Testing the treatment effect on competing causes of death in oncology clinical trials
© Rotolo and Michiels; licensee BioMed Central Ltd. 2014
Received: 4 December 2013
Accepted: 21 May 2014
Published: 29 May 2014
Chemotherapy is expected to reduce cancer deaths (CD), while possibly being harmful in terms of non-cancer deaths (NCD) because of toxicity. Peto’s log-rank test is popular in the medical literature, but its operating characteristics are barely known. We compared this test to the most common ones in the statistical literature: the cause-specific hazard test and Gray’s test on the hazard of the subdistribution. We investigated for the first time the impact of reclassifications of causes of death (CoD) after recurrences, and of misclassification of CoD.
We present a simulation study in which we varied the censoring rate and the correlation between CD and NCD times, we generated recurrence times to study the role of the reclassification of CoD, and we added 20% misclassified CoD. We considered four scenarios for the treatment effect: none; none for CD and negative for NCD; positive for CD and none for NCD; positive for CD and negative for NCD. We applied the three tests to a randomized clinical trial evaluating adjuvant chemotherapy in 1,867 patients with non-small-cell lung cancer.
Most often the three tests well preserved their nominal size, Gray’s test did not when the treatment had an effect on the competing CoD. With a high rate of misclassified CoD, Gray’s and the cause-specific tests lost much of their power, whereas the Peto’s test had the highest power. The cause-specific test had inflated size for NCD when the treatment was beneficial for CD with many misclassified CoD, but had the highest power for NCD when the treatment had no effect on CD, and had similar power to Peto’s test for CD when the treatment had no effect on NCD. Gray’s test performed best when the effect on the two CoD was opposite. The higher the censoring, the lower the rejection probabilities of all the tests and the smaller their differences.
In this first head-to-head comparison of the three tests, the cause-specific test often proved to be the most reliable. Comparing results with and without misclassification of the CoD, Peto’s test was the least influenced by the presence of such misclassification.
KeywordsCompeting risks Peto’s test Cause of death Cancer death Cumulative incidence function Cause-specific hazard Gray’s test
The analysis of survival data in the presence of competing risks has been a widely debated topic for many decades in both the statistical [1–5] and the medical literature [6–11]. Interest in the subject gained momentum in the 1990s, when two main approaches emerged: an approach based on the cause-specific hazard function and another based on the cumulative incidence function and its associated hazard of the subdistribution. For a detailed discussion see  for example. An issue of particular importance in clinical research is testing the effect of covariates – typically the treatment – on competing causes of death. Different solutions have been proposed. The most common ones in the statistical literature are the log-rank test for the cause-specific hazard [1, 13] and nonparametric and semiparametric tests for the cumulative incidence function (CIF) [14, 15]. However, Peto and the Early Breast Cancer Trialists’ Collaborative Group proposed the log-rank subtraction method in the context of oncology [16–19], which is quite popular in the clinical literature and especially in meta-analyses: see for instance references [20–24]. This test imputes deaths to the cancer whenever the cause is unknown or when they occur after a recurrence, whatever the recorded cause. It calculates cause-specific mortality as the difference between overall mortality and that attributable to other causes. The authors assert that this approach makes the test unbiased for the assessment of the effect on cancer mortality.
The comparison of different methods from a theoretical point of view and via simulation studies is being considered with increasing interest in the literature. Putter et al.  offered a detailed and insightful review of competing risks methodology. Dignam et al.  and Dignam and Kocherginsky  focused on point estimation of the treatment effect according to different modeling approaches. Pintilie  provided a simulation study, with independent variables of the times to death by cause, showing that the tests based on the cause-specific hazard – Wald, score and likelihood-ratio – have the correct size and power, in the absence of any effect on the competing event. Using simulations, Freidlin and Korn  compared the cause-specific log-rank test to Gray’s non-parametric test  for the CIF. They concluded that the former preserves its nominal size better and has greater power than the latter, even with positively correlated event times. Williamson et al.  extended these results showing that Gray’s test has greater power in the case of very different degrees of negative correlation between competing event times in the two treatment arms. Ruan and Gray  studied Peto’s test both analytically and in simulations with independent survival times. They proved that it has good properties when the rates of competing events are similar, whereas it has an inflated size and poor power otherwise.
For the first time we present in this article a simulation study to compare head-to-head Peto’s log-rank subtraction test to the log-rank test on the cause-specific hazard, and to Gray’s test based on the CIF in a broad set of clinical scenarios. In order to investigate the effect of different classifications of the cause of death established by Peto’s test, we used a simulation method that allows relapse times to be generated in addition to cancer-death (CD) and non-cancer-death (NCD) times. We simulated data with negative, null, and positive correlations, thereby covering an exhaustive range of dependence assumptions. This study is the first which investigates the impact of censoring and, most importantly, of misclassification of causes of death on the behaviour of these tests.
