American Journal of Epidemiology
ª The Author 2010. 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. 172, No. 9
DOI: 10.1093/aje/kwq246
Advance Access publication:
September 3, 2010
Practice of Epidemiology
Incidence Densities in a Competing Events Analysis
Nadine Grambauer*, Martin Schumacher, Markus Dettenkofer, and Jan Beyersmann
* Correspondence to Nadine Grambauer, Department of Medical Biometry and Statistics, University Medical Center Freiburg,
Stefan-Meier-Str. 26, 79104 Freiburg, Germany (e-mail: [email protected]).
Initially submitted January 19, 2010; accepted for publication June 30, 2010.
Epidemiologists often study the incidence density (ID; also known as incidence rate), which is the number of
observed events divided by population-time at risk. Its computational simplicity makes it attractive in applications,
but a common concern is that the ID is misleading if the underlying hazard is not constant in time. Another difficulty
arises if competing events are present, which seems to have attracted less attention in the literature. However,
there are situations in which the presence of competing events obscures the analysis more than nonconstant
hazards do. The authors illustrate such a situation using data on infectious complications in patients receiving stem
cell transplants, showing that a certain transplant type reduces the infection ID but eventually increases the
cumulative infection probability because of its effect on the competing event. The authors investigate the extent
to which IDs allow for a reasonable analysis of competing events. They suggest a simple multistate-type graphic
based on IDs, which immediately displays the competing event situation. The authors also suggest a more formal
summary analysis in terms of a best approximating effect on the cumulative event probability, considering another
data example of US women infected with human immunodeficiency virus. Competing events and even more
complex event patterns may be adequately addressed with the suggested methodology.
competing risks; event-specific hazard; incidence rate; model misspecification; proportional hazards; survival
analysis
Abbreviations: AIDS, acquired immunodeficiency syndrome; CIF, cumulative incidence function; HAART, highly active antiretroviral therapy; HIV, human immunodeficiency virus; ID, incidence density.
Incidence density (ID; also known as incidence rate), the
number of observed events divided by the population-time
at risk, is often calculated in applications. Examples include
studies of cardiovascular diseases (1, 2), human immunodeficiency virus (HIV) infection (3), and hospital-acquired
infection (4–6). IDs are attractive because they are computationally simple and allow for right-censoring as well as
left-truncation (7). Right-censoring is present if, for some
individuals, only the minimum time until the event is
known. For example, a patient surviving after a study has
been closed has a censored survival time. Left-truncation
occurs when individuals are not followed from time zero
but from some time afterward. For example, event times
are often left-truncated if the time scale of interest is age.
Under a constant hazards assumption, IDs are the standard
maximum likelihood estimators. However, investigators
such as Kraemer (8) recently emphasized that the validity
of ID analyses might be compromised if the underlying
hazards are not constant.
Another difficulty occurs if competing risks (or competing
events—both terms are used synonymously in the following)
are present (7, 9). In this case, when performing an ID analysis, the competing risks should be accounted for separately,
meaning that event-specific IDs should be calculated, which
is rarely found in the literature (refer, for example, to Glynn
and Rosner (2) and Dignam and Kocherginsky (10)). Lau
et al. (11) recently pointed out the relevance of competing
risks in epidemiologic studies. The authors introduced and
compared the standard semiparametric regression models for
competing risks. They illustrated the difficulties with pertinent risk set plots that display different estimation techniques.
Similar risk set plots have also been studied by Latouche (12).
1077
Am J Epidemiol 2010;172:1077–1084
1078 Grambauer et al.
Lau et al. concluded that some modeling approaches are
better suited to investigating the etiology of a disease,
whereas others provide a better synthesis in terms of an individual’s outcome.
In this paper, we profit from the computational simplicity
of IDs in that we are able to give a simple graphic illustration of the estimation results. Our graphic both displays
results in terms of the etiology of a disease and lends itself
to a synthesis interpretation. The graphic is useful in understanding how the expected proportion of the respective
competing events over the course of time evolves—even if
the constant hazards assumption fails.
