Chimeric antigen receptor T-cell (CAR-T) therapy is an exciting development in the field of cancer immunology and has received a lot of interest in recent years. Many time-to-event (TTE) endpoints related to relapse, disease progression, and remission are analyzed in CAR-T studies to assess treatment efficacy. Definitions of these TTE endpoints are not always consistent, even for the same outcomes (e.g., progression-free survival), which often stems from analysis choices regarding which events to consider as part of the composite endpoint, censoring or competing risk in the analysis. Subsequent therapies such as hematopoietic stem cell transplantation are common but are not treated the same in different studies. Standard survival analysis methods are commonly applied to TTE analyses but often without full consideration of the assumptions inherent in the chosen analysis. We highlight two important issues of TTE analysis that arise in CAR-T studies, as well as in other settings in oncology: the handling of competing risks and assessing the association between a time-varying (post-infusion) exposure and the TTE outcome. We review existing analytical methods, including the cumulative incidence function and regression models for analysis of competing risks, and landmark and time-varying covariate analysis for analysis of post-infusion exposures. We clarify the scientific questions that the different analytical approaches address and illustrate how the application of an inappropriate method could lead to different results using data from multiple published CAR-T studies. Codes for implementing these methods in standard statistical software are provided.

Translational Relevance

Many time-to-event (TTE) endpoints are analyzed in oncology clinical trials to assess treatment efficacy. Standard survival analysis methods are often applied without full consideration of the assumptions inherent in the chosen analysis. This article discusses some underappreciated statistical issues in these analyses and provides guidance on choosing appropriate analytical methods that address the scientific question of interest. Two important issues are highlighted; the handling of competing risks and assessing the association between a time-varying exposure and the TTE outcome. Concepts and methods are reviewed and illustrated with examples from studies of chimeric antigen receptor T-cell therapy. Better understanding of these statistical nuances will avoid bias in study analyses, facilitate critical review of emerging research in this field, and highlight differences in practice that may affect the comparability of results between studies.

Chimeric antigen receptor T-cell (CAR-T) therapy has been dramatically effective for some patients with relapsed/refractory B-cell acute lymphoblastic leukemia (B-ALL; refs. 1–8), chronic lymphocytic leukemia (CLL; refs. 9, 10), non-Hodgkin lymphoma (11, 12), and multiple myeloma (13, 14). Application of CAR-T therapy continues to increase, with the recent approval of multiple drugs by the FDA (https://www.fda.gov/vaccines-blood-biologics/cellular-gene-therapy-products/approved-cellular-and-gene-therapy-products). Several time-to-event (TTE) endpoints related to relapse, disease progression, and remission are analyzed in a single CAR-T study to fully assess treatment efficacy. Standard survival analyses are commonly used to analyze TTE endpoints, where the endpoint definition and censoring mechanism involve multiple outcomes. This has led to inconsistencies across trials in how a given event, such as receipt of an alternate anticancer therapy [e.g., hematopoietic stem cell transplantation (HSCT)] or non–disease-related death, is handled in analyses. Analytical issues related to competing risks and time-varying (post-infusion) exposures often arise in the CAR-T setting, but the statistical methods that directly address these issues are rarely applied in the published literature. When these issues are treated inadequately and inconsistently, the interpretation of study results would be compromised and the comparisons between studies (CAR-T or other treatment) would be difficult. Lack of familiarity with more advanced survival analysis methods and misconceptions about competing risks analyses are common in medical literatures (15–17) and have likely contributed to this inadequacy. In this rapidly evolving field, more careful consideration of these complex analytical issues is needed to better assess the different aspects of CAR-T efficacy. The goal of this paper is to address misconceptions regarding the analysis of TTE endpoints and provide guidance on choosing appropriate statistical methods to address different scientific questions of interest. We illustrate ideas with CAR-T clinical trials, but the issues discussed apply more generally to other oncology settings.

We explore several topics in this paper. First, there has been a general lack of consistency in definitions of TTE endpoints, which often stems from which events are incorporated into the composite outcome versus treated as censoring, or a competing risk. Moreover, while censoring is an important element for analyzing TTE endpoints, there is often a lack of appreciation of the assumptions necessary for the chosen analysis to produce valid inference for the efficacy measure of interest. Another analytical challenge often encountered in CAR-T studies is the estimation of the association of a time-varying exposure with a TTE endpoint; for example, whether post-infusion HSCT is associated with improved survival. Inappropriate use of standard methods could lead to immortal person-time bias and incorrect conclusions about the effect of the post-infusion exposure (18–22).

We will review the common TTE endpoints in CAR-T studies, highlighting some of the nuances involved in endpoint definitions that arise in different settings. We then discuss statistical approaches for competing risks analyses, followed by analytical considerations for different post-infusion events in CAR-T trials. We then focus on assessing the effect of a time-varying (post-infusion) exposure. We illustrate different statistical approaches using multiple CAR-T studies, with a focused discussion on the different scientific question that each approach addresses.

Common TTE endpoints

Like many oncology clinical trials, CAR-T trials often involve TTE endpoints, such as survival post-infusion. The analysis of TTE endpoints is complicated by the fact that the event of interest may not be observed on all subjects during the trial, and these subjects are said to be censored. Some standard survival analyses, such as the Kaplan–Meier (KM) survival curve, require that the event has only two possible outcomes (e.g., death or survival). To accommodate this requirement, time to first of multiple events is often created as a composite endpoint. For example, progression-free survival (PFS) is determined by the earliest occurring of death or disease progression. Censoring can also be a composite measure, such as subjects’ last follow-up, completion of the planned study follow-up, or being lost to follow-up. In some settings, events thought to be unrelated to the event of interest are censored. For example, non-relapse–related death (NRD) may be treated as censoring for relapse-free survival (RFS) endpoint (NCT02650414, NCT05037669). In CAR-T trials, censoring certain types of events that occur post-infusion may have unintended consequences on study estimates, as explained in the following sections.

