Sample Size Determination For Non-inferiority Clinical Trials With Time-to-event Data

Chapter1A comparative study of sample size determination in non-inferiority clinical trials with time-to-event data under exponential, Weibull and Gompertz distribution.BackgroundThe analysis of time-to-event data has evolved into a well-established application of advanced statistical methodology in medicine. In pharmaceutical field, the survival analysis methods are mainly used for finding a therapy for cardiovascular disease or tumor.Nowadays, it is too hard to find a therapy that has more superior efficacy than recognized effective one. In many times, the researchers may want to prove that an experimental therapy is not inferior to the standard therapy, usually in this case the experimental therapy is lower incidence rate of adverse events, less expensive, more convenient absorption pattern and so on Furthermore, New treatments may also provide alternatives for people who do not respond to available therapy and there may equally effective. Therefore, the non-inferiority clinical trials (NiCT) have often been discussed and studied. When taking the planning at the beginning of a study, the calculation of sample size is one of the most important steps. A proper samples provides reasonable power, usually between80%and90%to detect a clinically meaningful difference among groups. A mistaken sample size may lead a fail clinical trial. If the sample size is too small, important effects may not be detected, whereas the sample size that is too large is wasteful of social resources (study subjects, money and time) and unethically puts more participants at risk than necessary.There were some research results published recently. Therein, the focus of Rothmann’s work was to examine issues of retaining a proportion of active-control effect, meanwhile, some design considerations, the formulation of hypotheses, statistical methodology and the interpretation of active control NiCT are also mentioned in the paper. Sample size formulae for NiCT discussed by Chow, et al. and Crisp, et al. which are the expanding of Lachin’s and Foulkes’ work from traditional hypothesis testing to NiCT, have been adopted in the softwares, such as nQuery7.0, PASS12.0, etc. The power of the formula is biased in unbalanced design study and the bias increases as two arms are more unbalanced or the projected non-inferiority margin becomes farther from1. Hence, Jung et al. proposed a more accurate formula based on non-inferiority log-rank test. Afterwards, Jung and Chow derived a generalized log-rank test and its sample size formula, which is more flexible (any survival distributions for two arms and any accrual pattern) and applicable for both superiority and non-inferiority inference. This sample size formula is identical to that of the log-ran test by Schoenfeld if hazard ratio is1under the null hypothesis and is identical to that of Jung if the hazard ratio is1under the alternative hypothesis.All of above papers assumed that the survival time distribution is exponential. Although an exponential distribution may provide a reasonable approximation to the distribution of survival times over relatively short intervals, it typically does not adequately characterize the distribution of survival times over more substantial portions of the lifespan because of its property of constant hazard(or death rate) over time. There are a few of sample size formulae considering Weibull distribution in some branches of life data analysis methods. Heo et al. derived a sample size formula for comparing two group of Weibull distributed survival time the whole lifespan is of interest. And Gope proposed a calculation method for comparison of fatigue time following Weibull distribution in technology studies. A simulation based method for calculating group sequential trials under Weibull distribution is proposed by Jiang et al. All the researchers maintained that the Weibull distribution sample size is better than exponential one. But the exponential and Weibull distribution sample size formula for the non-inferiority clinical trials with the different failure time distribution has not been discussed in the literatures. Meanwhile, Gompertz distribution is another distribution whose hazard function is proportional, so The Gompertz sample size formula was also verified.AimsDiscuss and compare the sample size formulae under the exponential distribution, Weibull distribution and Gompertz distribution. Verify the applicability of three distribution sample size formulae under various distributions.MethodsSimulating the empirical powers from three sample size formulae in some parameter combinations. The increasing, constant and decreasing hazard rate and the high, mediate and low hazard rate are considered in the Weibull survival time simulations. In the simulation of Gompertz distribution time, only the increasing hazard rate is simulated, in which various change rate of hazard rate along with the time and incidence rates are considered. The simulation method is the standard time generating method, which is generating the corresponding distribution survival time and exponential censored time, respectively. The observed time is the minimum of the two times.10000independent trials were simulated for various combinations of several parameters. If the upper limit of95%two-tailed confidence interval of hazard ratio from Cox proportional hazard