As Lau et al. (11) point out, the course of this so-called
cumulative incidence function (CIF) is often difficult to predict from standard analyses. For instance, quite differential
effects on the CIF for the event of interest can be observed in
the ONKO-KISS study (13, 14), our first application example. ONKO-KISS considers patients with hematologic malignancies who have undergone peripheral blood stem cell
transplantation. After transplantation, patients enter a critical
condition called ‘‘neutropenia,’’ characterized by a low
number of white blood cells, which primarily avert infections. Bloodstream infection during neutropenia is a severe
complication. An event-specific ID analysis considers the
time until ‘‘bloodstream infection during neutropenia’’ or
‘‘end of neutropenia (without prior bloodstream infection),’’
whichever comes first, and the type of that first event.
Results are in line with the semiparametric analyses of
Beyersmann et al. (15), who found similar effects of female
gender and allogeneic transplants on the ID for bloodstream
infection but observed quite differential effects on the CIF
for the event of interest because of different effects on the
competing IDs.
To further cope with potentially difficult interpretation,
we develop in this paper a more formal summary analysis
of the different event-specific effects of one risk factor
based on IDs, which is similar to that given by Lau et al.
(11). To illustrate this analysis, we perform a second
competing risks analysis, considering data from the
Women’s Interagency HIV Study on HIV-infected US
women (refer, for example, to Lau et al. and Barkan
et al. (16)). Here, we find the parametric approach based
on IDs yielding competitive results compared with the
more complex parametric mixture model approach performed by Lau et al., which results in close to nonparametric ID-based CIF estimates.
This paper is organized as follows: The next section introduces competing risks and expands on the parametric
approach using event-specific IDs that allow one to perform
further inference easily. Consequently, the methodology is
illustrated in detail by means of the ONKO-KISS study;
results from ID analyses are connected to interpretation of
the estimated CIFs. A second data example from the
Women’s Interagency HIV Study is considered next to illustrate the potential of obtaining a summary analysis. Finally, we close with a discussion.
To make the proposed methodology easily applicable, we
provide corresponding functions for the software environment for statistical computing R (17) and a worked example
in the supplementary Web material (this information is
posted on the Journal’s Web site (http://aje.oxfordjournals.
org/)).
COMPETING RISKS AND STATISTICAL INFERENCE
Event-specific IDs and CIF
Competing risks model time to first event and event type
(18), whereas standard survival analysis considers time to
a single, usually combined, endpoint only. Inference for
competing risks can be based completely on the eventspecific hazard rates ah(t), where ah(t)dt ¼ P(T 2 dt,
event ¼ h j T t), T is the time until any first event, and
dt refers to the probability of experiencing event h in the
small time interval [t, t þ dt). These rates are similar to the
standard hazard rates but are marked with the respective
event-type h. For ease of presentation, we assume 2 competing risks, h 2 {1, 2}, but the findings can be transferred to
any number of competing risks. This approach does not
make any independent competing risks assumption, which
has been subject to pointed critique (19–21). Note, however,
that one would need to make an assumption about whether
and how one event-specific hazard changes if the aim were
inference for outcome probabilities when the other eventspecific hazard is modified or even supposed to vanish (20).
In assuming the event-specific hazards to be constant, it is
straightforward to estimate them by calculating the corresponding event-specific IDs:
number of type-h-events
:
âh ¼ P
patient-time at risk
ð1Þ
However, although parametric survival analysis provides
a powerful tool (refer, for example, to Cox et al. (22)), the
constant hazards assumption may seldom apply well
to medical research data (8). To determine whether the constant hazards assumption may still be appropriate for the
particular data, one may consider
the cumulative eventRt
specific hazards Ah ðtÞ ¼ 0 ah ðuÞdu by comparing its
parametric plug-in estimates with the corresponding nonparametric Nelson-Aalen estimates (18). The latter are defined as
Âh ðtÞ ¼
X
st
number of type-h-events at s
;
number of patients at risk before s
ð2Þ
where the sum is calculated over all observed event times s
of any type, s t.
Another important quantity in a competing risks setting is
the CIF for the particular event type h:
Fh ðtÞ ¼ RPðT t; event ¼ hÞ
ð3Þ
t
¼ 0 PðT > u Þ3ah ðuÞdu;
Ru
where PðT > uÞ ¼ expð 0 ða1 ðvÞ þ a2 ðvÞÞdvÞ is the survival probability, indicating that the CIF depends on all
event-specific hazards. For example, if interest focuses on
a certain event of interest 1, this is the expected proportion
of type 1 events over the course of time.
