Joint modeling of progression‐free and overall survival and computation of correlation measures

Abstract

In this paper, we derive the joint distribution of progression-free and overall survival as a function of transition probabilities in a multistate model. No assumptions on copulae or latent event times are needed and the model is allowed to be non-Markov. From the joint distribution, statistics of interest can then be computed. As an example, we provide closed formulas and statistical inference for Pearson's correlation coefficient between progression-free and overall survival in a parametric framework. The example is inspired by recent approaches to quantify the dependence between progression-free survival, a common primary outcome in Phase 3 trials in oncology and overall survival. We complement these approaches by providing methods of statistical inference while at the same time working within a much more parsimonious modeling framework. Our approach is completely general and can be applied to other measures of dependence. We also discuss extensions to nonparametric inference.Our analytical results are illustrated using a large randomized clinical trial in breast cancer. KEYWORDS event history analysis, illness-death model, multistate model, randomized clinical trials INTRODUCTIONAn established way to evaluate performance of therapies in clinical trials is time-to-event endpoints. While overall survival (OS) remains the gold standard for demonstrating clinical benefit, especially in oncology, alternative endpoints such as progression-free survival (PFS), are also accepted by Health Authorities. 1,2 PFS is not only considered a surrogate for OS but may provide clinical benefit by itself, eg, by delaying symptoms or subsequent therapies. 3 Further advantages of the use of PFS are shorter trial durations and the fact that, as opposed to OS, PFS is not confounded by the use of later line, ie, post-progression therapies, and PFS is less vulnerable to competing causes of death than OS. 4 Methodology has been developed to assess surrogacy of one endpoint for another 5 and these methodologies have been applied to a wide range of indications. 6 One important aspect in the assessment of surrogacy is to quantify the correlation between the surrogate and the real endpoint. 7 This aspect has received quite some attention in the literature lately. 8,9 These two papers consider Pearson's correlation coefficient and rely on an illness-death model without recovery (IDM) to model the association between PFS and OS. To specify the underlying statistical model for the likelihood function for parametric estimation, they use a latent failure time approach. One characteristic of the latent failure time approach is that it allows for PFS events after OS, see Section 2.3. Within this model formulation, Fleischer et al 8 then derive closed formulas for the survival functions S PFS and S OS for PFS and OS as well as the correlation coefficient Corr (PFS, OS ) by assuming an exponential distribution for all transition intensities in the underlying multistate model. Li and Zhang 9 generalized these results to Weibull transition intensitie...

A unified approach to power and sample size determination for log-rank tests under proportional and nonproportional hazards

“…The process is often modeled by the "illness-death" multistate model. 44,45 The process for the control arm in Figure 1(b) is more complex. Subjects may switch to the experimental treatment after disease progression with probability p s .…”

Multistate modeling of clinical hold in randomized clinical trials

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A pseudo-values regression model for non-fatal event free survival in the presence of semi-competing risks

“…We are aware that some authors question the use of latent times in copula model and suggest applying nonparametric methods and illness-death methods to estimate the association between progression free and overall survival and to estimate conditional probabilities of non-fatal event and death (Weber and Titman 2019;Xu et al 2010;Meller et al 2019). However, the copula based models, in particular the Clayton copula, has been proved to perform well in situation where the joint model of non-fatal and fatal event free survival is a realistically reflection of the clinical context (Weber and Titman 2019).…”