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Competing events in patients with malignant disease who are at risk for recurrent venous thromboembolism
Contemporary Clinical Trials 32 (2011) 829–833
Contents lists available at ScienceDirect
Contemporary Clinical Trials j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c o n c l i n t r i a l
Competing events in patients with malignant disease who are at risk for recurrent venous thromboembolism S. Parpia a, J.A. Julian a, L. Thabane b, A.Y.Y. Lee c, F.R. Rickles d, M.N. Levine a,⁎ a b c d
Ontario Clinical Oncology Group, Dept of Oncology, McMaster University, 711 Concession Street, 60 (G) Wing, Hamilton, ON, L8V 1C3, Canada Centre of Evaluation of Medicines, McMaster University, 50 Charlton Avenue East, Hamilton, ON, L8N 4A6, Canada Division of Hematology, University of British Columbia, Diamond Health Care Centre, 2775 Laurel Street, 10th Floor, Vancouver, BC, V5Z 1M9, Canada Katzen Cancer Research Center, George Washington University, 2150 Pennsylvania Ave NW, Washington, DC 20037, USA
a r t i c l e
i n f o
Article history: Received 22 March 2011 Revised 29 June 2011 Accepted 4 July 2011 Available online 18 July 2011 Keywords: Cancer Competing risks Cumulative incidence function Thrombosis
a b s t r a c t Patients with malignant disease enrolled in trials of thrombotic disorders may experience competing events such as death. The occurrence of a competing event may prevent the thrombotic event from being observed. Standard survival analysis techniques ignore competing risks, resulting in possible bias and distorted inferences. To assess the impact of competing events on the results of a previously reported trial comparing low molecular weight heparin (LMWH) with oral anticoagulant (OAC) therapy for the prevention of recurrent venous thromboembolism (VTE) in patients with advanced cancer, we compare the results from standard survival analysis with those from competing risk techniques which are based on the cumulative incidence function (CIF) and Gray's test. The Kaplan–Meier method overestimates the risk of recurrent VTE (17.2% in the OAC group and 8.7% in the LMWH group). Risk of recurrence using the CIF is 12.0% and 6.0% in the OAC and LMWH groups, respectively. Both the log-rank test (p = 0.002) and Gray's test (p = 0.006) suggest evidence in favor of LMWH. The overestimation of risk is 30% in each treatment group, resulting in a similar relative treatment effect; using the Cox model the hazard ratio (HR) is 0.48 (95% confidence interval [CI], 0.30 to 0.78) and HR = 0.47 (95% CI, 0.29 to 0.74) using the CIF model. Failing to account for competing risks may lead to incorrect interpretations of the probability of recurrent VTE. However, when the distribution of competing risks is similar within each treatment group, standard and competing risk methods yield comparable relative treatment effects. © 2011 Elsevier Inc. All rights reserved.
1. Introduction Thrombosis is a common occurrence in patients with malignant disease. There is considerable interest in research that evaluates therapies for the primary prevention of venous thromboembolism (VTE) in patients at high risk of VTE, and for the treatment of established VTE. Time-to-event analysis is one of the most common methods of analyzing data in
⁎ Corresponding author at: Ontario Clinical Oncology Group, Department of Oncology, McMaster University, 711 Concession Street, 60 (G) Wing, 1st Floor, Hamilton, ON, Canada. L8V 1C3. Tel.: + 1 905 527 2299x42176; fax: + 1 905 575 2639. E-mail address: [email protected] (M.N. Levine). 1551-7144/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.cct.2011.07.005
clinical trials. Kaplan–Meier (KM) estimates, the log-rank test and Cox proportional hazards model are standard methods used in trials where the outcome is the time from a starting point to an event of interest or to the last contact date for patients who remain event-free. However, in many situations there could be several events of interest, and the occurrence of one could prevent the occurrence or observation of others. These are often referred to as competing risk problems. Prentice et al. [1] define a competing risk problem as the study of any failure process in which there is more than one distinct cause or type of failure. Gooley et al. [2] define a competing risk as an event which precludes the observation of the event of interest or alters the probability of its occurrence.
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Competing risk is germane to thrombosis trials in patients with malignant disease. For example, in a thrombosis primary prevention trial, the primary event of interest is the first occurrence of VTE, whereas in a treatment trial, it is recurrent VTE. However, patients may die of their cancer or from side effects of treatment before experiencing an occurrence or recurrence, respectively. In this situation, death is considered a competing risk. Standard survival analysis techniques may not be valid in scenarios where the objective is to estimate the probability of failure from the primary event when competing risks are present. Failing to account for the presence of competing risks in an analysis may bias the results and, hence, lead to false interpretation of the data. The comparison of low molecular weight heparin versus oral anticoagulation therapy (CLOT) trial was an international, open label, multicenter randomized trial that investigated the efficacy and safety of dalteparin (Fragmin; Pfizer, New York, NY), a low molecular weight heparin (LMWH), with oral anticoagulant (OAC) therapy for 6 months compared with dalteparin alone for 6 months, for the prevention of recurrent venous thromboembolism (VTE) in 676 cancer patients with newly diagnosed VTE [3]. The concern is that death is a competing event and that it would impact the interpretation of the CLOT findings. During the six month post randomization period, considerably more patients died (39%) than experienced a VTE recurrence (12%), in part due to the enrollment of patients with advanced cancer. The occurrence of death prevents a patient from experiencing VTE and, therefore, is considered a competing risk. In this manuscript, we provide a brief overview of some methods to analyze competing risk data and apply them to the CLOT trial. To assess the impact of death as a competing risk on the results of the study, we compare the competing risk analysis results to those obtained by standard survival analysis methods.
at time t. More detailed reviews of survival analysis concepts are available [5–7].
