Biostatistics and Clinical Trial Methodology
Insights for Quantifying the Long-Term Benefit of Immunotherapy Using Quantile Regression
2Conservatoire National des Arts et Métiers, Paris, France
3Department of Drug Development and Innovation (D3i), Institut Curie, Paris, France
Immunotherapy has been approved to treat many tumor types. However, one characteristic of this therapeutic class is that survival benefit is due to late immune response, which leads to a delayed treatment effect. Quantifying the benefit, if any, of such treatment, will thus require other metrics than the usual hazard ratio and different approaches have been proposed to quantify the long-term response of immunotherapy.
In this paper, we suggest to use quantile regression for survival data to quantify the long-term benefit of immunotherapy. Our motivation is that this approach is not trial-specific and provides clinically understandable results without specifying arbitrary time points or the necessity to reach median survival, as is the case with other methods. We use reconstructed data from published Kaplan-Meier curves to illustrate our method.
Immunotherapy has been demonstrated to improve overall survival (OS) in several tumor types.1 However, one characteristic of this new therapeutic class is that the improved OS is mainly due to the durable responses obtained in a minority of patients. Indeed, the treatment effect may depend on time caused by the late immune response. This delayed effect of immunotherapy results in a late separation of the survival curves. Thus, the usual assumption of proportional hazard between treatment arms does not hold anymore. This has led to a debate on which was the more appropriate way to handle this nonproportionality.2
Having long-term responders in the recurrent and/or metastatic setting is the ultimate goal of cancer therapy in this situation. We previously showed that long-term responders are much more frequent using immunotherapy and especially immune checkpoint inhibitors than with other drugs (Pons-Tostivint et al, JCO Precis Oncol, 3:1-10, 2019).
Different approaches have been used to quantify long-term response to immunotherapy. The proposed approach used in this paper is a readily available tool for quantifying the benefit of immunotherapy and provides clinically interpretable results without specifying arbitrary time points or the necessity to reach median survival, as is the case with other methods.
We believe that these findings will be of interest in the clinical context to make therapeutic decisions.
We previously defined long-term responders as patients having an OS exceeding twice the median OS of the whole patient population. In a meta-analysis on aggregated data, the proportion of long-term responders using this criterion was significantly higher in patients treated with immunotherapy than those treated with other anticancer drugs.3 Yet, our definition was arbitrary, similar to other proposals that we detail thereafter. To quantify long-term survival benefit, the area under the Kaplan-Meier curve from the time of separation (also termed milestone) to a given time-horizon was proposed.4 Patients receiving immunotherapy lived on average 2.5 months longer than patients treated with chemotherapy. The area under the Kaplan-Meier curves is 8.3 and 10.8 within the chosen window of 36-72 months for chemotherapy and immunotherapy, respectively. Vivot et al5 used the ratio of the late-life expectancy, defined as the ratio of the area under the Kaplan-Meier OS curve from the median survival time to the end of follow-up in each arm, to quantify the long-term benefit of immunotherapy. These approaches are adequate when the proportional hazards assumption does not hold, but they require an arbitrary choice of a landmark time, a quantile or focus on median survival times, which is not always defined.
To complement these proposals, we propose to focus on the entire conditional distribution of survival times and estimate the effect of immunotherapy on each quantile of the survival times using a censored quantile regression model.6 We suggest using quantile regression to identify the upper quantile of the survival curves and to assess the treatment effect on the difference scale. Notably, Uno et al7 suggested to use the ratio or difference of percentile of the survival function between treatment arms. In fact, the difference of percentile of the survival function is in the time scale, facilitating the quantification of the benefit of a treatment arm over another. However, the method of Uno et al7 requires a choice of an arbitrary quantile. Censored quantile regression is an alternative to simply modeling conditional mean that can model the quantile of the response variable as a linear function of covariates and allows distinguishing among differential effects of covariates across conditional quantiles and thus can capture important population heterogeneity.8
The remainder of this paper is organized as follows. First, we provide details of the quantile regression model for right-censored data, the inference, and the goodness of fit. Next, we illustrate the methodology with a reconstructed dataset from a randomized trial in patients with non–small-cell lung cancer and present the results of this application. We close with a discussion. The data supplement, which contains R code, reconstructed data, is available in ref. 9.
