An Approach to Determine Sample Size and to Allocate Sample Size for a Specific Region in a Multiregional Trial for Survival (Time-to-Event) Data under Accelerated Failure Time Model

For the time-to-event outcome, current methods for sample determination are based on the proportional hazard model. However, if the proportionality assumption fails to capture the relationship between the hazard time and covariates, the proportional hazard model is not suitable to analyze survival data. The accelerated failure time (AFT) model is an alternative method to deal with survival data. In this paper, we address the issue that the relationship between the hazard time and the treatment effect is satisfied with the AFT model to design a multiregional trial. The log-rank test is employed to deal with the heterogeneous effect size among regions. The test statistic for the overall treatment effect is used to determine the total sample size for a multiregional trial, and the proposed criteria are used to rationalize partition sample size to each region.     

Sample Size Determination and Rational Partition for A Multi- Regional Trial for Survival (Time-To-Event) Data with Unrecognized Heterogeneity that Interacts with Treatment

To shorten the drug lag or the time lag for approval, simultaneous drug development, submission, and approval in the world may be desirable. Recently, multi-regional trials have attracted much attention from sponsors as well as regulatory authorities. Current methods for sample determination are based on the assumption that true treatment effect is uniform across regions. However, unrecognized heterogeneity among patients as ethnic or genetic factor will effect patients’ survival. In this article, we address the issue that the treatment effects with unrecognized heterogeneity that interacts with treatment are among regions to design a multi-regional trial. The log-rank test is employed to deal with the heterogeneous effect size among regions. The test statistic for the overall treatment effect is used to determine the total sample size for a multi-regional trial and the consistent trend is used to rationalize partition for sample size to each region.     

A Design For Survival Studies: A Multi-regional Trial With Unrecognized Heterogeneity

A bridging study defined by ICH E5 is usually conducted in the new region after the test product has been approved for commercial marketing in the original region due to its proven efficacy and safety. However, extensive duplication of clinical evaluation in the new region not only requires valuable development resources but also delay availability of the test product to the needed patients in the new regions. To shorten the drug lag or the time lag for approval, simultaneous drug development, submission, and approval in the world may be desirable. Recently, multi-regional trials have attracted much attention from sponsors as well as regulatory authorities. Current methods for sample determination are based on the assumption that true treatment effect is uniform across regions. However, unrecognized heterogeneity among patients as ethnic or genetic factor will effect patients’ survival. Using the simple log-rank test for analysis of treatment effect on survival in studies under heterogeneity may be severely underpowered. In this article, we address the issue that the treatment effects are different among regions to design a multi-regional trial. The optimal log-rank test is employed to deal with the heterogeneous effect size among regions. The test statistic for the overall treatment effect is used to determine the total sample size for a multi-regional trial and the consistent trend and the proposed criteria are used to rationalize partition sample size to each region.     

Cancer immunotherapy trial design with delayed treatment effect

A challenge arising in cancer immunotherapy trial design is the presence of a delayed treatment effect wherein the proportional hazard assumption no longer holds true. As a result, a traditional survival trial design based on the standard log‐rank test, which ignores the delayed treatment effect, will lead to substantial loss of statistical power. Recently, a piecewise weighted log‐rank test is proposed to incorporate the delayed treatment effect into consideration of the trial design. However, because the sample size formula was derived under a sequence of local alternative hypotheses, it results in an underestimated sample size when the hazard ratio is relatively small for a balanced trial design and an inaccurate sample size estimation for an unbalanced design. In this article, we derived a new sample size formula under a fixed alternative hypothesis for the delayed treatment effect model. Simulation results show that the new formula provides accurate sample size estimation for both balanced and unbalanced designs.     

Extended Hazard Regression Model for Reliability and Survival Analysis

We propose an extended hazard regression model which allows the spread parameter to be dependent on covariates. This allows a broad class of models which includes the most common hazard models, such as the proportional hazards model, the accelerated failure time model and a proportional hazards/accelerated failure time hybrid model with constant spread parameter. Simulations based on sub-classes of this model suggest that maximum likelihood performs well even when only small or moderate-size data sets are available and the censoring pattern is heavy. The methodology provides a broad framework for analysis of reliability and survival data. Two numerical examples illustrate the results.     

