Dynamic Risk Prediction via a Joint Frailty-Copula Model and IPD Meta-Analysis: Building Web Applications

Clinical risk prediction formulas for cancer patients can be improved by dynamically updating the formulas by intermediate events, such as tumor progression. The increased accessibility of individual patient data (IPD) from multiple studies has motivated the development of dynamic prediction formulas accounting for between-study heterogeneity. A joint frailty-copula model for overall survival and time to tumor progression has the potential to develop a dynamic prediction formula of death from heterogenous studies. However, the process of developing, validating, and publishing the prediction formula is complex, which has not been sufficiently described in the literature. In this article, we provide a tutorial in order to build a web-based application for dynamic risk prediction for cancer patients on the basis of the R packages joint.Cox and Shiny. We demonstrate the proposed methods using a dataset of breast cancer patients from multiple clinical studies. Following this tutorial, we demonstrate how one can publish web applications available online, which can be manipulated by any user through a smartphone or personal computer. After learning this tutorial, developers acquire the ability to build an online web application using their own datasets.

Copula-Based Estimation Methods for a Common Mean Vector for Bivariate Meta-Analyses

Traditional bivariate meta-analyses adopt the bivariate normal model. As the bivariate normal distribution produces symmetric dependence, it is not flexible enough to describe the true dependence structure of real meta-analyses. As an alternative to the bivariate normal model, recent papers have adopted “copula” models for bivariate meta-analyses. Copulas consist of both symmetric copulas (e.g., the normal copula) and asymmetric copulas (e.g., the Clayton copula). While copula models are promising, there are only a few studies on copula-based bivariate meta-analysis. Therefore, the goal of this article is to fully develop the methodologies and theories of the copula-based bivariate meta-analysis, specifically for estimating the common mean vector. This work is regarded as a generalization of our previous methodological/theoretical studies under the FGM copula to a broad class of copulas. In addition, we develop a new R package, “CommonMean.Copula”, to implement the proposed methods. Simulations are performed to check the proposed methods. Two real dataset are analyzed for illustration, demonstrating the insufficiency of the bivariate normal model.

Time to next treatment or death as a candidate surrogate endpoint for overall survival in advanced melanoma patients treated with immune checkpoint inhibitors: an insight from the phase III CheckMate 067 trial

Background

Time to next treatment or death (TNT-D) may be a patient-relevant endpoint in patients treated with immune checkpoint inhibitors. This study investigated TNT-D as a surrogate endpoint (SE) for overall survival (OS) in previously untreated advanced melanoma patients.

Methods

Patient-level data from the 60-month results of the CheckMate 067 randomised, controlled trial were used. Analyses were carried out for nivolumab monotherapy or nivolumab with ipilimumab versus ipilimumab monotherapy. The SE 1-step validation method based on a joint frailty-copula model was used where the country of enrolment was applied to define clusters. Kendall’s τ and the coefficient of determination (R²_trial) were estimated for respective measurements of association at the individual and cluster levels. The surrogate threshold effect, the maximum threshold hazard ratio for TNT-D that would translate into OS benefit, was estimated. A leave-one-out cross-validation analysis was carried out to evaluate model robustness.

Results

Fifteen clusters of data were generated from 945 patients. For both nivolumab-containing arms, the association between TNT-D and OS was deemed acceptable at the individual level (Kendall’s τ > 0.60) and strong at the cluster level, with R²_trial fairly close to 1, with narrow confidence intervals. The estimated surrogate threshold effects were 0.61 for nivolumab versus ipilimumab and 0.49 for nivolimub + ipilimumab versus ipilimumab. Cross-validation results showed minimum variation of the correlation measures and satisfactory predictive accuracy for the model.

Conclusion

Results suggest that TNT-D may be a valuable SE in previously untreated advanced melanoma patients treated with immune checkpoint inhibitors. Surrogacy analyses considering multiple randomised controlled trials are warranted for confirming these findings.

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This is the first study to assess the surrogacy properties of TNT-D for OS in immune checkpoint inhibitor-treated patients.

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TNT-D is a clinically relevant, pragmatic and often measurable endpoint that reflects the result of a therapeutic decision.

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TNT-D appears to be a promising SE for OS in advanced melanoma patients treated with immune checkpoint inhibitors.

