Clinical Infectious Diseases
INVITED ARTICLE
HEALTHCARE EPIDEMIOLOGY: Robert A. Weinstein, Section Editor
Handling Time-dependent Variables: Antibiotics and
Antibiotic Resistance
L. Silvia Munoz-Price,1,2 Jos F. Frencken,3,4 Sergey Tarima,1 and Marc Bonten3,5
1
Institute for Health and Society, and 2Department of Medicine, Medical College of Wisconsin, Milwaukee; 3Julius Center for Health Sciences and Primary Care, 4Department of Intensive Care
Medicine, and 5Department of Medical Microbiology, University Medical Center Utrecht, The Netherlands
Elucidating quantitative associations between antibiotic exposure and antibiotic resistance development is important. In the absence
of randomized trials, observational studies are the next best alternative to derive such estimates. Yet, as antibiotics are prescribed for
varying time periods, antibiotics constitute time-dependent exposures. Cox regression models are suited for determining such associations. After explaining the concepts of hazard, hazard ratio, and proportional hazards, the effects of treating antibiotic exposure
as fixed or time-dependent variables are illustrated and discussed. Wider acceptance of these techniques will improve quantification
of the effects of antibiotics on antibiotic resistance development and provide better evidence for guideline recommendations.
Keywords. time-dependent variables; Cox proportional hazards; hazard ratio; antibiotics.
In the field of hospital epidemiology, we are required to evaluate
the effect of exposures, such as antibiotics, on clinical outcomes
(eg, Clostridium difficile colitis or resistance development).
However, many of these exposures are not present throughout
the entire time of observation (eg, hospitalization) but instead
occur at intervals. These “fluctuating” variables are called
time-dependent variables, and their analyses should be performed by incorporating time-dependent exposure status in
the statistical models. This review provides a practical overview
of the methodological and statistical considerations required for
the analysis of time-dependent variables with particular emphasis on Cox regression models. Due to their relative ease of interpretation, we use antibiotic exposures as the core example
throughout the manuscript.
ANTIBIOTICS AND THEIR TIME DEPENDENCE
The global pandemic of antibiotic resistance represents a serious threat to the health of our population [1, 2]. The overuse
of antibiotics might be one of the most relevant factors associated with the rapid emergence of antibiotic resistance. Elucidating quantitative associations between antibiotic exposure and
antibiotic resistance development is, therefore, crucial for policy
making related to treatment recommendations and control
measures. As randomized controlled trials of antibiotic exposures are relatively scarce, observational studies represent the
next best alternative. Yet, as antibiotics are prescribed for
Received 30 December 2015; accepted 17 March 2016; published online 29 March 2016.
Correspondence: L. S. Munoz-Price, Medical College of Wisconsin, 8701 Watertown Plank
Rd, PO Box 26509, Milwaukee, WI 53226 ([email protected]).
Clinical Infectious Diseases® 2016;62(12):1558–63
© The Author 2016. Published by Oxford University Press for the Infectious Diseases Society
of America. All rights reserved. For permissions, e-mail [email protected].
DOI: 10.1093/cid/ciw191
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varying time periods, antibiotics constitute time-dependent
exposures.
To determine associations between antibiotic exposures and
the development of resistance or other clinical outcomes, most
peer-reviewed articles resort to the most simple approach: using
binary antibiotic variables (yes vs no) in their statistical analyses
[3–6]. Less frequently, antibiotics are entered in the model as
number of days or total grams of antibiotics received during
the observation period [7]. However, all of these 3 modalities
fail to account for the timing of exposures. Before expanding
on the principle of time-dependent variables, we need to review
other relevant concepts, such as hazard and hazard ratio (HR).
For illustration purposes, let us assume we are interested in
determining the development of antibiotic-resistant, gramnegative bacteria (AR-GNB), which happens to be recorded
on a daily basis.