2.2. Kaplan–Meier Method The trial was designed to compare the cumulative probability (or cumulative incidence) of recurrent VTE between the two treatment groups at 6 months. In general, this would be described by calculating the complement of the KM estimate (1-KM) [4]. The KM estimate is one of the most commonly used methods of statistical analysis for time-toevent data with a single event type [8]. However, it is widely recognized that the 1-KM estimate is not the probability of experiencing an event in the presence of competing risks [6,9–11]. The KM estimate for event-free survival assumes independence between failure types by censoring patients at the time of their competing event. By definition, the possibility that these patients will experience the event of interest beyond the time of censoring still exists. However, if a patient dies, they can no longer experience the event of interest, and their contribution to the estimation of the probability of failure for this event is zero [2]. Therefore, when competing risks are present, one may not be able to obtain unbiased estimates of the probability of recurrence using the KM method [6]. Moreover, the interpretation of 1-KM is altered to be the probability of an event occurring beyond a certain time point given that all other risks are ignored. In practice, this is unrealistic because it would involve all patients being free from any other events. Recognizing this issue, Kalbfleisch and Prentice [9] proposed the cumulative incidence function (CIF) to analyze competing risk data; this has become the preferred method of analysis for competing risk data [12].
2. Methods and materials
2.3. Competing risks method - cumulative incidence function
2.1. Basic concepts of survival analysis
The CIF can be described as the cumulative probability of an event occurring in the presence of other competing events [13]. It is the estimation of the hazard of the event of interest in the presence of all other failure types. The advantage of this method is that it does not require the assumption of independence of the event of interest and its competing failure events. Unlike in the KM method, the CIF does not censor patients at the time of their competing risk event, but instead treats them as failures in the event-free survival portion of its calculation [1,6,10,14].
Medical researchers are often interested in estimating the probability of a patient being event-free at a specific time t. The survival or event-free probability at time t, is the conditional probability of surviving to time t given that the patient is event-free at t-1. In practice, the survival probability for each time interval is calculated as the number of patients who survived that interval divided by the number of patients at risk at the beginning of that interval. The probability of surviving to time t is the product of surviving each of the preceding time intervals. This is analogous to the KM estimate of the survival function [4]. Similarly, the probability of an event at time t is the conditional probability of having an event time t, given that the individual is event-free at t-1. This can be estimated by the event-free probability at t-1 multiplied by the number of patients who experienced the event between t-1 and t, divided by the number of patients at risk. The cumulative incidence of the event at time t is the sum of the conditional probabilities of the event occurring during each of the previous time intervals. The cumulative incidence at time t is equivalent to one minus the probability of being event-free
2.4. Log-rank test and Gray's test The log-rank test is one of the most commonly used methods to compare survival between two groups [15]. When applied in a competing risk scenario, the test censors competing risk events at the time of the event and assumes that competing events are independent from the primary event. Alternatively, Gray [16] proposed a test which compares the CIF of the event of interest, and makes no such assumption. This test compares the hazards associated with the CIF, rather than that of the 1-KM [14].
S. Parpia et al. / Contemporary Clinical Trials 32 (2011) 829–833
2.5. Modeling of CIF Modeling of the hazard is used when the objective is to explore the effects of covariates on the probability of experiencing a specific outcome. The CIF is a function of the hazard of the event of interest as well as that of the competing risks events. The standard Cox model cannot be used for modeling the CIF because it ignores the hazard of the competing risk events by treating them as censored observations [6,17]. To directly estimate the effect of covariates on the CIF, Fine and Gray proposed a semi-parametric proportional hazards model which distinguishes between patients who have had a competing risk event from those who are alive and truly censored [18]. It is very possible that a covariate has a different effect on the CIF than on the causespecific hazard modeled by the Cox model [16]. 3. Analysis The first analysis ignored competing risks and this is equivalent to the analysis described in the CLOT manuscript. The analysis of efficacy was performed according to the intention-to-treat principle with the endpoint being time from randomization to first VTE. Patients were censored at death or at the time of their six-month follow-up visit, whichever occurred first. Six month cumulative incidence of recurrent VTE was estimated using the KM method and treatment groups were compared using the log-rank test. A Cox regression model was used to examine the impact of prognostic factors such as age (≥75 vs. b75 years), Eastern Cooperative Oncology Group (ECOG) status (2, 3 vs. 0, 1), current smoker (yes vs. no), and the presence/absence of metastases, in addition to treatment. The secondary outcome of probability of death from all causes was also calculated using the KM method; patients who were alive at 6 months were censored at that point. To account for the competing risk of death, the analysis of six month cumulative incidence of recurrence of VTE was estimated using the CIF. In this case, patients were censored at the end of the trial if they did not have a recurrent VTE and were alive at the end of the study period. Patients who died during the study were considered to have had a competing risk event. The CIF of the two treatment arms was compared using Gray's test. Fine and Gray's modeling method was used to investigate the effect of prognostic factors on the recurrence of VTE. The analysis ignoring competing risks was performed using the lifetest and phreg procedures in SAS 9.1 (Cary, NC). Gray's test was performed using the SAS macro gray.sas (available at http://www.mrc-bsu.cam.ac.uk/Software/ download.html, accessed in January 2010). Competing risk regression analysis was done using the stcrreg command in STATA 11 SE (College Station, TX). 4. Results
Table 1 Summary of prognostic factors and outcomes from the CLOT trial. Variable
Treatment group
Age (years): mean (sd) ECOG status: n (%) 0 1 2 3 Metastatic disease: n (%) Yes No Current smoker: n (%) Yes No Outcomes: n (%) Recurrent VTE Death
LMWH n = 338
OAC n = 338
62 (12)
63 (13)
80 (24) 135 (40) 118 (34) 5 (2)
63 (19) 150 (44) 122 (36) 3 (1)
223 (66) 115 (34)
232 (69) 106 (31)
33 (10) 305 (90)
42 (12) 296 (88)
27 (8) 130 (38)
53 (16) 136 (40)
LMWH = low molecular weight heparin, OAC = oral anticoagulant, ECOG = Eastern Cooperative Oncology Group.