If constant event-specific hazards are assumed, ah(t) ¼ ah,
the CIF of event type h is explicitly given as
Am J Epidemiol 2010;172:1077–1084
Incidence Densities in a Competing Events Analysis
Fh ðtÞ ¼
Rt
0 expð
ða1 þ a2 ÞuÞ3ah du
ah R t
expð ða1 þ a2 ÞuÞdu
a1 þ a2 0
ah
ð1 expð ða1 þ a2 ÞtÞÞ
¼
a1 þ a2
¼
ð4Þ
By plugging in both event-specific IDs, the CIF (equation 4)
can be estimated parametrically under a constant hazards
assumption.
Alternatively, one may estimate the CIF (equation 3) nonparametrically via the Aalen-Johansen estimator (18),
which generalizes the Kaplan-Meier estimator to the case
of more than one outcome state. For example, for 2 event
types, it is F̂1 ðtÞ þ F̂2 ðtÞ ¼ 1 Kaplan-MeierðtÞ ¼
P
P̂ðT > s Þ3DÂ:ðsÞ, where the sum is calculated over
st
all observed event times s of any type, s t, and DÂ:ðsÞ is
the increment of the Nelson-Aalen estimator (equation 2)
for the cumulative all-event hazard A.(s) = A1(s) þ A2(s).
Following this, the Aalen-Johansen estimate for the CIF
of event type h is
X
P̂ðT > s Þ3DÂh ðsÞ:
F̂h ðtÞ ¼
st
In the following section, we take the nonparametric CIF
estimate as a reference to judge the performance of the
parametric, ID-based CIF estimator, for example, by checking whether it is within the corresponding nonparametric
95% confidence intervals.
Regression analysis
To study potential risk factor effects, a standard approach
would be to use Cox proportional hazards models (refer, for
example, to Lau et al. (11), Kalbfleisch and Prentice (19),
and Cox (23)). In a competing events situation, Cox models
for all event-specific hazards have to be considered. In
addition, Poisson regression models may be used in a
person-time setting, which will typically provide maximum
likelihood estimates that closely agree with those of the Cox
model if the baseline event-specific hazards are assumed to
be piecewise constant (9, 24, 25).
However, even from a constant event-specific hazards (or
ID) analysis with, for example, a binary risk factor, the
course of the CIF may already be difficult to predict because
it is an involved function of both event-specific hazards. To
this end, for example, Lau et al. (11), Beyersmann et al.
(15), and Fine and Gray (26) discuss assessing a potential
covariate effect on the subdistribution hazard k(t), which is
implicitly defined
by a 1-to-1 relation with the CIF, Fh ðtÞ ¼
Rt
1 expð 0 kðuÞduÞ. This assessment may be performed
by using a Cox model for the subdistribution hazard (26).
However, if proportionality holds for the event-specific hazards, the proportional subdistribution hazards model will
generally be misspecified, since the latter hazard will be
time dependent (27) even if the former hazards are assumed
to be constant.
Am J Epidemiol 2010;172:1077–1084
1079
Still, the subdistribution analysis offers a summary analysis in that it estimates the least false parameter, a timeaveraged effect on the cumulative event probabilities
(27–30). The parameter is called least false in the sense that
it gives the best approximation within the misspecified model
class (here, proportional subdistribution hazards model) of
the true model (here, proportional event-specific hazards
models) in terms of the Kullback-Leibler distance (refer to
Claeskens and Hjort (29, chapter 2.2) for a formal definition
and discussion). As a solution to an explicit formula, this
time-averaged effect can be easily calculated numerically
for a given time interval (31). For further details, refer to
the Appendix and the supplementary Web material (R code).
APPLICATION TO BLOODSTREAM INFECTION DATA
Study population
The German multicenter ONKO-KISS surveillance study
population consisted of 1,699 patients undergoing peripheral blood stem cell transplantation for hematologic malignancies, observed during January 2000 and June 2004 (14).