The clinical problem motivating this study was the evaluation of the efficacy of adjuvant chemotherapy for patients with non-small-cell lung cancer in the International Adjuvant Lung Cancer Trial (IALT) [30, 31]. Its interest is to test whether chemotherapy has a beneficial effect on the occurrence of CD, taking into account the fact that patients can meanwhile die of other causes, and that an increased risk of NCD is possible in the treatment arm, due to chemotherapy toxicity.
In the next section, we present the test statistics of interest. Then, we provide details on the simulation study and its results. Finally, we present the IALT study and the results of the tests for CD and NCD.
Tests for competing causes of death
There are different approaches to dealing with duration data in the presence of competing events, such as cause-specific death and death from other causes. In a latent failure time perspective, there is a random variable T i for the time to each possible event, but only the time to the first event can be recorded. The hazard function of the marginal distribution of each T i is usually called the cause-specific hazard. Testing the treatment effect on the cause-specific hazard allows one to evaluate the net effect of covariates on each event; even though quite intuitive, this strategy has been criticized because it compares the treatment arms in terms of the risk of each event type while ignoring all the others. Gray  proposed an alternative approach, based on the CIF, which takes into account all types of events. As the hazard of the CIF incorporates information on all the competing risks, testing the effect of the treatment on the incidence of each type of event also reflects its effect on all the others. As variations of the risk of each event reverberate on the hazards of competitors, it is advised to consider their results in combination with the analysis of all cause-specific hazards . Moreover, due to its mathematical definition, the hazard of the CIF requires that patients who experience an event remain in the risk sets of the other types of event. For a detailed discussion of this topic, we refer to Section 3 of  and Chapters 4 to 6 of .
Consequently, there are also several approaches for testing the effect of a treatment on competing events. We restricted ourselves to considering three of the most popular ones: the cause-specific and the Gray tests, which receive most of the attention through methodological research, and the Peto test, which is quite common in the medical literature. We aim to compare them in several clinically relevant situations.
Peto ( Pe )
The log-rank subtraction test proposed and further described by Peto [16, 17] consists in a piecewise (with respect to time) version of the log-rank test, performed separately by cause of death. It is said to be a subtraction method because the quantities used to compute the test statistic are first calculated for overall mortality and for NCD. Those concerning CD are then obtained by taking the difference between the former two. Another relevant peculiarity of this approach is that all deaths due to an unknown cause and all those occurring after a relapse are ascribed to the cancer, even if explicitly declared as due to another cause.
Cause-specific ( CS )
Historically, the simplest and most naive approach adopted reflects the idea of considering only the relevant events for each cause of death, while treating all the competing events as independent censoring. This leads to the use of the log-rank test on the cause-specific hazard [1, 13], which is approximately equivalent to the score test of the cause-specific Cox model which itself is asymptotically equivalent to the Wald and likelihood-ratio tests in the same model.
Gray ( Gr )
Another popular approach in the context of competing risks is the one based on the CIF, for which the assumption of independence of the competing events is not required. The hazard associated with the CIF, called the hazard of the subdistribution, also takes into account the occurrence of competing events. In particular, when a subject experiences a competing event, his/her time to the relevant event is not censored and he/she remains in the risk set. Gray’s nonparametric test , used in our study, is asymptotically equivalent to the Wald test on the regression parameter in the Cox model of the hazard of the subdistribution  when there is no censoring.
In the example presented later on, we will show the cumulative risk and incidence curves for all, non-cancer and cancer deaths by treatment arm. They will be plotted by means of three methods corresponding to the three tests for the treatment effect. The first, corresponding to the cause-specific test, is the Nelson–Aalen method for the (cause-specific) cumulative risk [32, 33]. In the case of cause-specific risks, only deaths declared due to the cause of interest are considered as events by the Nelson–Aalen estimator, while all other deaths are censored (assuming non informative censoring). The plots in the second group are the Peto estimator of the (cause-specific) cumulative yearly rates; in these plots all deaths following a recurrence are classified as CD, as well as those of an unknown cause. NCDs preceded by a recurrence are censored when a recurrence occurs. According to the Peto method, first the survival probability is computed per year for the 2 arms combined. Then the survival probability for each arm is obtained by adding to it or subtracting from it a quantity which depends on the logarithm of the yearly risk ratio [16, 17]. Finally, the Aalen–Johansen estimates of the CIFs , corresponding to the Gray test, are plotted. It is noteworthy that, in the case of overall survival, one minus the CIF corresponds to the Kaplan–Meier estimate.