groups (Table 1). The primary efficacy outcome was symptomatic, recurrent VTE during the six month post-randomization period. Using the KM method, the cumulative incidence of recurrent VTE (Table 2, Fig. 1) is 17.2% in the OAC arm (53 events) and 8.7% in the LMWH arm (27 events). The log-rank test to compare the risk of recurrent VTE over time showed a statistically significant result (p = 0.002). Using the CIF method, we observe the cumulative incidence of recurrence to be 12.0% in the OAC arm and 6.0% in the LMWH arm (Fig. 2). Gray's test to compare the CIFs shows that the two curves are significantly different (p = 0.006). The probability of death from all causes is similar between the two treatment groups (Fig. 3). The two KM survival curves are close together and the log-rank test shows that the curves are not statistically significant (p = 0.43). The Cox model (Table 3) yields a hazard ratio (HR) of 0.48 for LMWH compared with OAC, with a 95% confidence interval (CI) of 0.30 to 0.78. Patients who have metastases (HR = 2.73, 95% CI; 1.20 to 3.43) as well as smokers (HR = 2.06, 95% CI; 1.47 to 4.10) are more likely to experience a recurrent VTE. In addition, older (i.e., N75 years of age) patients are less likely to have a recurrent VTE (HR = 0.39, 95% CI; 0.17 to 0.89). Fine and Gray's competing risk modeling method shows that the hazard ratio of the sub-distribution of risk of recurrent VTE is HR = 0.47 (95% CI; 0.29 to 0.74). Moreover, smokers (HR = 1.97, 95% CI; 1.16 to 3.40) and patients with
Table 2 Six month cumulative incidence of recurrent VTE, by treatment group using the Kaplan–Meier and competing risk methods. Method
Briefly, in the original trial, 676 patients with cancer and VTE were randomly assigned in a 1:1 ratio to receive either 6 months of treatment with dalteparin alone (LMWH arm), or dalteparin followed by oral anticoagulation (OAC arm). Baseline characteristics were similar in the two treatment
831
Kaplan–Meier Competing risk
Treatment group LMWH
OAC
8.7% 6.0%
17.2% 12.0%
p-value
0.002 (log-rank test) 0.006 (Gray's test)
LMWH = Low molecular weight heparin, OAC = oral anticoagulant.
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0.20
0.50 OAC
0.40
0.15 p = 0.002*
0.10 LMWH
0.05
Probability
Probability
OAC
LMWH
0.30 p = 0.53 *
0.20 0.10
0.00
0.00 0
30
60
90
120
150
180
210
0
Days since Randomization
metastases (HR = 2.60, 95% CI; 1.39 to 4.86) are at higher risk of recurrent VTE (Table 3). 5. Discussion In clinical trials of patients with thrombotic disorders, patients may be at risk of multiple individual failure types. The occurrence of one of these events may alter the probability of other failure types from being observed. In the CLOT study, the event of interest was recurrent VTE; however, patients were at high risk of death because of their advanced cancer disease. The concern in this study was that death is a competing risk and may have distorted the results of the trial. To assess the impact of death as a competing risk in the study, we compared two methods of analysis: (1) using the KM estimates and the log-rank test to analyze the data ignoring competing risks, and (2) using CIF estimates and Gray's test to account for death as a competing risk event. We observed that the KM method over estimates the risk of recurrent VTE compared with the competing risk analysis method using the CIF (Table 2). The reason for this is illustrated in the example presented in Table 4 where it is easy to see that the calculation of the KM and CIF are very similar. The difference can be seen in patients who have had a competing event; for example, patient 3. At this point, the KM method censors this observation, KM(t) = 0.90 × 8/8 = 0.90, whereas the CIF counts this observation as a failure, S(t) = 0.90 × 7/8= 0.79, which results in a larger survival probability estimate using the KM method. At the time of the next event (patient 4), this translates
OAC
Probability
90
120
p = 0.006 *
0.08 LMWH
0.02 0.00 30
60
90
120
150
210
into an over estimation or larger cumulative probability of the event occurring, 1-KM(t) = 0.23, compared to the CIF, CI(t) = 0.21. This over-estimation continues throughout the table and is amplified with the increasing number of competing risk events. This example demonstrates that the KM method will always over-estimate the risk of recurrence in the presence of competing risks. Even though an over estimation of the risk of recurrent VTE exists in both arms, the comparison of treatment groups using competing risk analysis produced results comparable to the analysis ignoring competing risks. The log-rank test comparing the two 1-KM curves, as well as the Gray's test comparing the CIFs, each shows a statistically significant result that the risk of recurrent VTE is significantly lower in the LMWH arm compared with the OAC arm. In addition, the hazard ratio for treatment and their confidence intervals from the two modeling methods are similar. This is primarily because the six month cumulative mortality is similar between the treatment arms as shown in Fig. 3. The survival curves show no difference at any time during the study period. This means that the censoring due to mortality is similar between the treatment groups. This observation is reasonable because neither treatment was expected to improve overall survival of the patients. The over estimation of risk of recurrent VTE is approximately 30% in both arms and this over estimation cancels out when a relative calculation is performed. Therefore, the analysis ignoring competing risks over estimates the risk of recurrent VTE, but
Characteristic
0
180
Fig. 3. Cumulative incidence of death from all causes by treatment group using the Kaplan–Meier method (LMWH = low molecular weight heparin, OAC= oral anticoagulant, *log-rank test p-value).