After transplantation, patients become neutropenic. One
purpose of the ONKO-KISS study was to investigate the
impact of risk factors for the occurrence of bloodstream
infection during neutropenia. Among other factors, the binary risk factor transplant type (either allogeneic or autologous) was recorded. A total of 913 (56.5%) of 1,616 patients
received allogeneic transplants, of whom 193 (21.1%) acquired bloodstream infection; of the patients with an autologous transplant, 126 (17.9%) acquired bloodstream
infection. Because patients may also leave the critical phase
without a prior bloodstream infection, either alive or dead,
this constitutes a competing risks setting. A total of 319
(19.7%) events were observed for bloodstream infection,
and 1,280 (79.2%) events were observed for ‘‘end of neutropenia without prior bloodstream infection (alive or
dead)’’; only 17 (1.1%) patients were censored. Follow-up
terminated with the end of neutropenia. Only 20 (1.2%) of
the patients died without having a prior bloodstream infection. To keep things as simple as possible, we do not consider them separately in this paper.
In the following section, we perform a competing risks
analysis with the primary endpoints of bloodstream infection and end of neutropenia. Following Beyersmann et al.
(15), we concentrate on the binary risk factor transplant type
(Z ¼ 1 for allogeneic, Z ¼ 0 for autologous) because it
provides an interpretationally challenging example.
Statistical analysis
The complete competing risks situation is best visualized
and explained by Figure 1, a multistate-type graphic, which
shows the competing risks situation for both allogeneic (left
panel) and autologous (right panel) transplant types; the
arrows, which are thicker for higher values, represent
the particular event-specific IDs according to the results of
the analyses given in Table 1. As evident from Figure 1,
allogeneic transplant has a reducing effect on both becoming infected and leaving the critical phase, but the reduction
1080 Grambauer et al.
Allogeneic
α^ 1 = 0.014
Autologous
BSI
α^ 1 = 0 . 0 2
BSI
α^ 2 = 0.09
End of
Neutropenia
Neutropenia
Neutropenia
α^ 2 = 0.051
End of
Neutropenia
Figure 1. Multistate-type graphic illustrating the competing-risks ONKO-KISS study data analysis by means of event-specific incidence densities.
The arrow thickness describes the particular amount of every incidence density. BSI, bloodstream infection.
is fairly different for the different event types. Compared
with the bloodstream-infection ID, the competing end-ofneutropenia ID is much more pronounced, especially for the
autologous groups. Receiving allogeneic transplants reduces the end-of-neutropenia ID more than it reduces the
bloodstream-infection ID (the exact amount of reduction is
given by ID ratio analysis in Table 1).
In addition to the instantaneous risk of experiencing one
of the competing events, embodied by the IDs, interest also
focuses on the expected proportion of the particular events
along time, that is, the CIF. This quantity may be estimated
parametrically by plugging in the event-specific IDs
(equation 4).
Applied to the ONKO-KISS data, these estimates are
given for the event of interest, bloodstream infection, by
the thick continuous lines in panel A of Figure 2, stratified
by transplant type. To judge the performance of the estimates, nonparametric Aalen-Johansen estimates for bloodstream infection (thick step curves) and corresponding
log-log transformed 95% confidence intervals (thin step
curves) are also depicted. Irrespective of the estimation
method, the estimated CIFs for bloodstream infection indicate that patients with allogeneic transplants initially ap-
pear to be at a lower risk of bloodstream infection; the CIFs
then cross, and eventually there are more patients with allogeneic transplants who acquire bloodstream infection than
there are patients with autologous transplants who
acquire bloodstream infection (15). Both approaches result
in the same plateau. Crossing of the CIFs results from the
higher reduction in allogeneic transplants on the end-ofneutropenia hazard, even amplified by the fact that the baseline end-of-neutropenia hazard is much more pronounced
compared with the bloodstream-infection hazard (also refer
to our proposed multistate-type graphic, Figure 1).