Hazard
The hazard (chance) is a risk that the clinical outcome will happen in a very short time period conditional that an individual
was event-free before. Sometimes hazard is explained as instantaneous risk that an event will happen in the very next moment
given that an individual did not experience this event before.
When data are observed on a daily basis, it is reasonable to
link the hazard to the immediate 24-hour period (daily hazards). This hazard is then calculated daily, so that in day 2
the hazard is calculated among patients who did not develop
the outcome on day 1, and in day 3 the hazard is calculated
among patients who did not develop the outcome on day 2,
etc. This hazard calculation goes on consecutively throughout
each single day of the observation period. For our antibiotic example, the daily hazard of AR-GNB acquisition is the probability of acquiring AR-GNB within the next 24 hours among
patients who have not yet acquired AR-GNB.
Hazard Ratio
As implied by its name, a HR is just a ratio of 2 hazards obtained to compare the hazard of one group against the hazard
of another. If the hazard of acquiring AR-GNB in the group
without antibiotic exposures is equal to 1% and the HR is
equal to 2, then the hazard of AR-GNB under antibiotic exposure would be equal to 2% (= 1% × 2).
Constant Hazard Ratio or Proportional Hazards
The popular proportional hazards assumption states that a HR
is constant throughout the observation time. If time to AR-GNB
acquisition is compared between groups based on their antibiotic exposures, then hazard functions for the antibiotic and no
antibiotic groups have to change proportionally in regard to
each other over time. For example, if hazards of acquiring
AR-GNB are 1.0%, 2.1%, and 1.4% for the first 3 days of hospitalization in the group without antibiotics and the HR describing the effect of antibiotics is equal to 2, then the daily hazards
for the antibiotic-exposed group would be 2.0%, 4.2%, and
2.8%. This is how the model assumes the HR remains constant
in time, or, in other words, hazards are proportional. Proportionality of hazards is an attractive feature of Cox proportional
hazards models because it allows interpreting the effects of covariates in a time-independent manner. Therefore, under the
proportional hazards assumption, we can state that antibiotic
exposure doubles the hazards of AR-GNB and this statement
is applicable for any day of hospitalization. If the proportional
hazard assumption does not hold, then the exposure to antibiotics may have distinct effects on the hazard of acquiring ARGNB, depending of the day of hospitalization.
Time-Fixed Versus Time-Dependent Variables
Now let us review the concept of “time-fixed” variables, which,
as the name implies, are opposite to time-dependent variables.
The status of time-fixed variables is not allowed to change in the
model over the observation time. Some variables, such as diabetes, are appropriately modeled as time-fixed, given that a patient
with diabetes will remain with that diagnosis throughout the
observation time. However, as previously stated, antibiotic exposures are far from being constant. Other examples of variables frequently misused as time-fixed, although intermittent
in real life, are mechanical ventilation, intensive care unit
(ICU) stay, and even the use of devices; the analyses of these
variables in future studies should ideally be performed mirroring their time-dependent behaviors. Going back to the previous
example, the effect of antibiotics given only on day 3 should not
change the hazards of AR-GNB on days 1 and 2, but solely the
hazard on day 3.
If we ignore the time dependency of antibiotic exposures
when fitting the Cox proportional hazards models, we might
end up with incorrect estimates of both hazards and HRs. Luckily, the traditional Cox proportional hazards model is able to
incorporate time-dependent covariates (coding examples are
shown in the Supplementary Data).