Cox model
Competing risks model
Hazard ratio (95% CI)
Hazard ratio (95% CI)
0.48 (0.30–0.78) 0.39 (0.17–0.89) 1.42 (0.97–2.38) 2.73 (1.20–3.43) 2.06 (1.47–4.10)
0.47 0.40 1.33 2.60 1.97
0.06 0.04
150
Table 3 Results from Cox and competing risks models to assess the influence of prognostic factors on recurrent VTE.
0.12 0.10
60
Days since Randomization
Fig. 1. Cumulative incidence of recurrent VTE by treatment group using the Kaplan–Meier method (LMWH = low molecular weight heparin, OAC= oral anticoagulant, *log-rank test p-value).
0.14
30
180
210
Days since Randomization Fig. 2. Cumulative incidence of recurrent VTE by treatment group using the competing risks method (LMWH = low molecular weight heparin, OAC= oral anticoagulant, *Gray's test p-value).
LMWH vs. OAC Age (yrs): ≥75 vs. b 75 ECOG: 2, 3 vs. 0, 1 Metastatic vs. non-metastatic Smoker vs. non-smoker
(0.29–0.74) (0.17–0.92) (0.84–2.11) (1.39–4.86) (1.16–3.40)
LMWH = low molecular weight heparin, OAC = oral anticoagulant, ECOG = Eastern Cooperative Oncology Group.
S. Parpia et al. / Contemporary Clinical Trials 32 (2011) 829–833
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Table 4 Example illustrating the difference between the Kaplan–Meier and competing risk methods in the presence of competing risks. Patient no.
Follow-up time (days)
Status
Kaplan–Meier method
Competing risk method using CIF
KM(t)
1-KM(t)
S(t)
CI(t)
– 1 2 3 4 5 6 7 8 9 10
0 7 11 15 18 23 27 31 33 35 36
– Ev Cen CR Ev Cen CR CR Ev Ev Cen
1 1 × 9/10 = 0.90 0.90 × 9/9 = 0.90 0.90 × 8/8 = 0.90 0.90 × 6/7 = 0.77 0.77 × 6/6 = 0.77 0.77 × 5/5 = 0.77 0.77 × 4/4 = 0.77 0.77 × 2/3 = 0.51 0.51 × 1/2 = 0.26 0.26 × 1/1 = 0.26
0 0 + (1 × 1/10) = 0.10 0.10 + (0.90 × 0/9) = 0.10 0.10 + (0.90 × 0/8) = 0.10 0.10 + (0.90 × 1/7) = 0.23 0.10 + (0.90 × 0/6) = 0.23 0.10 + (0.90 × 0/5) = 0.23 0.10 + (0.90 × 0/4) = 0.23 0.23 + (0.77 × 1/3) = 0.49 0.49 + (0.51 × 1/2) = 0.74 0.74 + (0.26 × 0/1) = 0.74
1 1 × 9/10 = 0.90 0.90 × 9/9 = 0.90 0.90 × 7/8 = 0.79 0.79 × 6/7 = 0.68 0.68 × 6/6 = 0.68 0.68 × 4/5 = 0.63 0.63 × 3/4 = 0.42 0.42 × 2/3 = 0.28 0.28 × 1/2 = 0.14 0.14 × 0/1 = 0.14
0 0 + (1 × 1/10) = 0.10 0.10 + (0.90 × 0/9) = 0.10 0.10 + (0.90 × 0/8) = 0.10 0.10 + (0.79 × 1/7) = 0.21 0.21 + (0.68 × 0/6) = 0.21 0.21 + (0.68 × 0/5) = 0.21 0.21 + (0.63 × 0/4) = 0.21 0.21 + (0.42 × 1/3) = 0.35 0.35 + (0.28 × 1/2) = 0.49 0.49 + (0.14 × 0/1) = 0.49
CIF = Cumulative incidence function; Status: Ev = Event, Cen = Censored, CR = Competing risk; KM(t) = Survival probability using Kaplan–Meier method, 1-KM (t) = Cumulative incidence of event using Kaplan–Meier method, S(t) = Survival probability using competing risk method, CI(t) = Cumulative incidence of event using competing risk method.
yields a relative treatment effect similar to that from the CIF model. The relative treatment effect from the standard method could be different from that of the competing risk method if the censoring distribution had a differential effect on the probability of recurrent VTE in the treatment groups; that is, if a difference in the mortality rates between the treatment groups was observed. If, for example, LMWH prolonged survival due its potential anti-cancer effects, the degree of over-estimation of risk of recurrence of VTE would be greater in the OAC arm than in the LMWH group which would lead to a relative effect estimate that is biased away from the null. This report summarizes some of the issues arising from competing risk analysis in thrombosis trials. Standard survival analysis may not be valid when the goal is to estimate the probability of an event in some situations, and alternative methods proposed by Prentice et al. [1] and Gray [16] should be used instead. It is difficult to assess the impact of competing risks unless a competing risk analysis is performed. When competing risks is a concern, it would be advisable to carry out a standard analysis as well as a competing risk analysis to assess the impact of competing risks. Should the results of the two analyses differ, the standard survival analysis version should be interpreted with caution. On the other hand, as in the study above, if the results are similar, one can be more confident in the results. This methodology is not limited to trials in thrombosis; it can be applied in other areas of medical research where patients may be subject to competing risks. 6. Conclusions Using standard and competing risks methods, we have performed some sensitivity analyses to assess the impact of competing events on the results of a previously reported trial comparing LMWH with OAC therapy for the prevention of recurrent VTE in patients with advanced cancer. Overall, the results show that failure to account for competing risks may yield misleading results about risk of recurrent VTE. However, when the distribution of competing risks is similar within
each treatment group, standard and competing risk methods yield comparable relative treatment effects.