The τ-th quantile of the OS is the smallest time where the OS exceeds τ. With explanatory covariate, Z, the τ-th conditional quantile of T given Z is . The quantile regression model relates QT (τ|Z) linearly to Z for each 0 < τ < 1.
where β0(τ), β1(τ) are the intercept and the effects of the treatment on the τ-th quantile of T, respectively. With a binary covariate denoting the treatment arm (eg, Z = 1 for immunotherapy), the difference will be our target of interest for quantifying the benefit, if any, of immunotherapy over standard of care.
In practice, T is not always observable because of loss of follow-up. To account for the right censoring, two approaches were considered for the inference of model (Eq 1). Portnoy10 developed, under random censoring, a recursively reweighted estimation procedure which a generalization of the Kaplan-Meier estimator. Peng and Huang6 developed, under the weaker assumption of conditionally independent censoring, a method relying on the Nelson-Aalen estimator using counting processes and martingale theory. These two approaches have been implemented in the R package quantreg.11
In the sequel, we will use the Peng and Huang's estimator since their approach enables us to conduct hypothesis testing, resampling inference, and model diagnosis. In fact, testing the null hypothesis H0: β1(τ) = 0, τ ∈ [0; 1] is of interest in our setting since it corresponds to no benefit of the experimental arm at the τ-th quantile. Furthermore, one strong assumption of the model (Eq 1) is the global linear relationship between conditional quantile and the covariates. Following the study of Peng and Huang,6 the assumption of the global linearity between covariates and conditional quantile of the model can be assessed using the martingale residuals. The plot of the fitted quantile value (x-axis) versus martingale residuals (y-axis) is next used to visually check the assumptions of the linearity. We were surprised by the lack of implementation of such a graphical procedure. We thus provide a R function in the reproducible code. The technical derivations of model (Eq 1) are deferred to the supplementary.
Finally, we should mention that in the case of a single binary covariate and a given quantile, the quantile regression is equivalent to the nonparametric approach that would compare the quantile of the survival function between two groups.12,13 This approach is currently implemented in the R package surv2sampleComp.14
We used the published Kaplan-Meier survival curves to simulate individual failure times. The algorithm developed by Guyot et al15 enables to emulate the survival time from the two arms of the clinical trial of Rittmeyer et al1 that comprises 850 patients with metastatic non–small-cell lung cancer that compared immunotherapy versus chemotherapy. The reliability of this method as quantified by the hazard ratio (HR) is illustrated in Figure 1. The estimated HR from reconstructed data is compatible with the published HR. The primary end point was OS compared between treatment groups. On the reconstructed data, the average follow-up was 8.78 months, and during the follow-up, 555 deaths were observed, leading to approximately 35% of right-censored observations. This censoring rate is consistent with the 30% observed rate in the actual trial. Usually, the 0.5 quantiles of the survival (eg, median) are reported in each arm: 14.13 months in the experimental arm and 9.78 months in the control arm.
In Figure 1, the survival curve levels off around the 0.72 quantile for the immunotherapy group, which corresponds to 24 months. This means that the conditional quantile in the immunotherapy group at any level higher than 0.72 would be estimated to be infinity. Therefore, we focus on the quantile levels from 0.1 to 0.60. In our study, the upper bound of the estimable quantile of survival time is 60%, which corresponds to 18.15 months in the immunotherapy arm and 12.69 months in the chemotherapy arm. At the quantile level 0.1, we notice that the coefficient is almost null because of the overlap of the survival curves at that level of the survival time. We then estimate for each available quantile, the benefit in months of immunotherapy treatment using censored quantile regression.