Investigation of disease-free survival as a surrogate endpoint for survival in cancer clinical trials

A stochastic model wiuh exponential components is used to describe our data collected from a phase III cancer clinical trial. Criteria which guarantee that disease-free survival (DFS) can be used as a surrogate for overall survival are explored under this model. We examine several colorectal adjuvant clinical trials and find that these conditions are not satisfied. The relationship between the hazard ratio of DFS for an active treatment versus a control treatment and the cumulative hazard ratio of survival for the same two treatments is then explored. An almost linear relationship is found such that a hazard ratio for DFS of less than a threshold R corresponds to a non-null treatment effect on survival The threshold value R is determined for our colorectal adjuvant trial data. Based on this relationship, a one-sided test of equal hazard rate of survival is equivalent to a test of hazard ratio of DFS small than R This approach assumes that recurrence information is unbiasedly and accurately assessed; an assumpion which is sometimes difficult to ensure for multicenter clinical trials, particularly for interim analyses.     

Testing the proportional odds model for interval-censored data

This paper discusses the goodness-of-fit test for the proportional odds model for K-sample interval-censored failure time data, which frequently occur in, for example, periodic follow-up survival studies. The proportional odds model has a feature that allows the ratio of two hazard functions to be monotonic and converge to one and provides an important tool for the modeling of survival data. To test the model, a procedure is proposed, which is a generalization of the method given in Dauxois and Kirmani [Dauxois JY, Kirmani SNUA (2003) Biometrika 90:913–922]. The asymptotic distribution of the procedure is established and its properties are evaluated by simulation studies     

Designing cancer immunotherapy trials with delayed treatment effect using maximin efficiency robust statistics

The indirect mechanism of action of immunotherapy causes a delayed treatment effect, producing delayed separation of survival curves between the treatment groups, and violates the proportional hazards assumption. Therefore using the log‐rank test in immunotherapy trial design could result in a severe loss efficiency. Although few statistical methods are available for immunotherapy trial design that incorporates a delayed treatment effect, recently, Ye and Yu proposed the use of a maximin efficiency robust test (MERT) for the trial design. The MERT is a weighted log‐rank test that puts less weight on early events and full weight after the delayed period. However, the weight function of the MERT involves an unknown function that has to be estimated from historical data. Here, for simplicity, we propose the use of an approximated maximin test, the V₀ test, which is the sum of the log‐rank test for the full data set and the log‐rank test for the data beyond the lag time point. The V₀ test fully uses the trial data and is more efficient than the log‐rank test when lag exits with relatively little efficiency loss when no lag exists. The sample size formula for the V₀ test is derived. Simulations are conducted to compare the performance of the V₀ test to the existing tests. A real trial is used to illustrate cancer immunotherapy trial design with delayed treatment effect.     

On computing estimates of a change-point in the Weibull regression hazard model

The hazard function describes the instantaneous rate of failure at a time t, given that the individual survives up to t. In applications, the effect of covariates produce changes in the hazard function. When dealing with survival analysis, it is of interest to identify where a change point in time has occurred. In this work, covariates and censored variables are considered in order to estimate a change-point in the Weibull regression hazard model, which is a generalization of the exponential model. For this more general model, it is possible to obtain maximum likelihood estimators for the change-point and for the parameters involved. A Monte Carlo simulation study shows that indeed, it is possible to implement this model in practice. An application with clinical trial data coming from a treatment of chronic granulomatous disease is also included.     

A class of mean residual life regression models with censored survival data

When describing a failure time distribution, the mean residual life is sometimes preferred to the survival or hazard rate. Regression analysis making use of the mean residual life function has recently drawn a great deal of attention. In this paper, a class of mean residual life regression models are proposed for censored data, and estimation procedures and a goodness-of-fit test are developed. Both asymptotic and finite sample properties of the proposed estimators are established, and the proposed methods are applied to a cancer data set from a clinic trial.     