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Contents lists available at ScienceDirect
Contemporary Clinical Trials j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c o n c l i n t r i a l
Competing events in patients with malignant disease who are at risk for recurrent venous thromboembolism S. Parpia a, J.A. Julian a, L. Thabane b, A.Y.Y. Lee c, F.R. Rickles d, M.N. Levine a,⁎ a b c d
Ontario Clinical Oncology Group, Dept of Oncology, McMaster University, 711 Concession Street, 60 (G) Wing, Hamilton, ON, L8V 1C3, Canada Centre of Evaluation of Medicines, McMaster University, 50 Charlton Avenue East, Hamilton, ON, L8N 4A6, Canada Division of Hematology, University of British Columbia, Diamond Health Care Centre, 2775 Laurel Street, 10th Floor, Vancouver, BC, V5Z 1M9, Canada Katzen Cancer Research Center, George Washington University, 2150 Pennsylvania Ave NW, Washington, DC 20037, USA
a r t i c l e
i n f o
Article history: Received 22 March 2011 Revised 29 June 2011 Accepted 4 July 2011 Available online 18 July 2011 Keywords: Cancer Competing risks Cumulative incidence function Thrombosis
a b s t r a c t Patients with malignant disease enrolled in trials of thrombotic disorders may experience competing events such as death. The occurrence of a competing event may prevent the thrombotic event from being observed. Standard survival analysis techniques ignore competing risks, resulting in possible bias and distorted inferences. To assess the impact of competing events on the results of a previously reported trial comparing low molecular weight heparin (LMWH) with oral anticoagulant (OAC) therapy for the prevention of recurrent venous thromboembolism (VTE) in patients with advanced cancer, we compare the results from standard survival analysis with those from competing risk techniques which are based on the cumulative incidence function (CIF) and Gray's test. The Kaplan–Meier method overestimates the risk of recurrent VTE (17.2% in the OAC group and 8.7% in the LMWH group). Risk of recurrence using the CIF is 12.0% and 6.0% in the OAC and LMWH groups, respectively. Both the log-rank test (p = 0.002) and Gray's test (p = 0.006) suggest evidence in favor of LMWH. The overestimation of risk is 30% in each treatment group, resulting in a similar relative treatment effect; using the Cox model the hazard ratio (HR) is 0.48 (95% confidence interval [CI], 0.30 to 0.78) and HR = 0.47 (95% CI, 0.29 to 0.74) using the CIF model. Failing to account for competing risks may lead to incorrect interpretations of the probability of recurrent VTE. However, when the distribution of competing risks is similar within each treatment group, standard and competing risk methods yield comparable relative treatment effects. © 2011 Elsevier Inc. All rights reserved.
1. Introduction Thrombosis is a common occurrence in patients with malignant disease. There is considerable interest in research that evaluates therapies for the primary prevention of venous thromboembolism (VTE) in patients at high risk of VTE, and for the treatment of established VTE. Time-to-event analysis is one of the most common methods of analyzing data in
⁎ Corresponding author at: Ontario Clinical Oncology Group, Department of Oncology, McMaster University, 711 Concession Street, 60 (G) Wing, 1st Floor, Hamilton, ON, Canada. L8V 1C3. Tel.: + 1 905 527 2299x42176; fax: + 1 905 575 2639. E-mail address: [email protected] (M.N. Levine). 1551-7144/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.cct.2011.07.005
clinical trials. Kaplan–Meier (KM) estimates, the log-rank test and Cox proportional hazards model are standard methods used in trials where the outcome is the time from a starting point to an event of interest or to the last contact date for patients who remain event-free. However, in many situations there could be several events of interest, and the occurrence of one could prevent the occurrence or observation of others. These are often referred to as competing risk problems. Prentice et al. [1] define a competing risk problem as the study of any failure process in which there is more than one distinct cause or type of failure. Gooley et al. [2] define a competing risk as an event which precludes the observation of the event of interest or alters the probability of its occurrence.
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Competing risk is germane to thrombosis trials in patients with malignant disease. For example, in a thrombosis primary prevention trial, the primary event of interest is the first occurrence of VTE, whereas in a treatment trial, it is recurrent VTE. However, patients may die of their cancer or from side effects of treatment before experiencing an occurrence or recurrence, respectively. In this situation, death is considered a competing risk. Standard survival analysis techniques may not be valid in scenarios where the objective is to estimate the probability of failure from the primary event when competing risks are present. Failing to account for the presence of competing risks in an analysis may bias the results and, hence, lead to false interpretation of the data. The comparison of low molecular weight heparin versus oral anticoagulation therapy (CLOT) trial was an international, open label, multicenter randomized trial that investigated the efficacy and safety of dalteparin (Fragmin; Pfizer, New York, NY), a low molecular weight heparin (LMWH), with oral anticoagulant (OAC) therapy for 6 months compared with dalteparin alone for 6 months, for the prevention of recurrent venous thromboembolism (VTE) in 676 cancer patients with newly diagnosed VTE [3]. The concern is that death is a competing event and that it would impact the interpretation of the CLOT findings. During the six month post randomization period, considerably more patients died (39%) than experienced a VTE recurrence (12%), in part due to the enrollment of patients with advanced cancer. The occurrence of death prevents a patient from experiencing VTE and, therefore, is considered a competing risk. In this manuscript, we provide a brief overview of some methods to analyze competing risk data and apply them to the CLOT trial. To assess the impact of death as a competing risk on the results of the study, we compare the competing risk analysis results to those obtained by standard survival analysis methods.
at time t. More detailed reviews of survival analysis concepts are available [5–7].