Common TTE endpoints in CAR-T studies include overall survival (OS), PFS, event-free survival (EFS), RFS, and duration of response (DOR). Table 1 summarizes their definitions, noting the variability for some clinical outcomes to be considered as events or censoring. For example, in the studies of childhood acute lymphoblastic leukemia (ALL), some publications considered subsequent malignancy such as myelodysplastic syndromes and acute myeloid leukemia as an event for EFS (2, 4), while others did not (3, 5, 23). For RFS, some studies counted only disease-related deaths as events and censored unrelated deaths (NCT02650414, NCT05037669). For all non-OS endpoints, receipt of new cancer therapy could be treated differently. In some studies it is considered as censoring (2–4, 11, 23), while in other studies it was ignored (not considered as event or censoring; refs. 1, 5–10, 12). Furthermore, the reason for new cancer therapy could impact how it is handled in the TTE endpoint definition. If the new cancer therapy is given because the disease progressed it can be considered as an event, but if it is given to prevent relapse while patients are still in remission (e.g., HSCT) it can be considered as censoring or as a competing risk (NCT02650414, NCT05037669).

Table 1.

Common TTE endpoints reported in CAR-T trials.

OSPFSEFSRFS/DOR
Analytic cohort All patients All patients All patients Responders onlya 
Time origin Infusion Infusion Infusion Response post-infusion 
Event Any death 
  • Disease Progression

  • Any deathc

  • Relapse

  • New cancer therapy variable by settingd

 
  • Treatment failureb

  • Any death

  • Relapse

  • New cancer therapy variable by settingd

  • Subsequent malignancy variable by settinge

 
  • Any deathc

  • Relapse

  • New cancer therapy variable by settingd

 
Potential censoring or competing risks Last contactf 
  • Last contactg

  • New cancer therapy variable by settingd

  • Subsequent malignancy

  • Unrelated deathc

 
  • Last contactg

  • New cancer therapy variable by settingd Subsequent malignancy variable by settinge

 
  • Last contactg

  • New cancer therapy variable by settingd

  • Subsequent malignancy

  • Unrelated deathc

 
Recommendation  
  • 1.  Consider competing risks rather than naïve censoring

  • 2.  Use all-cause death when death relationship to treatment is difficult to determine

  • 3.  For new cancer therapy, consider the reason of the therapy to possibly classify it as an event or competing risk

  • 4.  For subsequent malignancy, consider the relationship between the primary cancer and the risk of subsequent malignancy and the treatment effect of interest (e.g., specific effect on relapse only or a general effect)

 
OSPFSEFSRFS/DOR
Analytic cohort All patients All patients All patients Responders onlya 
Time origin Infusion Infusion Infusion Response post-infusion 
Event Any death 
  • Disease Progression

  • Any deathc

  • Relapse

  • New cancer therapy variable by settingd

 
  • Treatment failureb

  • Any death

  • Relapse

  • New cancer therapy variable by settingd

  • Subsequent malignancy variable by settinge

 
  • Any deathc

  • Relapse

  • New cancer therapy variable by settingd

 
Potential censoring or competing risks Last contactf 
  • Last contactg

  • New cancer therapy variable by settingd

  • Subsequent malignancy

  • Unrelated deathc

 
  • Last contactg

  • New cancer therapy variable by settingd Subsequent malignancy variable by settinge

 
  • Last contactg

  • New cancer therapy variable by settingd

  • Subsequent malignancy

  • Unrelated deathc

 
Recommendation  
  • 1.  Consider competing risks rather than naïve censoring

  • 2.  Use all-cause death when death relationship to treatment is difficult to determine

  • 3.  For new cancer therapy, consider the reason of the therapy to possibly classify it as an event or competing risk

  • 4.  For subsequent malignancy, consider the relationship between the primary cancer and the risk of subsequent malignancy and the treatment effect of interest (e.g., specific effect on relapse only or a general effect)

 

aThose who had a response at Day 28 in ALL studies, or at response time post-infusion in CLL and lymphoma studies.

be.g., nonresponse at Day 28.

cSome studies only consider disease or treatment related death as event and consider unrelated death as censoring (NCT02650414, NCT05037669).

dReceipt of new cancer therapy is considered as censoring in some studies (2–4, 11, 23), while ignored (not considered as event or censoring) in others (1, 5–10, 12). Furthermore, the reason for new cancer therapy could impact how it is handled, e.g., receipt of therapy that treats disease progression has been considered as an event, while receipt of new cancer therapy that is given when patients are in remission (e.g., HSCT) has been considered as censoring (or competing risks; NCT02650414, NCT05037669).

eSome childhood ALL studies include subsequent malignancy (such as myelodysplastic syndromes or acute myeloid leukemia) as an event (2, 4) while others do not (3, 5, 23).

fPer study visit, or EHR, or local doctor contact.

gOr last response evaluation.

Other TTE endpoints may be also of interest for specific diseases or treatments. For example, time to B-cell recovery (BCR) in studies of CD19 CAR-T treatment for B-ALL is of interest, as indicative of loss of functional persistence of CAR-T cells. Events considered include BCR and CD19-positive relapse, while patients with CD19-negative relapse are censored (4). Patients are also censored at last contact, death, HSCT or other new cancer therapy, and reinfusion. Note that censoring patients at reinfusion is specific to time to BCR endpoint. For other endpoints in Table 1, handling of reinfusion varies across studies.