STATISTICAL CONSIDERATIONS
Cox Model With Time-Fixed Versus Time-Dependent Exposure
We illustrate the analysis of a time-dependent variable using a cohort of 581 ICU patients colonized with antibiotic-sensitive gramnegative rods at the time of ICU admission [8]. In this cohort, the
independent variable of interest was exposure to antibiotics
(carbapenems, piperacillin-tazobactam, or ceftazidime), and the
outcome variable was time to acquisition of AR-GNB in the respiratory tract. In this study, a time-fixed variable for antibiotic exposures in the Cox regression model would have yielded an
incorrect hazard of AR-GNB acquisition (HR, 0.36; 95% confidence interval [CI], .19–.68). However, analyzing antibiotic
exposures as time-dependent variables resulted in a new hazard
markedly different than the former (HR, 0.99; 95% CI, .51–
1.93). The time-fixed model assumed that antibiotic exposures
were mutually exclusive (if subject received antibiotics then
subjects were analyzed as always on antibiotics), which is of
course not true. This is because a single patient may have
periods with and without antibiotic exposures. Ignoring timedependent exposures will lead to time-dependent bias (see Biases
section). Tables 1 and 2 illustrate the difference between timedependent and time-fixed analyses, by using Nelson-Aalen estimates of the daily hazards. These daily hazards were calculated
as the number of events (AR-GNB acquisition) divided by the
number of patients at risk at a particular day. In Table 1, antibiotic
exposures are treated as time-dependent variables; notice how the
number of patients at risk in the group exposed to antibiotics rises
and falls. This daily change in patients at risk occurs because the
number of patients exposed to antibiotics changes daily. Table 1
accurately represents these daily changes of patients at risk. In
Table 2, antibiotic exposures are treated as time-fixed variables:
all patients who ever receive antibiotics (111/581) are treated as
exposed for the entire study period, thereby greatly inflating the
risk set in the antibiotic-exposed group (while decreasing the risk
set in the unexposed group). To elaborate on the impact on the
hazard of these different analytic approaches, let us look at day
2. In the time-dependent analysis (Table 1), the hazard on day
2 is 2 / 24 = 0.083, whereas in the time-fixed analysis the hazard
is 2 / 111 = 0.018. Time-dependent bias has decreased the hazard
in the antibiotic-exposed group >4-fold. This underestimation of
the hazard in the antibiotic-exposed group is accompanied by an
overestimation of the hazard in the unexposed group. While the
calculations in our Cox model are naturally more complicated, the
essence remains the same: The time-fixed analysis incorrectly
labels patients as exposed to antibiotics.
Graphic Representation
Literature in the medical field frequently depicts Kaplan–Meier
curves, which are graphical representations of survival functions.
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Table 1.
Hazard Estimation Treating Antibiotic Exposure as a Time-Dependent Exposure
Not Exposed to Antibiotics
Time (Days Since
ICU Admission)
Exposed to Antibiotics
Patients at
Risk, No.
Events, No.
Hazard
Cumulative
Hazard
Patients at
Risk, No.
Events, No.
1
581
13
0.022
2
532
8
0.015
0.022
0
0
0
0
0.037
24
2
0.083
0.083
3
500
8
4
458
9
0.016
0.053
31
1
0.032
0.115
0.020
0.073
39
2
0.051
5
408
0.166
11
0.027
0.1
37
0
0
6
0.166
337
4
0.012
0.112
42
0
0
0.166
7
288
3
0.010
0.122
45
1
0.022
0.188
8
248
3
0.012
0.134
43
0
0
0.188
9
206
3
0.015
0.149
41
0
0
0.188
10
176
2
0.011
0.16
41
0
0
0.188
Hazard
Cumulative
Hazard
The table depicts daily and cumulative Nelson-Aalen hazard estimates for acquiring respiratory colonization with antibiotic-resistant gram-negative bacteria in the first 10 ICU days. The cohort of
581 ICU patients was divided into 2 groups, those with and those without exposure to antibiotics (carbapenems, piperacillin-tazobactam, or ceftazidime). Antibiotic exposure was treated as a
time-dependent variable and was allowed to change over time.
Abbreviation: ICU, intensive care unit.