2.2. Kaplan–Meier Method The trial was designed to compare the cumulative probability (or cumulative incidence) of recurrent VTE between the two treatment groups at 6 months. In general, this would be described by calculating the complement of the KM estimate (1-KM) [4]. The KM estimate is one of the most commonly used methods of statistical analysis for time-toevent data with a single event type [8]. However, it is widely recognized that the 1-KM estimate is not the probability of experiencing an event in the presence of competing risks [6,9–11]. The KM estimate for event-free survival assumes independence between failure types by censoring patients at the time of their competing event. By definition, the possibility that these patients will experience the event of interest beyond the time of censoring still exists. However, if a patient dies, they can no longer experience the event of interest, and their contribution to the estimation of the probability of failure for this event is zero [2]. Therefore, when competing risks are present, one may not be able to obtain unbiased estimates of the probability of recurrence using the KM method [6]. Moreover, the interpretation of 1-KM is altered to be the probability of an event occurring beyond a certain time point given that all other risks are ignored. In practice, this is unrealistic because it would involve all patients being free from any other events. Recognizing this issue, Kalbfleisch and Prentice [9] proposed the cumulative incidence function (CIF) to analyze competing risk data; this has become the preferred method of analysis for competing risk data [12].
2. Methods and materials
2.3. Competing risks method - cumulative incidence function
2.1. Basic concepts of survival analysis
The CIF can be described as the cumulative probability of an event occurring in the presence of other competing events [13]. It is the estimation of the hazard of the event of interest in the presence of all other failure types. The advantage of this method is that it does not require the assumption of independence of the event of interest and its competing failure events. Unlike in the KM method, the CIF does not censor patients at the time of their competing risk event, but instead treats them as failures in the event-free survival portion of its calculation [1,6,10,14].
Medical researchers are often interested in estimating the probability of a patient being event-free at a specific time t. The survival or event-free probability at time t, is the conditional probability of surviving to time t given that the patient is event-free at t-1. In practice, the survival probability for each time interval is calculated as the number of patients who survived that interval divided by the number of patients at risk at the beginning of that interval. The probability of surviving to time t is the product of surviving each of the preceding time intervals. This is analogous to the KM estimate of the survival function [4]. Similarly, the probability of an event at time t is the conditional probability of having an event time t, given that the individual is event-free at t-1. This can be estimated by the event-free probability at t-1 multiplied by the number of patients who experienced the event between t-1 and t, divided by the number of patients at risk. The cumulative incidence of the event at time t is the sum of the conditional probabilities of the event occurring during each of the previous time intervals. The cumulative incidence at time t is equivalent to one minus the probability of being event-free
2.4. Log-rank test and Gray's test The log-rank test is one of the most commonly used methods to compare survival between two groups [15]. When applied in a competing risk scenario, the test censors competing risk events at the time of the event and assumes that competing events are independent from the primary event. Alternatively, Gray [16] proposed a test which compares the CIF of the event of interest, and makes no such assumption. This test compares the hazards associated with the CIF, rather than that of the 1-KM [14].
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2.5. Modeling of CIF Modeling of the hazard is used when the objective is to explore the effects of covariates on the probability of experiencing a specific outcome. The CIF is a function of the hazard of the event of interest as well as that of the competing risks events. The standard Cox model cannot be used for modeling the CIF because it ignores the hazard of the competing risk events by treating them as censored observations [6,17]. To directly estimate the effect of covariates on the CIF, Fine and Gray proposed a semi-parametric proportional hazards model which distinguishes between patients who have had a competing risk event from those who are alive and truly censored [18]. It is very possible that a covariate has a different effect on the CIF than on the causespecific hazard modeled by the Cox model [16]. 3. Analysis The first analysis ignored competing risks and this is equivalent to the analysis described in the CLOT manuscript. The analysis of efficacy was performed according to the intention-to-treat principle with the endpoint being time from randomization to first VTE. Patients were censored at death or at the time of their six-month follow-up visit, whichever occurred first. Six month cumulative incidence of recurrent VTE was estimated using the KM method and treatment groups were compared using the log-rank test. A Cox regression model was used to examine the impact of prognostic factors such as age (≥75 vs. b75 years), Eastern Cooperative Oncology Group (ECOG) status (2, 3 vs. 0, 1), current smoker (yes vs. no), and the presence/absence of metastases, in addition to treatment. The secondary outcome of probability of death from all causes was also calculated using the KM method; patients who were alive at 6 months were censored at that point. To account for the competing risk of death, the analysis of six month cumulative incidence of recurrence of VTE was estimated using the CIF. In this case, patients were censored at the end of the trial if they did not have a recurrent VTE and were alive at the end of the study period. Patients who died during the study were considered to have had a competing risk event. The CIF of the two treatment arms was compared using Gray's test. Fine and Gray's modeling method was used to investigate the effect of prognostic factors on the recurrence of VTE. The analysis ignoring competing risks was performed using the lifetest and phreg procedures in SAS 9.1 (Cary, NC). Gray's test was performed using the SAS macro gray.sas (available at http://www.mrc-bsu.cam.ac.uk/Software/ download.html, accessed in January 2010). Competing risk regression analysis was done using the stcrreg command in STATA 11 SE (College Station, TX). 4. Results
Table 1 Summary of prognostic factors and outcomes from the CLOT trial. Variable
Treatment group
Age (years): mean (sd) ECOG status: n (%) 0 1 2 3 Metastatic disease: n (%) Yes No Current smoker: n (%) Yes No Outcomes: n (%) Recurrent VTE Death
LMWH n = 338
OAC n = 338
62 (12)
63 (13)
80 (24) 135 (40) 118 (34) 5 (2)
63 (19) 150 (44) 122 (36) 3 (1)
223 (66) 115 (34)
232 (69) 106 (31)
33 (10) 305 (90)
42 (12) 296 (88)
27 (8) 130 (38)
53 (16) 136 (40)
LMWH = low molecular weight heparin, OAC = oral anticoagulant, ECOG = Eastern Cooperative Oncology Group.