Handling competing risks

Two issues worth noting for the standard KM analysis are: first, when competing risks are treated as censoring, the KM method estimates the cumulative probability of an event in a hypothetical setting where competing risks do not exist (i.e., marginal survival); second, it assumes independent censoring, that censoring is not related to the risk of having the event, or in other words, if and when a patient is censored is not associated with when the patient has the event. While both conditions are reasonable for OS, where only the end of planned follow-up censors the outcome, they may not be reasonable for other endpoints. For example, if RFS is defined such that only relapse/disease-related death is considered as an event and unrelated death is treated as censoring, then the KM method estimates the cumulative probability of a relapse/disease-related death in a world where unrelated death does not exist, which is a theoretical quantity of questionable relevance. The KM curve for RFS can still be inappropriate if one treats unrelated death as an event but treats receipt of HSCT and other cancer therapy as censoring, because the independent censoring assumption might be violated. Some might argue that this KM analysis is valid because it is unclear whether the occurrence of HSCT (censoring) is related to the risk of relapse (and death); however, not knowing whether there is a relation is not the same as there is no relation, and it's possible that patients with better health condition (and thus a lower hazard of relapse/death) are more likely to receive HSCT. Thus, KM curves that treat clinical events, such as HSCT, as censoring could be misleading, as we demonstrate in an example later.

Alternatively, one can consider approaches that use a competing risk framework. A competing risk is defined as an event whose occurrence precludes the observation of the primary event of interest. In some settings, it is obvious that a competing risk exists (e.g., unrelated death is a competing risk for disease-related death), but it can be ambiguous in other settings. For example, because relapse may still occur after HSCT, one may think that HSCT does not ‘preclude’ the observation of relapse and thus does not meet the definition of a competing risk. However, relapse after HSCT could be viewed as a different kind of relapse, while the event of interest is the relapse without HSCT; therefore, the occurrence of HSCT does preclude observation of the relapse event of interest. Similarly, relapse and NRD could be considered as competing risks to each other. For the endpoints in Table 1, subjects who reach the end of their follow-up without having any event post-infusion are the only subjects for whom censoring is not impacted by a competing risk. Specifically, HSCT and other new cancer therapy are potential competing risks for endpoints, such as PFS, not only because subjects are typically taken off study at the time of referral/receipt of these therapies but more importantly those who received HSCT may have different risks of having events from those who did not. Though censoring and applying standard survival analysis appear convenient, there are many other analytical methods available to directly handle competing risks.

The competing risks framework has been extensively discussed in the statistical literature (15, 24–28) and several tutorials have appeared in the medical literature (16, 17, 29–34). A cumulative incidence function (CIF) can be generated for the event of interest and each competing event, instead of a single KM curve (29). Unlike the KM curve, the CIFs represent the separate cumulative probabilities of having had the primary event, a competing event(s), and no events in the presence of each other, and these three (or more) probabilities sum to 1. As an example, we consider the analysis of EFS (event includes relapse and nonresponse) using data from a published CAR-T study (6), which treated 35 adults with relapsed or refractory B-ALL. We apply two approaches: censoring patients at HSCT (using 1-KM for incidence estimate) or treating HSCT as a competing risk (CIF method). The KM and CIF methods yield substantially different results, with a 24-month incidence estimate of 0.83 [95% confidence interval (C), I 0.47–0.94] versus 0.64 (95% CI, 0.43–0.79) for KM versus CIF methods (Fig. 1). It is important to note that CIF curves can be more complex to interpret because the CIF for the primary event is influenced by the hazards of the primary event and competing event. A CIF for the primary event may appear to be low if the hazard of the competing event is high. Conversely, if a treatment reduces the hazard of the primary event, it then can be associated with an increased incidence of the competing event simply because there are more treated patients at risk for the competing event.

Figure 1.

Cumulative incidence of relapse/nonresponse, for the KM method by censoring at HSCT and using 1-KM to estimate the incidence, and CIF method by treating HSCT as a competing risk.

Figure 1.

Cumulative incidence of relapse/nonresponse, for the KM method by censoring at HSCT and using 1-KM to estimate the incidence, and CIF method by treating HSCT as a competing risk.

Close modal

Two popular regression models are also useful for estimating a treatment effect in the presence of competing risks: the cause-specific hazard (csH) model (27) and the Fine and Gray sub-distribution hazard (subH) model (26). Key elements and differences between these two models are summarized in Table 2. There is a misconception that the Fine and Gray model needs to be used whenever competing risks are present. However, the csH model is more appropriate for addressing etiologic/biological questions, which are generally the focus for a clinical trial (15–17, 24, 28, 35). Specifically, cause-specific hazard ratio (csHR) estimates the direct effect of the treatment/covariate on instantaneous risk of the primary event for those at risk; whereas sub-distribution hazard ratio (subHR) reflects a combination of the treatment's effect on the primary event and on the competing risk because both of these effects impact the probability of observing the primary event by a given time. Again using the data from Frey and colleagues (6), we apply both models to compare the risk of relapse/nonresponse between two infusion strategies (single-infusion N = 15 vs. fractionated-infusion N = 20), while treating HSCT as a competing risk. The estimated csHR is 6.09 (95% CI, 2.25–16.46; P = 0.004), suggesting that compared with the fractional-infusion strategy, the single-infusion strategy is associated with a significantly higher csH of nonresponse/relapse (Supplementary Table S1). The estimated subHR is 4.23 (95% CI, 1.8–9.94; P = 0.001), suggesting that compared with the patients treated with the fractional-infusion strategy, the patients treated with single-infusion strategy have significantly higher cumulative incidence of nonresponse/relapse. Although both estimates suggest a significant effect of infusion strategy, the magnitude of the effect is much stronger (44% increase) for csHR than subHR. It should also be emphasized that these two HRs are estimating different quantities; namely, the subHR cannot be interpreted as an effect of the infusion strategy on nonresponse/relapse alone, whereas the csHR can.