Differences between treatment arms can be plotted at increasing quantile for the illustrative purpose of a potential benefit (Fig 2). Yet, this plot does not account for multiple testing issues. We thus recommend to focus on a single attainable quantile (60% in our motivating example). The model diagnostic reveals that a moderate nonlinearity is present, especially for lower and upper quantiles (the figure is available in the supplementary). Yet, for the quantile of interest, for example, τ = 0.6, the linearity assumption holds. Thus, at the quantile level 0.6 of the survival time, patients treated with immunotherapy have a survival average benefit of 5.46 months compared with patients treated with chemotherapy (95% bootstrap CI, 2.57 to 9.02). In other words, on average, patients from the immunotherapy group have 60% of chance to survive 5.46 months (95% CI, 2.57 to 9.02) more than patients in the chemotherapy group.
At quantile 0.6, the estimated benefit obtained with surv2sample() is similar to that of the quantile regression. To illustrate the usefulness of the quantile regression, we consider the setting where in addition to the effect of treatment on the quantile of the survival, the effect of the expression of the biomarker PD-L1 and its interaction are investigated. The HR of the OS by PD-L1 gene expression was estimated at 0.69 (95% CI, 0.57 to 0.83) in the Data Supplement (online only).1 The details of the simulation are deferred to the supplementary, and the estimated coefficients are summarized in Table 1.
At the quantile τ = 0.6, the estimated effect is 1.73 (95% CI, 0.41 to 3.33) for immunotherapy, which indicates that patients treated with immunotherapy with a PD-L1 value of 0 have, on average, 60% of chance to survive 1.73 months compared with the patients treated with chemotherapy with the same PD-L1 value. Furthermore, patients treated with immunotherapy and with an increase of PD-L1 expression equal to 1 have, on average 60% of chance to survive 8.28 months (1.73 + [6.55 × 1]).
There had been a considerable effort to quantify the long-term survival in the presence of delayed or reverse treatment effect, relying notably on two-sample comparison of median survival time or restricted mean survival time.4,5,7,16
In this work, we illustrate a single randomized trial with reconstructed individual failure time, the merits of the quantile regression for survival data. This is a readily available tool for quantifying the long-term survival benefit of immunotherapy and provides clinically meaningful results without specifying arbitrary time points or the necessity to reach median survival. Furthermore, it informs of the treatment effect on the span of the OS. One appeal of this method is that it allows an effect that is quantile-specific. Of note, our illustrative example did not exhibit a plateau, where a survival curve levels off. The quantile regression can be useful to estimate the quantile that announces such a plateau.
The proposed quantile regression relates the outcome linearly to the explanatory variables. This can be a limitation, and this assumption should be checked. Moreover, one of the limits of the censored quantile regression is that it may fail when the censoring rate is high.
To improve the level of evidence of the benefit of immunotherapy over chemotherapy, it would be of interest to investigate how to conduct a meta-analysis on individual patient data that would synthesize the treatment effect at the trial level on the difference scale using quantile regression. A first route would be to consider a fixed-effect approach that would weigh properly the benefit of each trial. Yet, quantile regression for clustered censored data is not available. Thus, a one-stage inference, accounting for the correlation at the trial level, is not feasible for this regression model.
Conception and design: Bassirou Mboup, Aurélien Latouche
Collection and assembly of data: Bassirou Mboup, Aurélien Latouche
Data analysis and interpretation: All authors
Manuscript writing: All authors
Final approval of manuscript: All authors
Accountable for all aspects of the work: All authors
The following represents disclosure information provided by authors of this manuscript. All relationships are considered compensated unless otherwise noted. Relationships are self-held unless noted. I = Immediate Family Member, Inst = My Institution. Relationships may not relate to the subject matter of this manuscript. For more information about ASCO's conflict of interest policy, please refer to www.asco.org/rwc or ascopubs.org/po/author-center.
Open Payments is a public database containing information reported by companies about payments made to US-licensed physicians (Open Payments).
Honoraria: Novartis, Bristol-Myers Squibb, MSD, Merck Serono, Roche, Nanobiotix, GlaxoSmithKline, Celgene, Rakuten
Consulting or Advisory Role: Amgen, MSD, Bristol-Myers Squibb, Merck Serono, AstraZeneca, Nanobiotix, GlaxoSmithKline, Roche
Travel, Accommodations, Expenses: MSD, Bristol-Myers Squibb, AstraZeneca
No other potential conflicts of interest were reported.
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