However, the ID-based CIFs may be considered slightly
misspecified because they are not fully contained in the 95%
confidence intervals and therefore do not fully capture the
temporal dynamics. We find the reason for the differing time
dynamic to be the strong time-dependent course of the endof-neutropenia hazard (refer to the Nelson-Aalen estimates
in panel B of Figure 2), so the IDs might not offer an appropriate fit. Furthermore, according to the goodness-of-fit
plot in panel C of Figure 2, the end-of-neutropenia hazards
do not appear to be proportional; this curve should be an
approximately straight line with intercept zero under a proportional hazards model. However, even if the constant
Table 1. Event-specific Incidence Densities and Ratios Among Patients in the ONKO-KISS
Study, 2000–2004, Germany
Allogeneic
Autologous
Estimate
95% CI
Bloodstream infection
0.68
0.61, 0.76
End of neutropenia
0.55
0.52, 0.58
Estimate
95% CI
Estimate
95% CI
Bloodstream infection
0.014
0.012, 0.015
0.020
0.018, 0.023
End of neutropenia
0.051
0.048, 0.054
0.090
0.087, 0.097
Event-specific
incidence densities
Event-specific
incidence density
ratios (allogeneic
vs. autologous)
Abbreviation: CI, confidence interval.
Am J Epidemiol 2010;172:1077–1084
Incidence Densities in a Competing Events Analysis
B)
Nelson−Aalen Estimates
Estimated CIF for BSI
C)
6
0.25
0.20
0.15
0.10
0.05
0.00
5
EndNP
4
3
2
1
BSI
0
0
20
40
60
80
0
20
Days
40
60
Nelson−Aalen Estimate for
Allogeneic Transplant
A)
1081
6
5
EndNP
4
3
2
BSI
1
0
0
80
Days
1
2
3
4
5
6
Nelson−Aalen Estimate for
Autologous Transplant
Figure 2. A) Nonparametrically (thick step curves) and parametrically (thick lines) estimated cumulative incidence functions (CIFs) for bloodstream infection (BSI) according to allogeneic (dashed lines) and autologous (solid lines) transplant type, and nonparametric 95% confidence
intervals (thin lines, corresponding to transplant type). B) Estimated cumulative event-specific hazards for BSI (lower 2 curves) and end of
neutropenia (EndNP; upper 2 curves) for each transplant type. C) Goodness-of-fit plot for BSI and EndNP.
hazards assumption fails here, the event-specific IDs are still
useful because they offer good qualitative insights into the
competing risks situation.
In the following section, we consider another data example (provided in the Web material by Lau et al. (11)), in
which an ID analysis even offers good quantitative insight
in terms of a time-averaged effect on the CIF.
APPLICATION TO HIV DATA
death, or administrative censoring (September 28, 2006).
Among recorded covariates were history of injection drug
use at enrollment, age, CD4 nadir, and race. Following Lau
et al. (11), we considered the study sample of 1,164 women
who were alive, infected with HIV, and free of clinical AIDS
on December 6, 1995 (baseline), the day on which the first
protease inhibitor was approved by the US Federal Drug
Administration. The following statistical analysis concentrates on the effect of injection drug use on time until initiation of HAART without prior occurrence of clinical AIDS.
Study population
Statistical analysis
The multicenter Women’s Interagency HIV Study population consists of 2,058 HIV-positive and 567 HIV-negative
US women enrolled between October 1994 and November
1995. The aim of the study was to assess the impact of HIV
infection on US women. Details are given elsewhere (11,
16). Women were followed until the first of the following
events: initiation of highly active antiretroviral therapy
(HAART), acquired immunodeficiency syndrome (AIDS)/
From the multistate-type graphic (Figure 3, R code given
in the Web material), it is evident that history of injection
drug use reduces the HAART ID (ID ratio ¼ 0.744, 95%
confidence interval: 0.688, 0.800) but has a strong increasing effect on the AIDS/death ID (ID ratio ¼ 1.868, 95%
confidence interval: 1.675, 2.061). Especially here, the baseline HAART ID of interest is much more pronounced
Non-IDU
IDU
^ = 0.16
α
1
HAART
^ 1 = 0.22
α
HAART
^ 2 = 0.08
α
AIDS/Death
HIV
HIV
^ = 0.15
α
2
AIDS/Death
Figure 3. Multistate-type graphic illustrating the competing-risks Women’s Interagency HIV Study data analysis by means of event-specific
incidence densities. The arrow thickness describes the particular amount of every incidence density. AIDS, acquired immunodeficiency syndrome;
HAART, highly active antiretroviral therapy; HIV, human immunodeficiency virus; IDU, injection drug use. (This graphic is part of the worked
example outlined in the supplementary material posted on the Journal ’s Web site (http://aje.oupjournals.org/).)