Conditional copula models for correlated survival endpoints: Individual patient data meta-analysis of randomized controlled trials

Correlations among survival endpoints are important for exploring surrogate endpoints of the true endpoint. With a valid surrogate endpoint tightly correlated with the true endpoint, the efficacy of a new drug/treatment can be measurable on it. However, the existing methods for measuring correlation between two endpoints impose an invalid assumption: correlation structure is constant across different treatment arms. In this article, we reconsider the definition of Kendall's concordance measure (tau) in the context of individual patient data meta-analyses of randomized controlled trials. According to our new definition of Kendall's tau, its value depends on the treatment arms. We then suggest extending the existing copula (and frailty) models so that their Kendall's tau can vary across treatment arms. Our newly proposed model, a joint frailty-conditional copula model, is the implementation of the new definition of Kendall's tau in meta-analyses. In order to facilitate our approach, we develop an original R function condCox.reg(.) and make it available in the R package joint.Cox (https://CRAN.R-project.org/package=joint.Cox). We apply the proposed method to a gastric cancer dataset (3288 patients in 14 randomized trials from the GASTRIC group). This data analysis concludes that Kendall's tau has different values between the surgical treatment arm and the adjuvant chemotherapy arm (p-value<0.001), whereas disease-free survival remains a valid surrogate at individual level for overall survival in these trials.

A joint frailty-copula model for meta-analytic validation of failure time surrogate endpoints in clinical trials

In a meta-analysis framework, the classical approach for the validation of time-to-event surrogate endpoint is based on a two-step analysis. This approach often raises estimation issues. Recently, we proposed a one-step validation approach based on a joint frailty model. This approach was quite time consuming, despite parallel computing, due to individual-level frailties used to take into account heterogeneity in the data at the individual level. We now propose an alternative one-step approach for evaluating surrogacy, using a joint frailty-copula model. The model includes two correlated random effects treatment-by-trial interaction and a shared random effect associated with the baseline risks. At the individual level, the joint survivor functions of time-to-event endpoints are linked using copula functions. We used splines for the baseline hazard functions. We estimated parameters and hazard function using a semiparametric penalized marginal likelihood method, considering various numerical integration methods. Both individual-level and trial-level surrogacy were evaluated using Kendall's tau and coefficient of determination. The performance of the estimators was evaluated using simulation studies. The model was applied to individual patient data meta-analyses in advanced ovarian cancer to assess progression-free survival as a surrogate for overall survival, as part of the evaluation of new therapy. The model showed good performance and was quite robust regarding the integration methods and data variation, regardless of the surrogacy evaluation criteria. Kendall's Tau was better estimated using the Clayton copula model compared to the joint frailty model. The proposed model reduces the convergence and model estimation issues encountered in the two-step approach.

How to use frailtypack for validating failure-time surrogate endpoints using individual patient data from meta-analyses of randomized controlled trials

Background and Objective

The use of valid surrogate endpoints can accelerate the development of phase III trials. Numerous validation methods have been proposed with the most popular used in a context of meta-analyses, based on a two-step analysis strategy. For two failure time endpoints, two association measures are usually considered, Kendall’s τ at individual level and adjusted R2 (adjRtrial2) at trial level. However, adjRtrial2 is not always available mainly due to model estimation constraints. More recently, we proposed a one-step validation method based on a joint frailty model, with the aim of reducing estimation issues and estimation bias on the surrogacy evaluation criteria. The model was quite robust with satisfactory results obtained in simulation studies. This study seeks to popularize this new surrogate endpoints validation approach by making the method available in a user-friendly R package.

Methods

We provide numerous tools in the frailtypack R package, including more flexible functions, for the validation of candidate surrogate endpoints using data from multiple randomized clinical trials.

Results

We implemented the surrogate threshold effect which is used in combination with Rtrial2 to make decisions concerning the validity of the surrogate endpoints. It is also possible thanks to frailtypack to predict the treatment effect on the true endpoint in a new trial using the treatment effect observed on the surrogate endpoint. The leave-one-out cross-validation is available for assessing the accuracy of the prediction using the joint surrogate model. Other tools include data generation, simulation study and graphic representations. We illustrate the use of the new functions with both real data and simulated data.

Conclusion

This article proposes new attractive and well developed tools for validating failure time surrogate endpoints.

Meta-analysis of clinical trials with competing time-to-event endpoints

Recommendations for the analysis of competing risks in the context of randomized clinical trials are well established. Meta-analysis of individual patient data (IPD) is the gold standard for synthesizing evidence for clinical interpretation based on multiple studies. Surprisingly, no formal guidelines have been yet proposed to conduct an IPD meta-analysis with competing risk endpoints. To fill this gap, this work details (i) how to handle the heterogeneity between trials via a stratified regression model for competing risks and (ii) that the usual metrics of inconsistency to assess heterogeneity can readily be employed. Our proposal is illustrated by the re-analysis of a recently published meta-analysis in nasopharyngeal carcinoma, aiming at quantifying the benefit of the addition of chemotherapy to radiotherapy on each competing endpoint.

