competing risks; event-specific hazard; incidence rate; model misspecification; proportional hazards; survival analysis

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1 American Journal of Epidemiology ª The Author Published by Oxford University Press on behalf of the Johns Hopkins Bloomberg School of Public Health. All rights reserved. For permissions, please Vol. 172, No. 9 DOI: /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, Freiburg, Germany ( nadine.grambauer@imbi.uni-freiburg.de). Initially submitted January 19, 2010; accepted for publication June 30, 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

2 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 basedonids,whichissimilartothatgivenbylauetal. (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 ( 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 a h (t), where a h (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: â h ¼ number of type-h-events 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 eventspecific hazards A h ðtþ ¼ R t 0 a hðuþdu by comparing its parametric plug-in estimates with the corresponding nonparametric Nelson-Aalen estimates (18). The latter are defined as  h 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: F h ðtþ ¼PðT t; event ¼ hþ ¼ R t 0 PðT > u Þ3a ð3þ hðuþdu; where PðT > uþ ¼expð R u 0 ða 1ðvÞþa 2 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, a h (t) ¼ a h, the CIF of event type h is explicitly given as

3 Incidence Densities in a Competing Events Analysis 1079 F h ðtþ ¼ R t 0 expð ða 1 þ a 2 ÞuÞ3a h du ¼ a h R t 0 a 1 þ a expð ða 1 þ a 2 ÞuÞdu 2 ¼ a h ð1 expð ða 1 þ a 2 ÞtÞÞ a 1 þ a 2 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 ˆF 1 ðtþþ ˆF 2 ðtþ ¼1 Kaplan-MeierðtÞ ¼ P ˆPðT > 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) =A 1 (s) þ A 2 (s). Following this, the Aalen-Johansen estimate for the CIF of event type h is ˆF h ðtþ ¼ X ˆPðT > s Þ3DÂ h ðsþ: 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, F h ðtþ ¼ 1 expð R t 0kðuÞduÞ. 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. 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

4 1080 Grambauer et al. Allogeneic Autologous α^ 1 = BSI α^ 1 = 0.02 BSI Neutropenia Neutropenia α^ 2 = End of Neutropenia α^ 2 = 0.09 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 appear 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, , Germany Allogeneic Autologous Estimate 95% CI Estimate 95% CI Estimate 95% CI Event-specific incidence densities Bloodstream infection , , End of neutropenia , , Event-specific incidence density ratios (allogeneic vs. autologous) Bloodstream infection , 0.76 End of neutropenia , 0.58 Abbreviation: CI, confidence interval.

5 Incidence Densities in a Competing Events Analysis 1081 A) B) C) Estimated CIF for BSI Days Nelson Aalen Estimates EndNP BSI Days Nelson Aalen Estimate for Allogeneic Transplant BSI EndNP 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 Study population 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 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)/ 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. Statistical analysis 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 IDU Non-IDU α^1 = 0.16 HAART α^1 = 0.22 HAART HIV HIV α^2 = 0.15 AIDS/Death α^2 = 0.08 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 (

6 1082 Grambauer et al. A) B) C) Estimated CIF for HAART Years Nelson Aalen Estimates HAART Years AIDS/Death Nelson Aalen Estimate for IDU AIDS/Death HAART 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 important 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

7 Incidence Densities in a Competing Events Analysis 1083 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. REFERENCES 1. Chen PC, Sung FC, Chien KL, et al. Red blood cell distribution width and risk of cardiovascular events and mortality in a community cohort in Taiwan. Am J Epidemiol. 2010;171(2): Glynn RJ, Rosner B. 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