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.     

Weighted log‐rank test for time‐to‐event data in immunotherapy trials with random delayed treatment effect and cure rate

A cancer clinical trial with an immunotherapy often has 2 special features, which are patients being potentially cured from the cancer and the immunotherapy starting to take clinical effect after a certain delay time. Existing testing methods may be inadequate for immunotherapy clinical trials, because they do not appropriately take the 2 features into consideration at the same time, hence have low power to detect the true treatment effect. In this paper, we proposed a piece‐wise proportional hazards cure rate model with a random delay time to fit data, and a new weighted log‐rank test to detect the treatment effect of an immunotherapy over a chemotherapy control. We showed that the proposed weight was nearly optimal under mild conditions. Our simulation study showed a substantial gain of power in the proposed test over the existing tests and robustness of the test with misspecified weight. We also introduced a sample size calculation formula to design the immunotherapy clinical trials using the proposed weighted log‐rank test.     

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.     

A new two-sample test for choosing between log-rank and Wilcoxon tests with right-censored data

When differences of survival functions are located in early time, a Wilcoxon test is the best test, but when differences of survival functions are located in late time, using a log-rank test is better. Therefore, a researcher needs a stable test in these situations. In this paper, a new two-sample test is proposed and considered. This test is distribution-free. This test is useful for choosing between log-rank and Wilcoxon tests. Its power is roughly the maximal power of the log-rank test and Wilcoxon test.     

A weighted likelihood ratio test-based chart for monitoring process mean and variability

In this paper, a new single exponentially weighted moving average (EWMA) control chart based on the weighted likelihood ratio test, referred to as the WLRT chart, is proposed for the problem of monitoring the mean and variance of a normally distributed process variable. It is easy to design, fast to compute, and quite effective for diverse cases including the detection of the decrease in variability and individual observation case. The optimal parameters that can be used as a design aid in selecting specific parameter values based on the average run length (ARL) and the sample size are provided. The in-control (IC) and out-of-control (OC) performance properties of the new chart are compared with some other existing EWMA-type charts. Our simulation results show that the IC run length distribution of the proposed chart is similar to that of a geometric distribution, and it provides quite a robust and satisfactory overall performance for detecting a wide range of shifts in the process mean and/or variability.     

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.     

Group sequential monitoring based on the weighted log‐rank test statistic with the Fleming–Harrington class of weights in cancer vaccine studies

In recent years, immunological science has evolved, and cancer vaccines are now approved and available for treating existing cancers. Because cancer vaccines require time to elicit an immune response, a delayed treatment effect is expected and is actually observed in drug approval studies. Accordingly, we propose the evaluation of survival endpoints by weighted log‐rank tests with the Fleming–Harrington class of weights. We consider group sequential monitoring, which allows early efficacy stopping, and determine a semiparametric information fraction for the Fleming–Harrington family of weights, which is necessary for the error spending function. Moreover, we give a flexible survival model in cancer vaccine studies that considers not only the delayed treatment effect but also the long‐term survivors. In a Monte Carlo simulation study, we illustrate that when the primary analysis is a weighted log‐rank test emphasizing the late differences, the proposed information fraction can be a useful alternative to the surrogate information fraction, which is proportional to the number of events. Copyright © 2016 John Wiley & Sons, Ltd.     

Sample size determination for the weighted log‐rank test with the Fleming–Harrington class of weights in cancer vaccine studies

In recent years, immunological science has evolved, and cancer vaccines are available for treating existing cancers. Because cancer vaccines require time to elicit an immune response, a delayed treatment effect is expected. Accordingly, the use of weighted log‐rank tests with the Fleming–Harrington class of weights is proposed for evaluation of survival endpoints. We present a method for calculating the sample size under assumption of a piecewise exponential distribution for the cancer vaccine group and an exponential distribution for the placebo group as the survival model. The impact of delayed effect timing on both the choice of the Fleming–Harrington's weights and the increment in the required number of events is discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

A note about the identifiability of causal effect estimates in randomized trials with non-compliance

We show that assumptions that are sufficient for estimating an average treatment effect in randomized trials with non-compliance restrict the subgroup means for always takers, compliers, defiers and never takers to a two-dimensional linear subspace of a four-dimensional space. Implications and special cases are exemplified.     

A window-limited generalized likelihood ratio test for monitoring Poisson processes with linear drifts

There is gradually increasing attention devoted to the monitoring of Poisson process due to its wide applications in industry quality control and health-care surveillance. However, most of the study focuses on the case with step shifts in Poisson means. Relatively little attention has been paid to the case with linear drifts in Poisson means. This paper extends the window-limited generalized likelihood ratio (WGLR) test from the monitoring of normal means to Poisson processes, with focus on linear drifts. The comparison results with the adaptive cumulative sum (ACUSUM) charts and the weighted CUSUM (WCUSUM) charts show that the WGLR chart generally provides better detection performance than the other alternative methods in both the zero-state and steady-state cases.     