Mark-specific additive hazards regression with continuous marks

For survival data, mark variables are only observed at uncensored failure times, and it is of interest to investigate whether there is any relationship between the failure time and the mark variable. The additive hazards model, focusing on hazard differences rather than hazard ratios, has been widely used in practice. In this article, we propose a mark-specific additive hazards model in which both the regression coefficient functions and the baseline hazard function depend nonparametrically on a continuous mark. An estimating equation approach is developed to estimate the regression functions, and the asymptotic properties of the resulting estimators are established. In addition, some formal hypothesis tests are constructed for various hypotheses concerning the mark-specific treatment effects. The finite sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a data set from the first HIV vaccine efficacy trial is provided.     

Restricted mean survival time as a summary measure of time‐to‐event outcome

Many clinical research studies evaluate a time‐to‐event outcome, illustrate survival functions, and conventionally report estimated hazard ratios to express the magnitude of the treatment effect when comparing between groups. However, it may not be straightforward to interpret the hazard ratio clinically and statistically when the proportional hazards assumption is invalid. In some recent papers published in clinical journals, the use of restricted mean survival time (RMST) or τ ‐year mean survival time is discussed as one of the alternative summary measures for the time‐to‐event outcome. The RMST is defined as the expected value of time to event limited to a specific time point corresponding to the area under the survival curve up to the specific time point. This article summarizes the necessary information to conduct statistical analysis using the RMST, including the definition and statistical properties of the RMST, adjusted analysis methods, sample size calculation, information fraction for the RMST difference, and clinical and statistical meaning and interpretation. Additionally, we discuss how to set the specific time point to define the RMST from two main points of view. We also provide developed SAS codes to determine the sample size required to detect an expected RMST difference with appropriate power and reconstruct individual survival data to estimate an RMST reference value from a reported survival curve.     

Sample size re-estimation for survival data in clinical trials with an adaptive design

In clinical trials with survival data, investigators may wish to re-estimate the sample size based on the observed effect size while the trial is ongoing. Besides the inflation of the type-I error rate due to sample size re-estimation, the method for calculating the sample size in an interim analysis should be carefully considered because the data in each stage are mutually dependent in trials with survival data. Although the interim hazard estimate is commonly used to re-estimate the sample size, the estimate can sometimes be considerably higher or lower than the hypothesized hazard by chance. We propose an interim hazard ratio estimate that can be used to re-estimate the sample size under those circumstances. The proposed method was demonstrated through a simulation study and an actual clinical trial as an example. The effect of the shape parameter for the Weibull survival distribution on the sample size re-estimation is presented.     

Fitting and testing the Marshall–Olkin extended Weibull model with randomly censored data

The random censorship model (RCM) is commonly used in biomedical science for modeling life distributions. The popular non-parametric Kaplan–Meier estimator and some semiparametric models such as Cox proportional hazard models are extensively discussed in the literature. In this paper, we propose to fit the RCM with the assumption that the actual life distribution and the censoring distribution have a proportional odds relationship. The parametric model is defined using Marshall–Olkin's extended Weibull distribution. We utilize the maximum-likelihood procedure to estimate model parameters, the survival distribution, the mean residual life function, and the hazard rate as well. The proportional odds assumption is also justified by the newly proposed bootstrap Komogorov–Smirnov type goodness-of-fit test. A simulation study on the MLE of model parameters and the median survival time is carried out to assess the finite sample performance of the model. Finally, we implement the proposed model on two real-life data sets.     

Study design of single‐arm phase II immunotherapy trials with long‐term survivors and random delayed treatment effect

In the traditional study design of a single‐arm phase II cancer clinical trial, the one‐sample log‐rank test has been frequently used. A common practice in sample size calculation is to assume that the event time in the new treatment follows exponential distribution. Such a study design may not be suitable for immunotherapy cancer trials, when both long‐term survivors (or even cured patients from the disease) and delayed treatment effect are present, because exponential distribution is not appropriate to describe such data and consequently could lead to severely underpowered trial. In this research, we proposed a piecewise proportional hazards cure rate model with random delayed treatment effect to design single‐arm phase II immunotherapy cancer trials. To improve test power, we proposed a new weighted one‐sample log‐rank test and provided a sample size calculation formula for designing trials. Our simulation study showed that the proposed log‐rank test performs well and is robust of misspecified weight and the sample size calculation formula also performs well.     