Survival functions are calculated with the probabilities of remaining event-free throughout the observation. Kaplan–Meier
plots are a convenient way to illustrate 2 group comparisons that
do not require the proportionality of hazards assumption. However, a major limitation of the extended Cox regression model
with time-dependent variables is the absence of straightforward
relation between the hazard and survival functions [9]. Thus,
the standard way of graphically representing survival probabilities, the Kaplan–Meier curve, can no longer be applied. Several
attempts have been made to extrapolate the Kaplan–Meier
method to include time-dependent variables. Mathew et al
opted to categorize patients according to their final exposure
status, thereby acting as if the time-dependent exposure status
was known at baseline [10]. This method ignores the timedependency of the exposure and should not be used. Snapinn
Table 2.
et al proposed to extend the Kaplan–Meier estimator by updating the risk sets according to the time-dependent variable value
at each event time, similar to a method propagated by Simon
and Makuch [11, 12]. While this method may provide a realistic
graphical display of the effect of a time-dependent exposure, it
should be stressed that this graph cannot be interpreted as a
survival probability plot [13]. To avoid misinterpretation,
some researchers advocate the use of the Nelson-Aalen estimator, which can depict the effect of a time-dependent exposure
through a plot of the cumulative hazard [13, 14].
Nelson-Aalen cumulative hazards constitute a descriptive/
graphical analysis to complement the results observed in Cox
proportional hazards. In contrast to Cox models, NelsonAalen describes the behavior of cumulative hazards without imposing the proportionality assumption. Figures 1 and 2 show
Hazard Estimation Treating Antibiotic Exposure as a Time-Fixed Exposure
Not Exposed to Antibiotics
Time (Days Since
ICU Admission)
Exposed to Antibiotics
Patients at
Risk, No.
Events,
No.
1
470
13
0.028
0.028
111
0
0
0
2
445
8
0.018
0.046
111
2
0.018
0.018
3
422
8
0.019
0.065
109
1
0.009
0.027
4
392
9
0.023
0.088
105
2
0.019
0.046
5
346
11
0.032
0.120
99
0
0
0.046
6
282
4
0.014
0.134
97
0
0
0.046
7
240
3
0.013
0.147
93
1
0.011
0.057
8
207
3
0.014
0.161
84
0
0
0.057
9
173
3
0.017
0.178
74
0
0
0.057
10
146
2
0.014
0.192
71
0
0
0.057
Hazard
Cumulative
Hazard
Patients at
Risk, No.
Events,
No.
Hazard
Cumulative
Hazard
The table depicts daily and cumulative Nelson-Aalen hazard estimates for acquiring respiratory colonization with antibiotic-resistant gram-negative bacteria in the first 10 ICU days. The cohort of
581 ICU patients was divided into 2 groups, those with and those without exposure to antibiotics (carbapenems, piperacillin-tazobactam, or ceftazidime). Antibiotic exposure was treated as a
time-fixed variable and not allowed to change over time.
Abbreviation: ICU, intensive care unit.
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in left truncation of the data, also known as delayed entry [15,
16]. Let us assume that we restrict our study population to only
include patients who underwent admission to a particular unit
(eg, ICU). This restriction leads to left truncation as ICU admission can happen only after hospital admission [17, 18]. To correctly estimate the risk, patients with delayed entry should not
contribute to the risk set before study entry [19].
Partially Observed Measurements
Figure 1. Cumulative hazard of acquiring antibiotic-resistant gram-negative bacteria as calculated by the Nelson–Aalen method from a cohort of intensive care unit
patients colonized with antibiotic-sensitive gram-negative bacteria on admission
(n = 581). The exposure variable (no antibiotic exposure vs antibiotic exposure) is
treated as time-dependent.
the plots of the cumulative hazard calculated in Tables 1 and 2.
Note how antibiotic exposures analyzed as time-fixed variables
seem to have a protective effect on AR-GNB acquisition, similar
to the results of our time-fixed Cox regression analysis. This difference disappears when antibiotic exposures are treated as
time-dependent variables.