A flexible multi‐domain test with adaptive weights and its application to clinical trials

The design of a clinical trial is often complicated by the multi‐systemic nature of the disease; a single endpoint often cannot capture the spectrum of potential therapeutic benefits. Multi‐domain outcomes which take into account patient heterogeneity of disease presentation through measurements of multiple symptom/functional domains are an attractive alternative to a single endpoint. A multi‐domain test with adaptive weights is proposed to synthesize the evidence of treatment efficacy over numerous disease domains. The test is a weighted sum of domain‐specific test statistics with weights selected adaptively via a data‐driven algorithm. The null distribution of the test statistic is constructed empirically through resampling and does not require estimation of the covariance structure of domain‐specific test statistics. Simulations show that the proposed test controls the type I error rate, and has increased power over other methods such as the O'Brien and Wei‐Lachin tests in scenarios reflective of clinical trial settings. Data from a clinical trial in a rare lysosomal storage disorder were used to illustrate the properties of the proposed test. As a strategy of combining marginal test statistics, the proposed test is flexible and readily applicable to a variety of clinical trial scenarios.     

Sample size calculation for logrank test and prediction of number of events over time

We review and compare existing methods for sample size calculation based on the logrank statistic and recommend the method of Lakatos for its accuracy and flexibility in allowing time-dependent rates of event, loss to follow-up, and noncompliance. We extend the Lakatos method to allow a general follow-up scheme, to handle non-inferiority tests, and to predict the number of events over calendar time. We apply the Lakatos method to the simple nonproportional hazard situation of delayed treatment effect to facilitate the comparison of different weighting methods and to evaluate the performance of the maximum combination tests. We use simulation studies to confirm the validity of the Lakatos method and its extensions.     

A rational sequential probability ratio test control chart for monitoring process shifts in mean and variance

The sequential probability ratio test (SPRT) chart is a very effective tool for monitoring manufacturing processes. This paper proposes a rational SPRT chart to monitor both process mean and variance. This SPRT chart determines the sampling interval d based on the rational subgroup concept according to the process conditions and administrative considerations. Since the rational subgrouping is widely adopted in the design and implementation of control charts, the studies of the rational SPRT have a practical significance. The rational SPRT chart is designed optimally in order to minimize the index average extra quadratic loss for the best overall performance. A systematic performance study has also been conducted. From an overall viewpoint, the rational SPRT chart is more effective than the cumulative sum chart by more than 63%. Furthermore, this article provides a design table, which contains the optimal values of the parameters of the rational SPRT charts for different specifications. This will greatly facilitate the potential users to select an appropriate SPRT chart for their applications. The users can also justify the application of the rational SPRT chart according to the achievable enhancement in detection effectiveness.     

A simple test for the treatment effect in clinical trials with a sequential parallel comparison design and negative binomial outcomes

In placebo‐controlled, double‐blinded, randomized clinical trials, the presence of placebo responders reduces the effect size for comparison of the active drug group with the placebo group. An attempt to resolve this problem is to use the sequential parallel comparison design (SPCD). Although there are SPCDs with dichotomous or continuous outcomes, an SPCD with negative binomial outcomes—with which investigators deal eg, in clinical trials involving multiple sclerosis, where the investigators are still concerned about the presence of placebo responders—has not yet been discussed. In this article, we propose a simple test for the treatment effect in clinical trials with an SPCD and negative binomial outcomes. Through simulations, we show that the analysis method achieves the nominal type I error rate and power, whereas the sample size calculation provides the sample size with adequate power accuracy.     

An extensive power evaluation of a novel two-sample density-based empirical likelihood ratio test for paired data with an application to a treatment study of attention-deficit/hyperactivity disorder and severe mood dysregulation

In many case-control studies, it is common to utilize paired data when treatments are being evaluated. In this article, we propose and examine an efficient distribution-free test to compare two independent samples, where each is based on paired observations. We extend and modify the density-based empirical likelihood ratio test presented by Gurevich and Vexler [7] to formulate an appropriate parametric likelihood ratio test statistic corresponding to the hypothesis of our interest and then to approximate the test statistic nonparametrically. We conduct an extensive Monte Carlo study to evaluate the proposed test. The results of the performed simulation study demonstrate the robustness of the proposed test with respect to values of test parameters. Furthermore, an extensive power analysis via Monte Carlo simulations confirms that the proposed method outperforms the classical and general procedures in most cases related to a wide class of alternatives. An application to a real paired data study illustrates that the proposed test can be efficiently implemented in practice.