regression model is less than the non-inferiority margin, the non-inferiority is concluded. ResultsIn cases that k=1, the formulae under exponential distribution and Weibull distribution are completely the same. And the power is around0.8in each combination. When k is larger than1, if censoring rate is higher(say30%), exponential formula underestimates the sample size needed and the smallest empirical power is almost equal to0.7. Sample size of Weibull distribution lead a proper number of events so that the empirical power agrees with the predetermined power.The empirical powers of Gompertz distribution sample size formula are below0.8when the censoring rate is large (say30%) and over0.8when the censoring rate is small.In the cases that survival time is Gompertz, Weibull sample size is very close to that of Gompertz, which is around0.8. when the censoring rate is higher, the empirical power of exponential sample size is below the predetermined power0.8. The smallest is around0.2in the case of the largest censoring rate.ConclusionAt the stage of planning of a study, exponential and Gompertz assumption may cause waste of resources or underestimate the sample size needed. Therefore, we suggest that it is better to make a Weibull distribution survival time assumption only if there a straightforward evidence that the survival time is differently distributed.Chapter2Sample size formula for non-inferiority clinical trials with competing risksBackgroundIn the studies that more than1event can occur, sometimes all the events can be reasonably combined in a composite endpoint as, for example, cancer-related death and non-cancer-related death can be combined into overall survival (OS). In this case, standard survival techniques, as Kaplan-Meier estimations of OS probability, can be employed. However, It has been widely mentioned in the literatures that simply using the Kaplan-Meier method for estimating survival rate of one of the events in presence of competing risks, in which the competing events are treated as censoring, would overestimate the hazard rates for the one event so that the summation of the incidence rate of all events at one time is larger than1. If the independence assumptions are held, the Cox proportional hazards model and KM estimator can still be used, but the interpretation of the results is different.Some planning methods for clinical trials in presence of competing risks were published recently. Pintile derived a sample size formula with the independence assumption of event of interest and competing causes., Latouche raise a sample size determination for proportional subdistribution hazard ratio model, which can be used for dealing with the two covariates with dependence. Then Latouche proposed another sample size formula for Renyi-type test for competing risks. Sample size for Weibull distributed survival time of event of interest was discussed in Maki’s work, in which the formulae were based on the cause-specific hazard ratio. The newest edition of PASS(?)12.0have involved the sample size calculation for log-rank test accounting for competing risks, which is based on Pintile’s work.Non-inferiority clinical trials considering competing risks (NiCTCR) are needed if we want to demonstrate the non-inferiority in the presence of competing risks. And NiCTCR have already appealed some researchers as wel1. Yet, the procedures for planning a NiCTCR have not been put forward, of which Determination of sample size is unavoidable in design procedure. So we would like to develop a sample size formula for NiCTCR in this paper.AimsPresent a formula for sample size determination in the non-inferiority clinical trials in presence of competing risks, in which confidence interval of sub-hazard ratio from Fine-Gray proportional hazard ratio model is used for inferring.MethodsSurvival time for event of interest and competing events are assumed to be Weibull distributed, which is conditional on that the corresponding event occurs, with the same shape parameter across two groups. The Fisher information is calculated for the variance of the Wald test for the parameter in the Fine and Gray proportional sub-hazard model. In order to simplify the variance, the sub-density function and censoring distribution are approximately identical across two groups under the alternative hypothesis of non-inferiority framework. Accordingly, the sample size formula is derived from the Fisher information of Wald test, in which the cumulative incidence rate are calculated by the numerical integral.The validity of the presented formula is investigated through10000numerical Monte Carlo simulations. The increasing, constant and decreasing hazard rates and the high, mediate and low hazard rates are considered in generating the survival time. The upper limit of95%confidence interval of sub-hazard ratio from the Fine and Gray model is used for testing the non-inferiority hypothesis.Resultsthe incidence rate of censoring and competing events are simulated from70%to more than20%. The dotted line represents the Type I error, which is around the0.025(0.021-0.028). With the data generate under the alternative hypothesis, all the empirical powers are approximately equal to the predetermined power0.8(0.791-0.813). So the proposed sample size formula performs very well in simulated data.ConclusionThe proposed sample size formula performs well for non-inferiority clinical trials in presence of competing risks.