Table 2.

Comparison of the csH and Fine and Gray regression models for competing risks analysis.

csH modelFine and Gray subH model
Hazard modeled csH: hazard of event of interest occurring in those still at risk (i.e., those not yet censored or experience any type of event) subH: hazard of event of interest occurring in those still at risk and those who already experienced the competing event 
Measure of covariate effect 
  • csHR; Interpretable;

  • csHR estimates the direct effect of the covariate on instantaneous risk of the primary event.

 
  • subHR;

  • Not directly interpretable;

  • Better to interpret as the covariate effects on the risk of having the event of interest at the end of the study (i.e., cumulative incidence);

  • subHR reflects a combination of the covariate's effect on the primary event and on the competing risk.

 
Questions addressed csHR addresses etiologic/biological questions, such as whether or not a specific treatment reduces the relative hazard of an event subHR addresses prognostic questions, such as whether the treated patients are less likely to ever experience the event of interest 
Example csHR = 1, suggesting no direct treatment effect on the hazard of the outcome subHR < 1, suggesting treated patients have lower cumulative incidence of the event of interest than the control patients, because they have higher hazard of the competing event and thus have less opportunities to experience the event of interest 
Notes csH model has the same form as the Cox PH model, but the interpretation of the resulting estimated HR (i.e., cause-specific) is different due to the competing risks  
csH modelFine and Gray subH model
Hazard modeled csH: hazard of event of interest occurring in those still at risk (i.e., those not yet censored or experience any type of event) subH: hazard of event of interest occurring in those still at risk and those who already experienced the competing event 
Measure of covariate effect 
  • csHR; Interpretable;

  • csHR estimates the direct effect of the covariate on instantaneous risk of the primary event.

 
  • subHR;

  • Not directly interpretable;

  • Better to interpret as the covariate effects on the risk of having the event of interest at the end of the study (i.e., cumulative incidence);

  • subHR reflects a combination of the covariate's effect on the primary event and on the competing risk.

 
Questions addressed csHR addresses etiologic/biological questions, such as whether or not a specific treatment reduces the relative hazard of an event subHR addresses prognostic questions, such as whether the treated patients are less likely to ever experience the event of interest 
Example csHR = 1, suggesting no direct treatment effect on the hazard of the outcome subHR < 1, suggesting treated patients have lower cumulative incidence of the event of interest than the control patients, because they have higher hazard of the competing event and thus have less opportunities to experience the event of interest 
Notes csH model has the same form as the Cox PH model, but the interpretation of the resulting estimated HR (i.e., cause-specific) is different due to the competing risks  

We present another example comparing results of the two competing risks regressions, using a CAR-T combined cohort including children and young adults with relapsed/refractory B-ALL treated in one of five clinical trials (2–4, 23, 36) conducted at the Children's Hospital of Philadelphia (CHOP). This CHOP CAR-T cohort has been used for evaluating outcomes of patients with central nervous system relapse/refractory B-ALL (37) and assessing impact of high-risk cytogenetics on CAR-T therapy efficacy (38). A full analysis for this cohort is currently underway and therefore for illustrations in this paper we took a random sample of 103 individuals (50% from the full cohort) and did not formally evaluate/adjust for potential confounders. Suppose, we are interested in comparing patients with refractory disease versus non-refractory disease on the time to nonresponse/relapse/death, while considering receipt of HSCT, receipt of other new cancer therapy, and occurrence of subsequent malignancy as competing risks. With our random sample, 44 patients had refractory disease and 59 had non-refractory disease at baseline. In the csH regression model, the estimated csHR is 2.29 (95% CI, 1.18–4.44; P = 0.014), suggesting that compared with non-refractory disease, refractory disease status is associated with a significantly higher csH of nonresponse/relapse/death (Supplementary Table S1). The estimated subHR is 1.51 (95% CI, 0.8–2.85; P = 0.209), suggesting that compared with the patients with non-refractory disease, the patients with refractory disease do not have a significantly higher cumulative incidence of nonresponse/relapse/death. This is because that although refractory disease is associated with a significantly higher hazard of nonresponse/relapse/death, it is also associated with a higher hazard of the competing risks, so that the cumulative incidence of nonresponse/relapse/death appears to be similar in the two disease groups. Again, subHR reflects a combination of the covariate's effect on the primary event (nonresponse/relapse/death) and on the competing risks (HSCT, other new cancer therapy, and subsequent malignancy). In this example, the strength of evidence for the covariate's effect is different depending on the regression model used. Similar observations have also been noted in pervious publications in cancer studies (33) and veterinary studies (16).