Testing the efficacy of the therapy on the time to CD and NCD is the focus of the researcher’s interest. Sometimes, the classification of causes of deaths implicitly requires the occurrence of a recurrence (Rec). Although the treatment evaluation is done directly on times to CD and NCD, the tests differ in the manner of classifying causes of death. In particular, the Peto test requires information on the times to recurrence.
Here we present different scenarios concerning the treatment effect. Figure A.1 in the Additional file 1 shows, for the first scenario, the correlations obtained between the event times, depending on ρ, the correlation of the underlying normal random variables: the relation is roughly linear and setting the parameter ρ can be considered almost equivalent to setting the correlation between CD and NCD times. On the other hand, this does not affect the correlation between CD and recurrence times, which can be shown to be constantly . In this respect, no difference exists between the scenarios. In order to investigate the properties of the tests in a wide range of situations, we chose five values for ρ, covering very negative and very positive dependence, passing through weak and no dependence: -0.75, -0.375, 0, 0.375, 0.75.
a null effect on both CD and NCD (H R CD = H R NCD = 1),
a null effect on CD (H R CD = 1) and an increased NCD risk (H R NCD = 1.25),
a reduction of the risk of CD (H R CD = 0.8) and a null effect on NCD (H R NCD = 1),
a reduction of the risk of CD (H R CD = 0.8) and an increased NCD risk (H R NCD = 1.25).
The International Adjuvant Lung Cancer Trial
Causes of deaths by treatment arm in the IALT study
(Of which 71 after a relapse)
Deaths from unknown cause
(Of which 26 after a relapse)
The Ethics Committee of Kremlin-Bicêtre hospital in France France (Comité de Protection des Personnes Île-de-France VII) approved the protocol on January 9, 1995. When the study began in 1995, informed consent was obtained from each patient according to the regulations of the participating country; in 1999, all participants were required to give written informed consent.
Results and discussion
As described above, we considered four scenarios for the treatment effect, five possible degrees of dependence between the times to CD and NCD, three possible proportions of censoring, and presence or absence of misclassification of the cause of death. For each of these 4 × 5 × 3 × 2 = 120 situations, 10 000 data sets of size 1000 were generated. The three tests were performed for each of them and the empirical rejection probabilities at a 5% nominal size were computed across the 10 000 replications. The null hypothesis of no treatment effect holds in scenarios 1 and 2 for CD and in scenarios 1 and 3 for NCD. In these cases the empirical rejection probabilities stand for the empirical size of the tests. On the contrary, in all the other situations, the hypothesis does not hold and the rejection probabilities represent the empirical power of the tests. Of note, the rate of miclassified causes of death (20%) is quite high with respect to clinical real life, but it is useful in this context to study its role in a somehow extreme situation.
In the null scenario, i.e. in the absence of any treatment effect on both causes of death, all the tests have empirical rejection probabilities that are very close to the nominal size of 5% (range: 0.04 – 0.06; Additional file 1: Table A.1 and Figure A.2) and their use is equivalent. Furthermore, none of censoring, correlation between causes of death, and misclassification of causes of death (Additional file 1: Table A.2 and Figure A.3) affect the results.
Results of the three tests for the treatment effect on CD and NCD , in the IALT study
(p – val)
(p – val)
The difference in survival in favor of the treatment arm, which is non-significant for overall survival, is significant or borderline for CD, with p-values ranging from 0.033 to 0.064. This suggests that the effects on the risks of CD and NCD are in opposite directions, and that they compensate each other, at least partially, when all deaths due to any cause are considered together. Gr, based on the CIF, detects a statistically significant difference at a 5% level (p = 0.033), whereas the other two are borderline but not significant (p = 0.054 for Pe, p = 0.064 for CS). Most likely, the net increase (i.e. in the cause-specific hazard) in the risk of NCD in the chemotherapy arm contributes to reducing the incidence of CD in that group, amplifying the reduction in the risk of CD when measured in terms of the CIF, although the differences between the test statistics are small.
Both the CS and the Pe tests treat death from other causes as independent censoring, which is not realistic in most practical situations. Gr does not require such an assumption but on the other hand its estimated effect on each competing event reflects also the effect on the others. Thus, both the approaches have a possible drawback, but none of the two prevailed clearly in the simulation study: assuming independent censoring can be a serious issue in the case of strong correlation, whereas using the hazard of the subdistribution can be misleading whenever the treatment changes the hazard of only one of the competing events.