Am J Epidemiol 2010;172:1077–1084
1082 Grambauer et al.
B)
0.6
0.4
0.2
0.0
0
2
4
6
8
10 12
C)
2.5
2.5
2.0
1.5
HAART
1.0
AIDS/Death
0.5
0.0
Nelson−Aalen Estimate
for IDU
0.8
Nelson−Aalen Estimates
Estimated CIF for HAART
A)
2.0
HAART
1.5
AIDS/Death
1.0
0.5
0.0
0
2
Years
4
6
8
Years
10 12
0.0
0.5
1.0
1.5
2.0
2.5
Nelson−Aalen Estimate
for Non−IDU
Figure 4. A) Nonparametrically (thick step curves) and parametrically (thick lines) estimated cumulative incidence functions (CIFs) for highly
active antiretroviral therapy (HAART) according to injection drug use (IDU; solid lines) and non-IDU (dashed lines), and nonparametric 95%
confidence intervals (thin lines, corresponding to IDU and non-IDU, respectively). B) Estimated cumulative event-specific hazards for HAART
(upper 2 curves) and acquired immunodeficiency syndrome (AIDS)/death (lower 2 curves) according to IDU and non-IDU. C) Goodness-of-fit plot
for HAART and AIDS/death.
compared with the baseline AIDS/death ID, which contributes even more to the estimated CIFs of interest (Figure 4,
panel A). Here, the CIF according to history of injection
drug use follows a course below the one without such a history for most of the time, meaning that the expected proportion of those receiving HAART is smaller for individuals
with an injection drug use history.
The Nelson-Aalen estimates of the cumulative eventspecific hazards (Figure 4, panel B), especially for HAART,
show a slight time-dependent course. However, at least the
proportional hazards assumption appears to hold rather well
because both curves in panel C of Figure 4 are approximately linear, indicating that the event-specific ID analysis
here is reasonable. Because of this latter fact, the ID-based
CIF estimates are close to the Aalen-Johansen estimates.
In addition, Lau et al. (11, refer to Figure 3 or Table 4)
report a time-averaged effect on HAART-CIF based on either a proportional subdistribution hazards model or their
parametric mixture model for nonproportional hazards (with
subsequent averaging). Our proposed summary analysis—the least false parameter determined using event-specific IDs (refer to the Appendix and Web material)—results
in a similar estimate of 0.496, corresponding to 0.61 ¼
exp(–0.496) (bootstrapped 95% confidence interval: 0.52,
0.7), which equals the respective estimate reported by Lau
et al. (11, Figure 3).
DISCUSSION
The ID is often calculated because it is computationally
simple and allows for right-censoring as well as lefttruncation. However, it is also questionable because of potentially nonconstant hazards; piecewise IDs (32) or even
semiparametric approaches, for example, proportional hazards models (23), might be preferred. However, in this paper, we point out that, although an ID analysis might be
questionable, the analysis might be even more obscured
when potentially existing competing events are not taken
into account. When dealing with competing risks, it is im-
portant to analyze event-specific IDs for every event instead
of considering just the ID of interest. However, accounting
for 2 or more risks simultaneously makes interpretation
much more difficult.
We examined 2 different data examples to address 2
different issues. First, analysis of the ONKO-KISS data
demonstrated that a covariate may have quite differential
effects on the competing hazards, possibly leading to a nonproportional (i.e., crossing) course of the respective estimated CIFs of interest. In addition, the Nelson-Aalen
estimates of the competing hazard showed a pronounced
time dependency, suggesting that an ID analysis is questionable. This was reflected in the ID-based CIF estimates,
which missed the time dynamic of the Aalen-Johansen
estimates. Nevertheless, the ID-based CIFs still reflected
the empirical course, and the eventual proportion of infected patients, that is, the plateau of the nonparametric
CIFs, was well captured. Here, although the constant hazards assumption failed, the ID analysis offered qualitative
insight into the underlying process. Furthermore, interpretation was facilitated by the proposed multistate-type
graphic that illustrates the underlying data structure by
displaying the existing risks and its particular IDs. The
graphic offers qualitative insights and lends itself to a synthesis interpretation. Competing events may be adequately
addressed with the suggested methodology. Moreover, it
can straightforwardly be extended to more general multistate models (20), which are realized as a series of nested
competing risks experiments.
