Quantifying the average of the time-varying hazard ratio via a class of transformations

Methods for non-proportional hazards in clinical trials: A systematic review

For the analysis of time-to-event data, frequently used methods such as the log-rank test or the Cox proportional hazards model are based on the proportional hazards assumption, which is often debatable. Although a wide range of parametric and non-parametric methods for non-proportional hazards has been proposed, there is no consensus on the best approaches. To close this gap, we conducted a systematic literature search to identify statistical methods and software appropriate under non-proportional hazard. Our literature search identified 907 abstracts, out of which we included 211 articles, mostly methodological ones. Review articles and applications were less frequently identified. The articles discuss effect measures, effect estimation and regression approaches, hypothesis tests, and sample size calculation approaches, which are often tailored to specific non-proportional hazard situations. Using a unified notation, we provide an overview of methods available. Furthermore, we derive some guidance from the identified articles.

A robust hazard ratio for general modeling of survival-times

Hazard ratios (HR) associated with the well-known proportional hazard Cox regression models are routinely used for measuring the impact of one factor of interest on a time-to-event outcome. However, if the underlying real model does not fit with the theoretical requirements, the interpretation of those HRs is not clear. We propose a new index, gHR, which generalizes the HR beyond the underlying survival model. We consider the case in which the study factor is a binary variable and we are interested in both the unadjusted and adjusted effect of this factor on a time-to-event variable, potentially, observed in a right-censored scenario. We propose non-parametric estimations for unadjusted gHR and semi-parametric regression-induced techniques for the adjusted case. The behavior of those estimators is studied in both large and finite sample situations. Monte Carlo simulations reveal that both estimators provide good approximations of their respective inferential targets. Data from the Health and Lifestyle Study are used for studying the relationship of the tobacco use and the age of death and illustrate the practical application of the proposed technique. gHR is a promising index which can help facilitate better understanding of the association of one study factor on a time-dependent outcome.

CauchyCP: A powerful test under non-proportional hazards using Cauchy combination of change-point Cox regressions

Non-proportional hazards data are routinely encountered in randomized clinical trials. In such cases, classic Cox proportional hazards model can suffer from severe power loss, with difficulty in interpretation of the estimated hazard ratio since the treatment effect varies over time. We propose CauchyCP, an omnibus test of change-point Cox regression models, to overcome both challenges while detecting signals of non-proportional hazards patterns. Extensive simulation studies demonstrate that, compared to existing treatment comparison tests under non-proportional hazards, the proposed CauchyCP test (a) controls the type I error better at small α levels (<0.01); (b) increases the power of detecting time-varying effects; and (c) is more computationally efficient than popular methods like MaxCombo for large-scale data analysis. The superior performance of CauchyCP is further illustrated using retrospective analyses of two randomized clinical trial datasets and a pharmacogenetic biomarker study dataset. The R package CauchyCP is publicly available on CRAN.

A marginal estimate for the overall treatment effect on a survival outcome within the joint modeling framework

Joint models for longitudinal and survival data are increasingly used and enjoy a wide range of application areas. In this article, we focus on the application of joint models on clinical trial data with special interest in the treatment effect on the survival outcome. Within a joint model, the estimated treatment effect on the survival outcome is an aggregate comprising the indirect treatment effect through the longitudinal outcome and the direct treatment effect on the survival outcome. This overall treatment effect is, however, conditional on random effects, and therefore has a subject‐specific interpretation. The conditional interpretation arises from the shared random effects between the longitudinal and survival process in combination with the nonlinear link function of the survival model. The overall treatment effect is, therefore, not valid for population‐based inference, which is the goal for most clinical trials. We propose a method to obtain a marginal estimate of the overall treatment effect on the survival outcome in a joint model. Additionally, we extend our proposal to allow for different parameterizations for the association between the longitudinal and survival outcome. The proposed method is demonstrated on data of a clinical study on the effect of synbiotic on the gut microbiota of cesarean delivered infants, where we estimate the marginal overall treatment effect on the risk of eczema or atopic dermatitis using longitudinal information on fecal bifidobacteria.

Bayesian design of clinical trials using joint models for longitudinal and time-to-event data

Joint models for longitudinal and time-to-event data are increasingly used for the analysis of clinical trial data. However, few methods have been proposed for designing clinical trials using these models. In this article, we develop a Bayesian clinical trial design methodology focused on evaluating the treatment's effect on the time-to-event endpoint using a flexible trajectory joint model. By incorporating the longitudinal outcome trajectory into the hazard model for the time-to-event endpoint, the joint modeling framework allows for non-proportional hazards (e.g., an increasing hazard ratio over time). Inference for the time-to-event endpoint is based on an average of a time-varying hazard ratio which can be decomposed according to the treatment's direct effect on the time-to-event endpoint and its indirect effect, mediated through the longitudinal outcome. We propose an approach for sample size determination for a trial such that the design has high power and a well-controlled type I error rate with both operating characteristics defined from a Bayesian perspective. We demonstrate the methodology by designing a breast cancer clinical trial with a primary time-to-event endpoint and where predictive longitudinal outcome measures are also collected periodically during follow-up.

Estimation of treatment effects and model diagnostics with two-way time-varying treatment switching: an application to a head and neck study

Treatment switching frequently occurs in clinical trials due to ethical reasons. Intent-to-treat analysis without adjusting for switching yields biased and inefficient estimates of the treatment effects. In this paper, we propose a class of semiparametric semi-competing risks transition survival models to accommodate two-way time-varying switching. Theoretical properties of the proposed method are examined. An efficient expectation-maximization algorithm is derived to obtain maximum likelihood estimates and model diagnostic tools. Existing software is used to implement the algorithm. Simulation studies are conducted to demonstrate the validity of the model. The proposed method is further applied to data from a clinical trial with patients having recurrent or metastatic squamous-cell carcinoma of head and neck.

Quantifying time-varying cause-specific hazard and subdistribution hazard ratios with competing risks data

Various non-proportional hazard models have been developed in the literature for competing risks data. The regression coefficients under these models, however, typically cannot be compared directly. We propose new methods to quantify the average of the time-varying cause-specific hazard ratios and subdistribution hazard ratios through two general classes of transformations and weight functions that are chosen to reflect the relative importance of the hazard ratios in different time periods. We further propose an L∞ -norm type of test statistic that incorporates the test statistics for all possible pairs of the transformation function and weight function under consideration. Extensive simulations are conducted under various settings of the hazards and demonstrate that the proposed test performs well under all settings. An application to a clinical trial in follicular lymphoma is examined in detail.