An alternative trial-level measure for evaluating failure-time surrogate endpoints based on prediction error

To validate a failure-time surrogate for an established failure-time clinical endpoint such as overall survival, the meta-analytic approach is commonly used. The standard correlation approach considers two levels: the individual level, with Kendall's τ measuring the rank correlation between the endpoints, and the trial level, with the coefficient of determination R2 measuring the correlation between the treatment effects on the surrogate and on the final endpoint. However, the estimation of R2 is not robust with respect to the estimation error of the trial-specific treatment effects.

The alternative proposed in this article uses a prediction error based on a measure of the weighted difference between the observed treatment effect on the final endpoint and a model-based predicted effect. The measures can be estimated by cross-validation within the meta-analytic setting or external validation on a set of trials. Several distances are presented, with varying weights, based on the standard error of the observed treatment effect and of its predicted value. A simulation study was conducted under different scenarios, varying the number and the size of the trials, Kendall's τ and R2. These measures have been applied to individual patient data from a meta-analysis of trials in advanced/recurrent gastric cancer (20 randomized trials of chemotherapy, 4069 patients).

The distance-based measures appeared to be robust with respect to different values of simulation parameters in several scenarios (such as Kendall's τ, size and number of clinical trials). The absolute prediction error can be an alternative to the trial-level R2 for evaluation of candidate time-to-event surrogates.

One-step validation method for surrogate endpoints using data from multiple randomized cancer clinical trials with failure-time endpoints

A surrogate endpoint can be used instead of the most relevant clinical endpoint to assess the efficiency of a new treatment. Before being used, a surrogate endpoint must be validated based on appropriate methods. Numerous validation approaches have been proposed with the most popular used in a context of meta-analysis, based on a two-step analysis strategy. For two failure-time endpoints, two association measurements are usually used, Kendall's τ at the individual level and the adjusted coefficient of determination ( R t r i a l , a d j 2 ) at the trial level. However, R t r i a l , a d j 2 is not always available due to model estimation constraints. We propose a one-step validation approach based on a joint frailty model, including both individual-level and trial-level random effects. Parameters have been estimated using a semiparametric penalized marginal log-likelihood method, and various numerical integration approaches were considered. Both individual- and trial-level surrogacy were evaluated using a new definition of Kendall's τ and the coefficient of determination. Estimators' performances were evaluated using simulation studies and satisfactory results were found. The model was applied to individual patient data meta-analyses in gastric cancer to assess disease-free survival as a surrogate for overall survival, as part of the evaluation of adjuvant therapy.

surrosurv: An R package for the evaluation of failure time surrogate endpoints in individual patient data meta-analyses of randomized clinical trials

BACKGROUND AND OBJECTIVE

Surrogate endpoints are attractive for use in clinical trials instead of well-established endpoints because of practical convenience. To validate a surrogate endpoint, two important measures can be estimated in a meta-analytic context when individual patient data are available: the R_indiv² or the Kendall's τ at the individual level, and the R_trial² at the trial level. We aimed at providing an R implementation of classical and well-established as well as more recent statistical methods for surrogacy assessment with failure time endpoints. We also intended incorporating utilities for model checking and visualization and data generating methods described in the literature to date.

METHODS

In the case of failure time endpoints, the classical approach is based on two steps. First, a Kendall's τ is estimated as measure of individual level surrogacy using a copula model. Then, the R_trial² is computed via a linear regression of the estimated treatment effects; at this second step, the estimation uncertainty can be accounted for via measurement-error model or via weights. In addition to the classical approach, we recently developed an approach based on bivariate auxiliary Poisson models with individual random effects to measure the Kendall's τ and treatment-by-trial interactions to measure the R_trial². The most common data simulation models described in the literature are based on: copula models, mixed proportional hazard models, and mixture of half-normal and exponential random variables.

RESULTS

The R package surrosurv implements the classical two-step method with Clayton, Plackett, and Hougaard copulas. It also allows to optionally adjusting the second-step linear regression for measurement-error. The mixed Poisson approach is implemented with different reduced models in addition to the full model. We present the package functions for estimating the surrogacy models, for checking their convergence, for performing leave-one-trial-out cross-validation, and for plotting the results. We illustrate their use in practice on individual patient data from a meta-analysis of 4069 patients with advanced gastric cancer from 20 trials of chemotherapy.

CONCLUSIONS

The surrosurv package provides an R implementation of classical and recent statistical methods for surrogacy assessment of failure time endpoints. Flexible simulation functions are available to generate data according to the methods described in the literature.