In contrast, when we considered the Women’s Interagency HIV Study data, the ID-based CIF estimates were
close to their nonparametric counterparts for most of
the time, a consequence of the underlying proportional
course of the respective cumulative hazards. Although at
least one hazard appeared to be time dependent, the ID
analysis here is an appropriate approach while concurrently
also being the simplest. It was also highlighted that a subdistribution hazard analysis, although potentially misspecified, offered a summary analysis in terms of an average
Am J Epidemiol 2010;172:1077–1084
Incidence Densities in a Competing Events Analysis
effect on the CIF, similar to Lau et al. (11). This so-called
least false parameter can be easily estimated numerically
whenever all event-specific IDs are known, together with
an estimate of the censoring distribution (33). Current work
is allowing the least false parameter to also take lefttruncation into account.
Note that our approach is not restricted to binary covariates, which we used to illustrate the multistate-type
graphic. However, continuous covariates can also be considered, following, for example, Aalen et al. (18), Makuch
(34), and Shen and Fleming (35). That is, in a more involved
regression setup, one would need to compute the individual
ID predictions, which could then be averaged and subsequently used in the multistate-type graphic.
ACKNOWLEDGMENTS
Author affiliations: Department of Medical Biometry and
Statistics, Institute of Medical Biometry and Medical Informatics, University Medical Center Freiburg, Freiburg, Germany (Nadine Grambauer, Martin Schumacher, Jan
Beyersmann); Freiburg Center for Data Analysis and Modeling, University of Freiburg, Freiburg, Germany (Nadine
Grambauer, Jan Beyersmann); and Institute of Environmental Medicine and Hospital Epidemiology, University
Medical Center Freiburg, Freiburg, Germany (Markus
Dettenkofer).
This work was supported by the Deutsche Forschungsgemeinschaft (DFG Forschergruppe FOR 534). The funding
sources had no involvement in this manuscript.
Conflict of interest: none declared.
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APPENDIX
This appendix briefly addresses the meaning of the least false parameter as a time-averaged effect on the CIF and how it can
be easily obtained in the setting considered in this paper.
The subdistribution hazard is build such that it directly relates to the CIF for the event of interest. Therefore, interest is
directed toward a potential covariate effect on this quantity. By assuming event-specific IDs to hold, the Cox model for the
subdistribution hazard is generally misspecified. However, the maximizer, ĉ, of the partial log-likelihood, ‘n(c), for the
subdistribution hazard is still asymptotically consistent for the least false parameter c* (28, 36).
In our considerations, assuming a binary covariate Z with p ¼ P(Z ¼ 1), the least false parameter is the solution to the
equation b(c) ¼ 0, with
bðcÞ ¼ pð1 pÞ3
Z
N
0
ð1 F1 ðu; Z ¼ 0ÞÞf1 ðu; Z ¼ 1Þ expðcÞð1 F1 ðu; Z ¼ 1ÞÞf1 ðu; Z ¼ 0Þ
PðC > uÞdu;
ð1 pÞð1 F1 ðu; Z ¼ 0ÞÞ þ p expðcÞð1 F1 ðu; Z ¼ 1ÞÞ
where F1(u;Z) is the CIF for the event of interest 1 according to covariate Z, f1(u;Z) ¼ dF1(u;Z)/du its derivative, and P(C > u)
is the censoring survival probability (31).
Whenever all event-specific IDs and an estimate of the censoring survival probability (33) are available, the least false
parameter can be estimated numerically by plugging in the respective estimates and using, for example, the function uniroot of
the software environment for statistical computing R (17); the respective R code is provided in the Web material. Corresponding standard errors can be easily obtained via the bootstrap.
Am J Epidemiol 2010;172:1077–1084