Author affiliations: Center for Environmental Health and
Technology, Brown University, Providence, Rhode Island
(Shirley V. Wang, Gregory A. Wellenius); Department of
Biostatistics, Harvard School of Public Health, Boston,
Massachusetts (Brent A. Coull); Department of Epidemiology, Harvard School of Public Health, Boston, Massachusetts (Joel Schwartz, Murray A. Mittleman); Department of
Environmental Health, Harvard School of Public Health,
Boston, Massachusetts (Joel Schwartz); and Cardiovascular
Epidemiology Research Unit, Beth Israel Deaconess Medical Center, Boston, Massachusetts (Murray A. Mittleman).
This work was supported by grants ES015774,
ES009825, and ES017125 from the National Institute of
Environmental Health Sciences.
The contents of this article are solely the responsibility of
the authors and do not necessarily represent the official
views of the National Institute of Environmental Health
Sciences or the National Institutes of Health.
Conflict of interest: none declared.
13.
14.
15.
16.
17.
18.
19.
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the baseline risk is constant within the reference window, the
likelihood function is given by
LðbÞ ¼
The case-crossover study design compares the exposure
of cases in interval ti with the exposures from a set of reference periods. Using the notation of Lu and Zeger (28), we
denote the event interval by ti, the set of reference periods by
W(ti), and the exposure for subject i during the event interval
as Xiti . Assuming that the n subjects are independent and that
expðb#Xiti Þ
:
i¼1 Rj2Wti exp b#Xij
ðA1Þ
Taking account of the shared exposure for all events on
a given day, Xiti can be notationally represented by Xd,
which represents the exposure associated with day d of
the study. Using this notation, equation A1 is equivalent to
LðbÞ ¼
D
Y
expðb#nd Xd Þ
h
i nd ;
d¼1 Rj2WðdÞ expðb#ðXÞ Þ
dj
ðA2Þ
where nd denotes the number of cases observed on day d.
Equation A2 is identical to the partial likelihood function
used by the Breslow approach for handling tied failure times
in the setting of the proportional hazards model (29).
In contrast, the partial likelihood function specified by the
‘‘discrete’’ ties option in SAS (29) is given by
LðbÞ ¼
APPENDIX
n
Y
D
Y
expðb#nd Xd Þ
;
exp b#sq
R
d¼1 q2Qd
ðA3Þ
where Qd is the set of all possible collections of nd cases
from an expanded risk set constructed from nd replications
of the correct risk set W(d), and sq is the sum of the exposures associated with the nd cases in collection q. Generally,
the denominators of equations A2 and A3 will not be equal,
suggesting that use of the likelihood function shown in
equation A3 will result in biased estimates of b.
Am J Epidemiol. 2011;174(1):118–124