More consideration for handling intercurrent events such as HSCT

Our prior discussion on handling HSCT is for the studies with no planned follow-up after HSCT. In studies where follow-up continues after HSCT, other analytical options can be considered. Table 3 summarizes four possible strategies. The first two strategies are for the setting where follow-up stops at the time of HSCT. In Strategy 1, patients are censored at the time of HSCT, and the estimand (the quantity a study wants to estimate) is survival or probability an event occurs under CAR-T therapy in a hypothetical setting where HSCT does not exist. A standard KM analysis in this case has the implicit assumption of independent censoring and could be inappropriate as discussed above, thus is not generally recommended. In Strategy 2, HSCT is treated as a competing risk and no independent censoring assumption for HSCT is imposed. This strategy estimates the cumulative incidence of the event without HSCT. If follow-up does not stop at HSCT and events occurring post HSCT are collected, two additional strategies could be considered. In Strategy 3, HSCT is simply ignored in the outcome definition and KM analysis is then used. This strategy is appropriate if there is no interest in separating the effects associated with post–CAR-T treatments like HSCT, with the purpose to estimate the survival probability for a “treatment strategy” of receiving either CAR-T or CAR-T plus HSCT. In this strategy, patients who undergo HSCT and those who do not are not distinguished and the efficacy of CAR-T therapy alone cannot be assessed. In contrast, Strategy 4 is for the setting where the effect of HSCT is of interest, so the goal is to provide an estimate or compare survival probability with versus without HSCT. We discuss this in more detail in the next section. It is important to note that the estimands in the four strategies are all different and thus the appropriate choice is tied to the study question of interest. Because published studies (CAR-T or other therapy) use various strategies, the results are not directly comparable.

Table 3.

Various strategies for handling HSCT.

Strategy 1Strategy 2Strategy 3Strategy 4
Censor at HSCTHSCT as a competing riskIgnore HSCTConsider HSCT as a TVC
Study design; follow-up Stops at HSCT Stops at HSCT Continues after HSCT Continues after HSCT 
Estimand (the quantity a study wants to estimate) Survival/event probability in a hypothetical setting where HSCT does not exist Cumulative incidence of event without HSCT Survival/event probability under a treatment strategy (CAR-T plus possible HSCT) Survival/event probability with vs. without HSCT (CAR-T vs. CAR-T plus HSCT) 
Statistical method KM analysis CIF curve KM analysis Extended KM curve for TVC 
Implication Standard KM analysis assumes the occurrence of HSCT is not related to the hazard of the event; Cannot estimate survival post HSCT Cannot estimate survival post HSCT No interest in separating patients by post–CAR-T treatments Can estimate HSCT effect on survival 
Strategy 1Strategy 2Strategy 3Strategy 4
Censor at HSCTHSCT as a competing riskIgnore HSCTConsider HSCT as a TVC
Study design; follow-up Stops at HSCT Stops at HSCT Continues after HSCT Continues after HSCT 
Estimand (the quantity a study wants to estimate) Survival/event probability in a hypothetical setting where HSCT does not exist Cumulative incidence of event without HSCT Survival/event probability under a treatment strategy (CAR-T plus possible HSCT) Survival/event probability with vs. without HSCT (CAR-T vs. CAR-T plus HSCT) 
Statistical method KM analysis CIF curve KM analysis Extended KM curve for TVC 
Implication Standard KM analysis assumes the occurrence of HSCT is not related to the hazard of the event; Cannot estimate survival post HSCT Cannot estimate survival post HSCT No interest in separating patients by post–CAR-T treatments Can estimate HSCT effect on survival 

Examples in CAR-T studies

The effect of a post-infusion exposure on treatment efficacy is of particular interest in CAR-T studies. One example commonly arises in the setting of ALL; after a CAR-T induced remission, subjects may receive a HSCT, frequently within the first year after infusion (5, 6), and a fundamental question is whether HSCT improves survival. In another example, for B-ALL, one might want to estimate the association between the occurrence of post-infusion BCR and relapse (1, 4). Other studies also are interested in comparing survival for responders versus nonresponders (9, 11, 12) where the response is determined post-infusion. We discuss three possible approaches for estimating the effect of time-varying (post-infusion) exposures.

Possible analytical approaches for time-varying exposures

Three analytical approaches are summarized in Table 4. The “naïve” approach includes all patients, defines the exposed group as those who ever had the exposure during follow-up, and the time origin for the TTE endpoint remains as the time of infusion. The exposure is analyzed as if it is a baseline variable (a variable defined at the time of infusion), so standard analyses could be applied. However, this approach introduces immortal person-time bias, caused by using future information to define a variable in the past (18–22). Because the patients must survive (or be event-free), until they experience the exposure, this approach artificially makes the exposed patients appear to have better survival than the unexposed. This could lead to an incorrect conclusion of a protective exposure effect when there is truly no effect (such as falsely concluding that HSCT prolongs survival), or an incorrect conclusion of no exposure effect while there is truly a positive association (such as falsely concluding that BCR is not associated with relapse). Despite this bias, the naïve approach is still used due to its simplicity, even in publications in top-tier journals (1, 5, 9, 12).

Table 4.

Possible analytical approaches for estimating the effect of a post-infusion exposure/covariate on TTE endpoints.

Approach 1Approach 2Approach 3
Naïve ever/neverLandmarkExposure as TVC
Analytic cohort All patients Those who are still event-free and under observation by the landmark time All patients 
Time origin Infusion Landmark time Infusion 
Exposure group Defined by ever having the exposure using information obtained post-infusion Defined by ever having the exposure by the landmark time Time-varying; a person moves to the exposure group at the time the exposure occurs 
Analysis method When competing risks absent:
  • KM