Left Truncation
When analyzing time to event data, it is important to define
time zero—that is, the time from which we start analyzing behaviors of hazards. In healthcare epidemiology, this time zero
will often be the time of hospital admission. If the time of
study entry is after time zero (eg, unit admission), this results
The extended Cox regression model requires a value for the
time-dependent variable at each time point (eg, each day of observation) [16]. Antibiotic exposure should be available and determined on a daily basis. These data are readily available in
hospitals that use electronic medical records, especially in the
inpatient setting. However, daily antibiotic exposures could be
challenging to obtain in other settings, such as in ambulatory
locations, which would bias the analysis. Besides daily antibiotic
exposures, other relevant exposures might have different frequency of measurements (eg, weekly). For instance, a recent article evaluated colonization status with carbapenem-resistant
Acinetobacter baumannii as a time-dependent exposure variable; this variable was determined using weekly rectal cultures
[6]. Given the lack of daily testing, the exact colonization status
might not be known at the time of the event, which in the last
example corresponded to the development of carbapenemresistant A. baumannii clinical infections. The colonization
status used for estimation in the model will depend on how
the researcher has organized the data; often the last available covariate value will be used. This can lead to attenuated regression
coefficients [20]. Other options are to use the value closest to the
event time (not necessarily the last recorded value) or to use linear interpolation of the covariate value. More sophisticated
methods are also available, such as joint modeling of the
time-dependent variable and the time-to-event outcomes [21].
Competing Risks
So far we have ignored the possibility of competing risks. For
instance, a patient exposed to antibiotics may either die or be
discharged before the acquisition of AR-GNB can be demonstrated. Ignoring such competing events will lead to biased results [22]. Discussion of the specifics is beyond the scope of this
review; please see suggested references [23, 24].
Biases
Figure 2. Cumulative hazard of acquiring antibiotic-resistant gram-negative bacteria as calculated by the Nelson–Aalen method from a cohort of intensive care unit
patients colonized with antibiotic-sensitive gram-negative bacteria on admission
(n = 581). The exposure variable (no antibiotic exposure vs antibiotic exposure) is
treated as time-fixed.
Biases occur due to systematic errors in the conduct of a study. In
cohort studies, there are 2 main biases associated with lack of timing consideration of exposure variables: length bias and immortal
time bias (also referred as time-dependent bias). Immortal time
bias occurs when exposure variables are considered independent
of their timing of occurrence, and consequently are assumed to
exist since study entry (time-fixed). This bias is prevented by coding these exposure variables in a way such that timing of occurrences is taken into consideration (time-dependent variables). A
2004 publication reviewed studies in leading journals that used
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survival analyses [25]. They found that out of all studies that
should have used time-dependent variables, only 40.9% did so.
Therefore, time-dependent bias has the potential of being rather
ubiquitous in the medical literature.
As clearly described by Wolkewitz et al [19], length bias occurs when there is no accounting for the difference between
time zero and the time of study entry. This bias is prevented
by the use of left truncation, in which only the time after
study entry contributes to the analysis. In the specific case of
antibiotics, we will need future studies to establish the appropriate timing of variable entry given the delayed effects of antibiotics on the gut microbiome.
Delayed Effects and Other Considerations
We should emphasize that in this manuscript we analyze the
hypothesized immediate effect of antibiotic exposures (today’s
antibiotic exposure impacts today’s hazard). However, this analysis does not account for delayed effects of antibiotic exposures
(today’s exposure affects hazards after today). The delayed effect
of antibiotics can be analyzed within proportional hazards
models, but additional assumptions on the over-time distribution of the effect would need to be made. Additionally, antibiotic exposures before time zero might have an impact on the
hazards during the observation period (eg, by altering the gut
microbiome). Given the lack of publications describing these
longitudinal changes, researchers would need to hypothesize
how antibiotic exposures might affect the chances of acquiring
AR-GNB in days to follow. This is an area of uncertainty that
deserves future work.