groups (Table 1). The primary efficacy outcome was symptomatic, recurrent VTE during the six month post-randomization period. Using the KM method, the cumulative incidence of recurrent VTE (Table 2, Fig. 1) is 17.2% in the OAC arm (53 events) and 8.7% in the LMWH arm (27 events). The log-rank test to compare the risk of recurrent VTE over time showed a statistically significant result (p = 0.002). Using the CIF method, we observe the cumulative incidence of recurrence to be 12.0% in the OAC arm and 6.0% in the LMWH arm (Fig. 2). Gray's test to compare the CIFs shows that the two curves are significantly different (p = 0.006). The probability of death from all causes is similar between the two treatment groups (Fig. 3). The two KM survival curves are close together and the log-rank test shows that the curves are not statistically significant (p = 0.43). The Cox model (Table 3) yields a hazard ratio (HR) of 0.48 for LMWH compared with OAC, with a 95% confidence interval (CI) of 0.30 to 0.78. Patients who have metastases (HR = 2.73, 95% CI; 1.20 to 3.43) as well as smokers (HR = 2.06, 95% CI; 1.47 to 4.10) are more likely to experience a recurrent VTE. In addition, older (i.e., N75 years of age) patients are less likely to have a recurrent VTE (HR = 0.39, 95% CI; 0.17 to 0.89). Fine and Gray's competing risk modeling method shows that the hazard ratio of the sub-distribution of risk of recurrent VTE is HR = 0.47 (95% CI; 0.29 to 0.74). Moreover, smokers (HR = 1.97, 95% CI; 1.16 to 3.40) and patients with
Table 2 Six month cumulative incidence of recurrent VTE, by treatment group using the Kaplan–Meier and competing risk methods. Method
Briefly, in the original trial, 676 patients with cancer and VTE were randomly assigned in a 1:1 ratio to receive either 6 months of treatment with dalteparin alone (LMWH arm), or dalteparin followed by oral anticoagulation (OAC arm). Baseline characteristics were similar in the two treatment
831
Kaplan–Meier Competing risk
Treatment group LMWH
OAC
8.7% 6.0%
17.2% 12.0%
p-value
0.002 (log-rank test) 0.006 (Gray's test)
LMWH = Low molecular weight heparin, OAC = oral anticoagulant.
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0.20
0.50 OAC
0.40
0.15 p = 0.002*
0.10 LMWH
0.05
Probability
Probability
OAC
LMWH
0.30 p = 0.53 *
0.20 0.10
0.00
0.00 0
30
60
90
120
150
180
210
0
Days since Randomization
metastases (HR = 2.60, 95% CI; 1.39 to 4.86) are at higher risk of recurrent VTE (Table 3). 5. Discussion In clinical trials of patients with thrombotic disorders, patients may be at risk of multiple individual failure types. The occurrence of one of these events may alter the probability of other failure types from being observed. In the CLOT study, the event of interest was recurrent VTE; however, patients were at high risk of death because of their advanced cancer disease. The concern in this study was that death is a competing risk and may have distorted the results of the trial. To assess the impact of death as a competing risk in the study, we compared two methods of analysis: (1) using the KM estimates and the log-rank test to analyze the data ignoring competing risks, and (2) using CIF estimates and Gray's test to account for death as a competing risk event. We observed that the KM method over estimates the risk of recurrent VTE compared with the competing risk analysis method using the CIF (Table 2). The reason for this is illustrated in the example presented in Table 4 where it is easy to see that the calculation of the KM and CIF are very similar. The difference can be seen in patients who have had a competing event; for example, patient 3. At this point, the KM method censors this observation, KM(t) = 0.90 × 8/8 = 0.90, whereas the CIF counts this observation as a failure, S(t) = 0.90 × 7/8= 0.79, which results in a larger survival probability estimate using the KM method. At the time of the next event (patient 4), this translates
OAC
Probability
90
120
p = 0.006 *
0.08 LMWH
0.02 0.00 30
60
90
120
150
210
into an over estimation or larger cumulative probability of the event occurring, 1-KM(t) = 0.23, compared to the CIF, CI(t) = 0.21. This over-estimation continues throughout the table and is amplified with the increasing number of competing risk events. This example demonstrates that the KM method will always over-estimate the risk of recurrence in the presence of competing risks. Even though an over estimation of the risk of recurrent VTE exists in both arms, the comparison of treatment groups using competing risk analysis produced results comparable to the analysis ignoring competing risks. The log-rank test comparing the two 1-KM curves, as well as the Gray's test comparing the CIFs, each shows a statistically significant result that the risk of recurrent VTE is significantly lower in the LMWH arm compared with the OAC arm. In addition, the hazard ratio for treatment and their confidence intervals from the two modeling methods are similar. This is primarily because the six month cumulative mortality is similar between the treatment arms as shown in Fig. 3. The survival curves show no difference at any time during the study period. This means that the censoring due to mortality is similar between the treatment groups. This observation is reasonable because neither treatment was expected to improve overall survival of the patients. The over estimation of risk of recurrent VTE is approximately 30% in both arms and this over estimation cancels out when a relative calculation is performed. Therefore, the analysis ignoring competing risks over estimates the risk of recurrent VTE, but
Characteristic
0
180
Fig. 3. Cumulative incidence of death from all causes by treatment group using the Kaplan–Meier method (LMWH = low molecular weight heparin, OAC= oral anticoagulant, *log-rank test p-value).