  • Cox PH model

 
When competing risks absent:
  • KM

  • Cox PH model

 
When competing risks absent:
  • Extended KM

  • Extended Cox PH model w. TVC

 
 When competing risks present:
  • CIF

  • csH model

  • Fine and Gray model

 
When competing risks present:
  • CIF

  • csH model

  • Fine and Gray model

 
When competing risks present:
  • csH model w. TVC

  • Fine and Gray model w. TVC

 
Estimate of the exposure effect Biased estimate of the effect on the risk of outcome after the infusion due to immortal person-time bias; Unbiased estimate of the effect of exposure pre-landmark on the risk of outcome after the landmark conditional on being event-free and censoring-free at the landmark Unbiased estimate of the effect on the risk of outcome after the infusion 
 The immortal person-time refers to the time between infusion and the occurrence of exposure, among the patients who eventually get exposed.   
Notes Should be avoided. Choice of landmark time can complicate the interpretation of the estimated exposure effect if there are too many additional individuals with a post-landmark exposure, or too-many excluded individuals who failed or were censored before the landmark time. More complex analytically but most appropriate for exposure effect of interest. 
 Bias increases with increasing risk of events shortly after infusion May be acceptable given its simplicity, if the exposure occurs in a short time window with few intercurrent events Advantages include allowing the exposure status to change more than one time (e.g., from unexposed to exposed and then unexposed again), and being able to evaluate the exposure effects within a given time interval (e.g., BCR within 6 months) by considering a patient exposed only if the exposure occurs within 6 months and not counting any exposure afterwards. 
   Caution is needed, for extended KM method due to its problematic causal interpretation (41), and for Fine and Gray model w. TVC because some TVCs are not measurable after competing risks occurred (43). 
Approach 1Approach 2Approach 3
Naïve ever/neverLandmarkExposure as TVC
Analytic cohort All patients Those who are still event-free and under observation by the landmark time All patients 
Time origin Infusion Landmark time Infusion 
Exposure group Defined by ever having the exposure using information obtained post-infusion Defined by ever having the exposure by the landmark time Time-varying; a person moves to the exposure group at the time the exposure occurs 
Analysis method When competing risks absent:
  • KM

  • Cox PH model

 
When competing risks absent:
  • KM

  • Cox PH model

 
When competing risks absent:
  • Extended KM

  • Extended Cox PH model w. TVC

 
 When competing risks present:
  • CIF

  • csH model

  • Fine and Gray model

 
When competing risks present:
  • CIF

  • csH model

  • Fine and Gray model

 
When competing risks present:
  • csH model w. TVC

  • Fine and Gray model w. TVC

 
Estimate of the exposure effect Biased estimate of the effect on the risk of outcome after the infusion due to immortal person-time bias; Unbiased estimate of the effect of exposure pre-landmark on the risk of outcome after the landmark conditional on being event-free and censoring-free at the landmark Unbiased estimate of the effect on the risk of outcome after the infusion 
 The immortal person-time refers to the time between infusion and the occurrence of exposure, among the patients who eventually get exposed.   
Notes Should be avoided. Choice of landmark time can complicate the interpretation of the estimated exposure effect if there are too many additional individuals with a post-landmark exposure, or too-many excluded individuals who failed or were censored before the landmark time. More complex analytically but most appropriate for exposure effect of interest. 
 Bias increases with increasing risk of events shortly after infusion May be acceptable given its simplicity, if the exposure occurs in a short time window with few intercurrent events Advantages include allowing the exposure status to change more than one time (e.g., from unexposed to exposed and then unexposed again), and being able to evaluate the exposure effects within a given time interval (e.g., BCR within 6 months) by considering a patient exposed only if the exposure occurs within 6 months and not counting any exposure afterwards. 
   Caution is needed, for extended KM method due to its problematic causal interpretation (41), and for Fine and Gray model w. TVC because some TVCs are not measurable after competing risks occurred (43). 

To address this issue, the landmark method was proposed (18) and has been applied in some CAR-T studies (6, 11). In this approach, a fixed time after infusion is selected as the landmark time, which becomes the new origin for the TTE endpoint. Patients who experienced the event or were censored before the landmark are excluded from the analysis and any findings are only applicable to the sub-cohort included in the analysis. The exposure group is defined by the information accumulated up to the landmark and any exposure changes occurring afterwards are ignored. Standard analyses could be used, and an unbiased estimate can be obtained for the effect of exposure pre-landmark on the risk of developing the outcome after the landmark, conditional on being event-free and censoring-free at the landmark. One key element is the choice of landmark time. If the landmark time is too early, then there may be many individuals exposed after the landmark but are grouped into the unexposed group. However, if the landmark time is too late, many individuals with early events pre-landmark are excluded, which result in a loss of statistical power and inability to generalize the findings. Therefore, the landmark time should be carefully chosen in the context of the specific study as a compromise of these two issues. With different landmark times, the numbers of patients included in the analyses are different and thus the interpretations of the results are also different. One scenario where landmark analysis is useful is when the exposure occurs in a relatively short time after infusion, in which case the two issues above become minimal. For example, in the comparison of survival for responders versus nonresponders in studies of CAR-T therapy for ALL, the responder status is determined by 28 days after infusion. With Day 28 selected as the landmark time, no patient has post-landmark exposure (responder status), and most patients are maintained in the analysis (because death or censoring before Day 28 is rare).