The main innovation and the motivation of the present work was to study the operating characteristics of the test by Peto, which is largely used in the medical literature, though almost absent in statistical publications. We aimed at comparing the test by Peto to the most common ones in the statistical literature, i.e. the test on the cause-specific hazard and the test on the hazard of the subdistribution by Gray. These two tests have already been compared head to head previously (see notably  and ). The main reason for this is that, despite the fact that these two tests address different questions, these are closely linked to each other and in our experience the interest of physicians in a clinical trial is somewhere in-between. Furthermore, to the best of our knowledge, the behavior of these tests in presence of misclassification of the cause of death had never been studied before; we think that the knowledge of such an aspect for the three tests is of primary importance for their practical use.
As our aim was to compare the tests across objectively characterized scenarios, we also investigated how the power and level of the tests could depend on the correlation between times to death from different causes, which has a precise clinical meaning. For example, positive correlation corresponds to comorbidity, which is quite common in advanced diseases. Negative correlation, too, is interesting as this could correspond to the effect of a standard of care therapy with different modalities that impact both disease control and toxicity. In the adjuvant context for lung cancer, for instance, all patients undergo surgery, either segmentectomy, or lobectomy or pneumonectomy: the greater the portion of lung resected, the lower the risk of relapses (and then of CD) but the higher the risk of pulmonary complications (and then of NCD).
Testing the treatment effect on the cause-specific death rate requires paying attention to the effect on the competing events. We considered three popular tests among several existing ones: a test based on recurrence data proposed by Peto, the cause-specific test and the cumulative incidence test proposed by Gray.
We performed a simulation study in four clinically relevant scenarios, with negatively correlated, uncorrelated and positively correlated event times, and with two censoring proportions in addition to complete data. We also generated recurrence times in order to bring to the fore the effects of classifying the cause of death in different ways. The recurrence times, conditional on the time to cancer deaths, followed a uniform distribution, which we considered a reasonable hypothesis. Further, we compared results to those obtained in the case of a high rate of misclassified causes of death.
All the three tests adequately preserved their nominal size when the treatment was completely ineffective. Gr seemed to be the most reliable in the situation of a therapy that reduced the risk of CD and increased that of NCD, provided that causes of death are correctly recorded; otherwise, it performed substantially worse and Pe should be recommended. In all the other situations Gr had the poorest performances, both in terms of the preservation of the nominal size and in terms of power.
CS should be preferred whenever the treatment is expected to be ineffective against the risk of CD and possibly harmful in terms of NCD. A cancer treatment is required to be effective against the risk of CD but not against that ofNCD. In that case, Pe was comparable to CS, except that CS had very high size for NCD in the presence of a high rate of misrecorded causes of death. In our study, Pe did not outperform its competitors in any situation in which the causes of death were correctly classified, whereas it was often the most reliable when the misclassification rate was high.
No clear pattern linked to the dependence between time variables emerged from our study. Censoring always reduces the rejection probabilities of all the tests, notably under the alternative hypothesis. Consequently, the tests are less and less powerful as censoring increases and their differences are less and less pronounced as well.
In the IALT study, the three tests suggested possible harm due to toxicity; Gr was firmly in favor of a benefit versus the risk of CD, whereas CS and Pe were borderline. We showed how the natural graphical representations for the three tests are the Nelson–Aalen estimate of the cumulative cause-specific hazard, the cumulative yearly rates as estimated by Peto, and the Aalen–Johansen estimate of the cumulative incidence function.
This study is the first to compare the operating characteristics of the log-rank test by Peto to those of the two best established tests in the statistical literature. The method used to simulate the data is innovative in that it takes into account the occurrence of recurrences and, at the same time, it is capable of generating both negatively and positively dependent times. This allowed us to study the effect of the reclassification of the causes of death proposed by Peto, without the requirement of assuming independence between CD and NCD. To keep things simple, we chose not to generate times to death from unknown causes. In such cases, multiple imputations or inverse probability weighting techniques exist (see for instance ).
- CD :
Cumulative incidence function
- CS :
- Gr :
International Adjuvant Lung cancer Trial
- NCD :
- Pe :
- Rec :
The authors are grateful to Thierry Le Chevalier, Institut Gustave-Roussy, Villejuif, France, and the IALT Collaborative Group for providing the IALT data set. They further thank Boris Freidlin, Paula Williamson and Ruwanthi Kolamunnage Dona for their valuable elucidations and for sharing their R code for generating correlated survival times. They are grateful to Ariane Dunant for her precious work of critical and attentive revision. The authors thank Vanessa Rousseau for her R code on the Peto method and Lorna Saint Ange for editing the article.
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