Cox model
Competing risks model
Hazard ratio (95% CI)
Hazard ratio (95% CI)
0.48 (0.30–0.78) 0.39 (0.17–0.89) 1.42 (0.97–2.38) 2.73 (1.20–3.43) 2.06 (1.47–4.10)
0.47 0.40 1.33 2.60 1.97
0.06 0.04
150
Table 3 Results from Cox and competing risks models to assess the influence of prognostic factors on recurrent VTE.
0.12 0.10
60
Days since Randomization
Fig. 1. Cumulative incidence of recurrent VTE by treatment group using the Kaplan–Meier method (LMWH = low molecular weight heparin, OAC= oral anticoagulant, *log-rank test p-value).
0.14
30
180
210
Days since Randomization Fig. 2. Cumulative incidence of recurrent VTE by treatment group using the competing risks method (LMWH = low molecular weight heparin, OAC= oral anticoagulant, *Gray's test p-value).
LMWH vs. OAC Age (yrs): ≥75 vs. b 75 ECOG: 2, 3 vs. 0, 1 Metastatic vs. non-metastatic Smoker vs. non-smoker
(0.29–0.74) (0.17–0.92) (0.84–2.11) (1.39–4.86) (1.16–3.40)
LMWH = low molecular weight heparin, OAC = oral anticoagulant, ECOG = Eastern Cooperative Oncology Group.
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Table 4 Example illustrating the difference between the Kaplan–Meier and competing risk methods in the presence of competing risks. Patient no.
Follow-up time (days)
Status
Kaplan–Meier method
Competing risk method using CIF
KM(t)
1-KM(t)
S(t)
CI(t)
– 1 2 3 4 5 6 7 8 9 10
0 7 11 15 18 23 27 31 33 35 36
– Ev Cen CR Ev Cen CR CR Ev Ev Cen
1 1 × 9/10 = 0.90 0.90 × 9/9 = 0.90 0.90 × 8/8 = 0.90 0.90 × 6/7 = 0.77 0.77 × 6/6 = 0.77 0.77 × 5/5 = 0.77 0.77 × 4/4 = 0.77 0.77 × 2/3 = 0.51 0.51 × 1/2 = 0.26 0.26 × 1/1 = 0.26
0 0 + (1 × 1/10) = 0.10 0.10 + (0.90 × 0/9) = 0.10 0.10 + (0.90 × 0/8) = 0.10 0.10 + (0.90 × 1/7) = 0.23 0.10 + (0.90 × 0/6) = 0.23 0.10 + (0.90 × 0/5) = 0.23 0.10 + (0.90 × 0/4) = 0.23 0.23 + (0.77 × 1/3) = 0.49 0.49 + (0.51 × 1/2) = 0.74 0.74 + (0.26 × 0/1) = 0.74
1 1 × 9/10 = 0.90 0.90 × 9/9 = 0.90 0.90 × 7/8 = 0.79 0.79 × 6/7 = 0.68 0.68 × 6/6 = 0.68 0.68 × 4/5 = 0.63 0.63 × 3/4 = 0.42 0.42 × 2/3 = 0.28 0.28 × 1/2 = 0.14 0.14 × 0/1 = 0.14
0 0 + (1 × 1/10) = 0.10 0.10 + (0.90 × 0/9) = 0.10 0.10 + (0.90 × 0/8) = 0.10 0.10 + (0.79 × 1/7) = 0.21 0.21 + (0.68 × 0/6) = 0.21 0.21 + (0.68 × 0/5) = 0.21 0.21 + (0.63 × 0/4) = 0.21 0.21 + (0.42 × 1/3) = 0.35 0.35 + (0.28 × 1/2) = 0.49 0.49 + (0.14 × 0/1) = 0.49
CIF = Cumulative incidence function; Status: Ev = Event, Cen = Censored, CR = Competing risk; KM(t) = Survival probability using Kaplan–Meier method, 1-KM (t) = Cumulative incidence of event using Kaplan–Meier method, S(t) = Survival probability using competing risk method, CI(t) = Cumulative incidence of event using competing risk method.
yields a relative treatment effect similar to that from the CIF model. The relative treatment effect from the standard method could be different from that of the competing risk method if the censoring distribution had a differential effect on the probability of recurrent VTE in the treatment groups; that is, if a difference in the mortality rates between the treatment groups was observed. If, for example, LMWH prolonged survival due its potential anti-cancer effects, the degree of over-estimation of risk of recurrence of VTE would be greater in the OAC arm than in the LMWH group which would lead to a relative effect estimate that is biased away from the null. This report summarizes some of the issues arising from competing risk analysis in thrombosis trials. Standard survival analysis may not be valid when the goal is to estimate the probability of an event in some situations, and alternative methods proposed by Prentice et al. [1] and Gray [16] should be used instead. It is difficult to assess the impact of competing risks unless a competing risk analysis is performed. When competing risks is a concern, it would be advisable to carry out a standard analysis as well as a competing risk analysis to assess the impact of competing risks. Should the results of the two analyses differ, the standard survival analysis version should be interpreted with caution. On the other hand, as in the study above, if the results are similar, one can be more confident in the results. This methodology is not limited to trials in thrombosis; it can be applied in other areas of medical research where patients may be subject to competing risks. 6. Conclusions Using standard and competing risks methods, we have performed some sensitivity analyses to assess the impact of competing events on the results of a previously reported trial comparing LMWH with OAC therapy for the prevention of recurrent VTE in patients with advanced cancer. Overall, the results show that failure to account for competing risks may yield misleading results about risk of recurrent VTE. However, when the distribution of competing risks is similar within
each treatment group, standard and competing risk methods yield comparable relative treatment effects.
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