A third approach is to consider the exposure as a time-varying covariate (TVC). As a TVC, the exposure status of an individual changes from unexposed before the exposure occurs, to exposed afterwards. Because the exposure status is not defined using future exposure and is allowed to change over time, the immortal person-time bias is avoided. The time origin for the TTE endpoint remains at the infusion and analyses include all patients. When competing risks are absent, an extended KM method (39), log-rank test, and Cox proportional hazard (PH) model (40) have been proposed. With the extended KM method, a patient's contribution to KM estimates shifts from the unexposed group to the exposed group at the time of exposure. Therefore, in contrast to the standard KM curve where the number of patients at risk always decreases over time, in the extended KM curve, the number of patients at risk in the exposed group could increase. However, this approach has been criticized for its problematic causal interpretation, so caution is needed when using it (41). Similarly, the extended Cox model accommodates the TVC by incorporating the two different periods (exposed and unexposed) of the exposed patients, and the corresponding event status during those periods. For specific case of a TVC that is binary and only changes status once during follow-up, one could also generate survival curves based on the fitted extended Cox model (42). When competing risks are present, the extension of the standard CIF for a TVC is ongoing and not yet available. However, the csH model can easily incorporate a TVC in a similar manner to the Cox PH model, although generating CIF curves based on the fitted csH model is challenging (43). The Fine and Gray subH model also has the ability to incorporate a TVC. However, its use requires caution in this setting, because this approach requires specifying values of the TVC after a competing risk has occurred, because these subjects are kept in the risks set, but some TVCs are not measurable/meaningful for those who already had a competing risk (43–47).

Illustrative examples

We use two examples to illustrate the three approaches described above. First, we revisited the CHOP CAR-T cohort and evaluated the effect of post-infusion HSCT on OS. As noted earlier, we took a random sample of the full cohort and did not formally adjust for potential confounders. In this random sample of N = 103 patients, 29 patients received HSCT at some points during follow-up, and the time to HSCT ranges from 2 to 39 months post-infusion with a median time of 7.7 months. For the landmark approach, we conducted two versions of analyses. In the first version, we selected 8 months as the landmark time which captured 16 HSCT of a total of 29 HSCT and the patients without an HSCT by 8 months are in the no-HSCT group. Sixteen patients who died or censored by the landmark were excluded from the analysis. In the second version, we selected 15 months as the landmark time to capture more HSCT (21 of 29) but more patients (25 who died or censored by the landmark) were excluded in this version of analysis. As shown in Figure 2, the naïve approach suggests a protective effect of HSCT that approaches statistical significance (P = 0.085). In the landmark analysis V1, the KM curves start at month 8 and suggest no HSCT effect (P = 0.849); in the landmark analysis V2, the KM curves start at month 15 and show little to no evidence of a difference between the groups, with an observed OS that appears slightly worse for HSCT group (P = 0.212). Note, the two versions of landmark analysis yield different results, demonstrating that choice of landmark time and patient exclusion in landmark analysis can influence the findings. The TVC analysis includes all patients and thus is more applicable to the general study population than the landmark analysis. The results from the TVC approach are similar to those from the landmark analyses, suggesting no association between HSCT and OS (P = 0.689), in contrast to the naïve approach that showed an artificial protect effect of HSCT due to the immortal time bias. Results of the Cox regression models were summarized in Supplementary Table S2. In summary, the naïve Cox approach, which suffers from immortal bias, suggests that HSCT is associated with a better OS (HR = 0.49; 95% CI, 0.22–1.12; P = 0.092), while other regression approaches suggested no associations.

Figure 2.

Estimated survival curves of OS by HSCT, using CHOP CAR-T cohort. A, Naïve approach, N = 103. B, Landmark approach V1, with 8 months as the landmark time, N = 87. C, Landmark approach V2, with 15 months as the landmark time, N = 78. D, TVC approach, N = 103.

Figure 2.

Estimated survival curves of OS by HSCT, using CHOP CAR-T cohort. A, Naïve approach, N = 103. B, Landmark approach V1, with 8 months as the landmark time, N = 87. C, Landmark approach V2, with 15 months as the landmark time, N = 78. D, TVC approach, N = 103.

Close modal

The second example considers data from a trial of huCART19, a humanized CD19 CAR-T product, in children and young adults with relapsed/refractory B-ALL (4). As part of the analysis, the association between post-infusion BCR and RFS was evaluated in CAR-naïve patients who respond to huCART19 (N = 40). In this analysis, receipt of HSCT or other alternative treatment are considered as competing risks (note that NRD was not a competing risk because there was no NRD in this cohort). A csH model is used to examine the association of post-infusion BCR and relapse, with the three analytical approaches described above: the naïve, landmark, and TVC approach. The landmark time of 6 months is selected on the basis of clinical interest. A total of 15 of 40 patients had BCR, but only 8 had BCR before 6 months; the remaining 7 patients who had BCR after 6 months are included in the ‘No BCR’ group in the landmark analysis and BCR post 6 months is ignored. Nine patients are excluded for the landmark analysis (5 had relapse and 4 had competing events pre-landmark), and the csHR is not estimable because ‘BCR’ group had no events post-landmark. The TVC approach suggests a very strong positive association between BCR and relapse (csHR = 4.6; 95% CI, 1.1–18.8; P = 0.034), but the naïve approach fails to identify this association due to the immortal time bias (csHR = 1.7; 95% CI, 0.5–5.5; P = 0.380; Supplementary Table S2).

Stata and R markdown code for the analyses are provided in the Appendix.

In this paper, we review statistical issues encountered in the analyses of TTE endpoints in CAR-T oncology trials, including handling of different types of clinical outcomes in endpoint definitions, alignment of the analytical approach to the scientific question of interest when there are competing risks, and estimation of the effect of a time-varying (post-infusion) exposure. Although various methods have been proposed in the statistical literature, applying them in the context of CAR-T trials and matching the appropriate statistical method with the scientific question is not always straightforward. As we demonstrated in the examples, these issues are not well understood and thus compromised approaches have been used in the field in the past. This paper aims to raise awareness of these issues and provides practical guidance for researchers to ensure transparency, validity, and reproducibility of analyses of TTE endpoints in CAR-T studies. Many of the analytical issues that arose in the CAR-T trials occur more generally in clinical oncology studies. We also provide code for implementing these methods in standard statistical software.