Limitations
The proportional hazards Cox model using time-dependent variables should be applied with caution as there are a few potential
model violations that may lead to biases. For example, the presence of time-varying HRs is one source of such bias [26]. Researchers should also be careful when using a Cox model in
the presence of time-dependent confounders. If these confounders are influenced by the exposure variables of interest, then controlling these confounders would amount to adjusting for an
intermediate pathway and potentially leading to selection bias
[27]. Other analysis techniques, such as marginal structural models using inverse probability weighting, can be utilized to estimate
the causal effect of a time-dependent exposure in the presence of
time-dependent confounders [28]. These techniques usually require some strong assumptions that may be difficult to ascertain.
Further discussion into causal effect modeling can be found in a
report by O’Hagan and colleagues [29].
EXAMPLES IN THE LITERATURE
There are only a couple of reports that looked at the impact of
time-dependent antibiotic exposures. In 2015, Jongerden and
colleagues published a retrospective cohort of patients cultured
at the time of ICU admission and twice a week thereafter [30].
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Their analysis aimed to determine the effect of time-dependent
antibiotic exposures on the acquisition of gram-negative rods. A
total of 250 patients acquired colonization with gram-negative
rods out of 481 admissions. Although the use of time-fixed
analysis (Kaplan–Meier survival curves) detected a difference
in days to acquisition of gram-negative rods between antibiotic-exposed and nonexposed patients (6 days vs 9 days, respectively; log-rank: .0019), these differences disappeared using
time-dependent exposure variables.
In 2015, Noteboom and colleagues published a retrospective
cohort performed across 16 Dutch ICUs aimed at determining
the impact of antibiotic exposures on the development of antibiotic resistance in preexisting gram-negative rod isolates [31].
Antibiotic exposures were treated as time-dependent variables
within Cox hazard models. After adjusting for subject-level variables and the receipt of selective decontamination, the only
variable found to be significantly associated to the development
of resistance was time-dependent carbapenem exposure (adjusted HR, 4.2; 95% CI, 1.1–15.6).
Stevens et al published in 2011 a retrospective cohort of
patients admitted from 1 January to 31 December 2005 [32].
Exposure variables consisted of cumulative defined daily antibiotic doses (DDDs). Therefore, as observation time progressed,
DDDs increased in an additive pattern based on daily exposures. Patients were followed for up to 60 days after discharge
for the development of the outcome variable: C. difficile–
positive stool toxins. Time-dependent exposures to quinolones,
vancomycin, β-lactamase inhibitor combinations, cephalosporins, and sulfonamides increased the risk of a positive C. difficile toxin. Time was modeled in the analysis given that the
antibiotic exposures changed cumulatively in a daily basis. However, this analysis assumes that the effect of antibiotic exposures
is equally significant on the day of administration than later during admission (eg, on day 20 after antibiotic administration).
This is a slightly different approach than the one used in the previous 2 examples, where time-dependent antibiotic exposure
changed in a binary fashion from zero (days before antibiotic
was administered) to 1 (days after antibiotic was administered).
CONCLUSIONS
Although antibiotic use clearly is a driving force for the emergence of antibiotic resistance, accurate quantification of associations between antibiotic exposure and antibiotic resistance
development is difficult. Randomized trials would be the optimal design, but in real life we usually have to work with data
(which are frequently incomplete) from observational studies.
Analysis is then complicated by the time-varying exposure to
antibiotics and the possibilities for bias. Application of Cox regression models with, at times, complex use of time-dependent
variables (eg, antibiotic exposure) will improve quantification of
the effects of antibiotics on antibiotic resistance development
and provide better evidence for guideline recommendations.
Supplementary Data
Supplementary materials are available at http://cid.oxfordjournals.org.
Consisting of data provided by the author to benefit the reader, the posted
materials are not copyedited and are the sole responsibility of the author, so
questions or comments should be addressed to the author.
Note
Potential conflicts of interest. L. S. M.-P. has received speaking fees
from ECOLAB and Xenex, and consultancy fees from Xenex and Clorox.
All other authors report no potential conflicts. All authors have submitted
the ICMJE Form for Disclosure of Potential Conflicts of Interest. Conflicts
that the editors consider relevant to the content of the manuscript have been
disclosed.
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