Exponentiated Weibull regression for time-to-event data

The Weibull, log-logistic and log-normal distributions are extensively used to model time-to-event data. The Weibull family accommodates only monotone hazard rates, whereas the log-logistic and log-normal are widely used to model unimodal hazard functions. The increasing availability of lifetime data with a wide range of characteristics motivate us to develop more flexible models that accommodate both monotone and nonmonotone hazard functions. One such model is the exponentiated Weibull distribution which not only accommodates monotone hazard functions but also allows for unimodal and bathtub shape hazard rates. This distribution has demonstrated considerable potential in univariate analysis of time-to-event data. However, the primary focus of many studies is rather on understanding the relationship between the time to the occurrence of an event and one or more covariates. This leads to a consideration of regression models that can be formulated in different ways in survival analysis. One such strategy involves formulating models for the accelerated failure time family of distributions. The most commonly used distributions serving this purpose are the Weibull, log-logistic and log-normal distributions. In this study, we show that the exponentiated Weibull distribution is closed under the accelerated failure time family. We then formulate a regression model based on the exponentiated Weibull distribution, and develop large sample theory for statistical inference. We also describe a Bayesian approach for inference. Two comparative studies based on real and simulated data sets reveal that the exponentiated Weibull regression can be valuable in adequately describing different types of time-to-event data.     

A weighted log-rank test and associated effect estimator for cancer trials with delayed treatment effect

The standard log-rank test has been extended by adopting various weight functions. Cancer vaccine or immunotherapy trials have shown a delayed onset of effect for the experimental therapy. This is manifested as a delayed separation of the survival curves. This work proposes new weighted log-rank tests to account for such delay. The weight function is motivated by the time-varying hazard ratio between the experimental and the control therapies. We implement a numerical evaluation of the Schoenfeld approximation (NESA) for the mean of the test statistic. The NESA enables us to assess the power and to calculate the sample size for detecting such delayed treatment effect and also for a more general specification of the non-proportional hazards in a trial. We further show a connection between our proposed test and the weighted Cox regression. Then the average hazard ratio using the same weight is obtained as an estimand of the treatment effect. Extensive simulation studies are conducted to compare the performance of the proposed tests with the standard log-rank test and to assess their robustness to model mis-specifications. Our tests outperform the G^ρ,γ class in general and have performance close to the optimal test. We demonstrate our methods on two cancer immunotherapy trials.     

Cancer immunotherapy trial design with long-term survivors

Cancer immunotherapy often reflects the improvement in both short-term risk reduction and long-term survival. In this scenario, a mixture cure model can be used for the trial design. However, the hazard functions based on the mixture cure model between two groups will ultimately crossover. Thus, the conventional assumption of proportional hazards may be violated and study design using standard log-rank test (LRT) could lose power if the main interest is to detect the improvement of long-term survival. In this paper, we propose a change sign weighted LRT for the trial design. We derived a sample size formula for the weighted LRT, which can be used for designing cancer immunotherapy trials to detect both short-term risk reduction and long-term survival. Simulation studies are conducted to compare the efficiency between the standard LRT and the change sign weighted LRT.     

Accelerated failure time models for the analysis of competing risks

Competing risks are common in clinical cancer research, as patients are subject to multiple potential failure outcomes, such as death from the cancer itself or from complications arising from the disease. In the analysis of competing risks, several regression methods are available for the evaluation of the relationship between covariates and cause-specific failures, many of which are based on Cox’s proportional hazards model. Although a great deal of research has been conducted on estimating competing risks, less attention has been devoted to linear regression modeling, which is often referred to as the accelerated failure time (AFT) model in survival literature. In this article, we address the use and interpretation of linear regression analysis with regard to the competing risks problem. We introduce two types of AFT modeling framework, where the influence of a covariate can be evaluated in relation to either a cause-specific hazard function, referred to as cause-specific AFT (CS-AFT) modeling in this study, or the cumulative incidence function of a particular failure type, referred to as crude-risk AFT (CR-AFT) modeling. Simulation studies illustrate that, as in hazard-based competing risks analysis, these two models can produce substantially different effects, depending on the relationship between the covariates and both the failure type of principal interest and competing failure types. We apply the AFT methods to data from non-Hodgkin lymphoma patients, where the dataset is characterized by two competing events, disease relapse and death without relapse, and non-proportionality. We demonstrate how the data can be analyzed and interpreted, using linear competing risks regression models.