We discussed why some events traditionally treated as censoring, such as HSCT, are better treated as competing risks for certain scientific questions of interest. Recent publications have started to consider competing risks in their analyses (4, 8). When reporting the results, CIF should be presented for both the primary and the competing event, as the CIF estimate is influenced by the hazard of the competing event. Moreover, there is a common misconception that the Fine and Gray model should be used for the competing risks regression, and this has even been requested by journal reviewers in the authors’ experience. However, the choice of regression model should be dictated by the scientific question. We noted that the csH model is a better choice for most study questions in a clinical trial setting, where the direct etiologic effect of a therapy on the hazard of an event is of primary interest (15–17, 24, 28, 35). A related issue is the power calculation when designing the trial, if competing risks to the primary event of interest are anticipated. Statistical methods have been proposed for testing hypothesis either about csHR (48, 49) or subHR (50, 51). Power calculation based on csHR is implemented in PASS (52) and it requires users to specify not only a csHR but also the cumulative incidence of the primary and competing events. The calculated power is typically different for the method based on csHR versus subHR (53), and which method yields higher power depends on the specific setting of treatment's association with the primary and competing events. Thus, it is important to choose the power calculation method that matches with the study question and the planned analysis. Finally, although we focused on competing risks analysis as the alternative approach when independent censoring assumption is questionable, it is not the only approach. When the study is interested in estimating/comparing marginal survival, advanced methods that use inverse probability weighting or multiple imputations could also be used to account for nonindependent censoring (34).

For estimating the effect of a post-infusion exposure, a naïve approach that treats the time-varying exposure as if it is measured at baseline is often used in this field, but this approach could provide misleading results, as illustrated in the examples. The landmark approach is useful but has limitations, especially when the outcome can occur quickly, and the timing of the post-baseline exposure is heterogeneous. The TVC approach is therefore recommended. When competing risks are absent, both survival curves and regression techniques for TVC are established and can be implemented in standard software. However, the extended KM curve approach with TVC is controversial, as its causal interpretation is problematic (41) and it relies on strong technical assumptions (54). Also, the extended KM approach can be used only for certain types of TVC; some recent work has been published to generate survival curves for subjects with more complicated TVC patterns (55). When competing risks are present, regression techniques for TVC are available, but constructing the CIF curve with TVC is more complicated. We are not aware of any existing method or software to accomplish this and work in this area is ongoing. Finally, for power calculation with TVC, some work has been published recently (56, 57) although the proposed methods are not yet available in established software.

The concepts and principles we discussed are applicable to studies of other treatments and diseases beyond those that arise in CAR-T trials. This paper focuses on statistical considerations for analyses of TTE endpoints, and there are other important issues related to designs of clinical trials, such as intent-to-treat versus as-treated analysis, a prior-defined versus post hoc analysis, etc. Finally, in the recent statistical literature, there has been discussion that some of the standard survival analyses do not have a causal interpretation, including the standard HR from a Cox model (58–60). Research is ongoing in this area; for example, for estimating causal survival functions in the setting of TVC, some causal inference approaches have been proposed (41, 61–63). A detailed discussion of this topic is beyond the scope of this paper. In all cases, care should be taken in identifying the statistical method that best applies to the specific clinical setting and addresses the scientific question of interest.

S.L. Maude reports grants and personal fees from Novartis Pharmaceuticals and Wugen outside the submitted work; in addition, S.L. Maude has a patent for PCT/US2017/044425: Combination Therapies of Car and PD-1 Inhibitors pending and licensed to Novartis Pharmaceuticals. D.T. Teachey reports grants and personal fees from Beam Therapeutics; grants from NeoImmuneTech; and personal fees from Sobi during the conduct of the study; in addition, D.T. Teachey has a patent for Biomarkers for cytokine release syndrome and CART-38 for hematologic malignancies pending. N.V. Frey reports other support from Novartis and Kite and other support from Sana Biotechnology outside the submitted work. D.L. Porter reports grants and other support from Novartis and other support from Kite/Gilead, BMS, Janssen, DeCART, Mirror Biologics, and Angiocrine Bioscience outside the submitted work; in addition, D.L. Porter has a patent for CAR T cells for CD19-positive malignancies issued, licensed, and with royalties paid from Novartis and a patent for CAR T cells for CD19-positive malignancies issued, licensed, and with royalties paid from Tmunity Therapeutics. S.A. Grupp reports grants and other support from Novartis during the conduct of the study. S.A. Grupp also reports grants from Kite, Vertex, Jazz, and Servier and other support from Roche, GSK, CBMG, Eureka, Vertex/CRISPR, AmerisourceBergen, Janssen/Johnson & Johnson, Jazz, Adaptimmune, Cellectis, Juno, Vertex, Allogene, and Cabaletta outside the submitted work; in addition, S.A. Grupp has a patent for Toxicity management for antitumor activity of CARs, WO 2014011984 A1 licensed and with royalties paid from University of Pennsylvania. P.A. Shaw reports a patent for PCT/US2016/050112 licensed to Novartis. No disclosures were reported by the other authors.

Y. Li and W.T. Hwang are supported by the NIH, Cancer Center Support Grant P30-CA16520. The authors thank the clinical research teams of the Children's Hospital of Philadelphia Cancer Immunotherapy Program and the University of Pennsylvania Center for Cellular Immunotherapies.

Note: Supplementary data for this article are available at Clinical Cancer Research Online (http://clincancerres.aacrjournals.org/).

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Supplementary data