Sample Size Estimation Method For Survival Trials With Truncated Gaussian Mixture Distribution Accrual

BackgroundIn survival trials with fixed trial length,patient accrual pattern affects sample size estimation by influencing the expected event rate during the trial.Influenced by many factors,the patient accrual rate may increase and decrease alternately,especially in survival trials with long enrollment period and large sample size.However,the existing accrual models can not represent fluctuated patient accrual rate,which may lead to sample size estimation bias.ObjectiveTo estimate the sample size accurately in survival trials with fluctuated patient accrual rate,in this paper,we adopted truncated Gaussian Mixture Distribution(GMD)to represent the fluctuated patient accrual rate and discussed the sample size estimation method,parameter setting and application of the truncated GMD accrual model.Methods1.Under the assumption of exponential distributed survival time and drop out time,we calculated the expected incidence of the event of interest in survival trials via triple integral when patient entry time followed truncated GMD,uniform distribution,linear distribution,piecewise constant distribution,piecewise linear distribution and truncated exponential distribution,respectively.The corresponding sample size formulas were derived further.In the context of a survival trial,we calculated the sample sizes for different patient accrual models based on the deduced formulas and generated the random numbers of patient entry time,survival time and drop out time by SAS software.Through 5000 Monte Carlo simulations,we obtained the empirical trial power under the estimated sample size and compared it with the expected trial power when calculating the sample size.2.Based on the derived sample size formula for truncated GMD accrual,we explored the impact of mean,standard deviation and weight allocation of truncated GMD on the sample size required in survival trials.3.We used SAS software to generate 500,1000,and 2000 random numbers of different distributions to simulate patient entry times of historical trials.Expectation maximization(EM)algorithm was adopted to estimate the truncated GMD parameters based on the simulated data.Then we demonstrated the goodness of fit of the estimated distribution to the simulated data by plotting the frequency distribution histogram of the simulated data and the probability density curve of the estimated truncated GMD in the same coordinate system.At the same trial background,we calculated and compared the required sample sizes in real patient accrual model and estimated truncated GMD accrual model.4.In the scenarios that patients entered into the trial late,early or relatively evenly throughout the accrual duration,we chose different patient accrual distributions inconsistent with the actual accrual model and estimated the sample size calculation bias and power change under different trial lengths.Results1.Under the assumption of exponential distributed survival time and drop out time,the sample size formulas under eight different accrual models were derived and compiled as SAS macro program %SScal.Trial designers only need to input the total trial length,accrual duration,accrual model and parameters,median survival time of the control therapy,hazard ratio,patient allocation between groups,type I error and the expected trial power,the sample size required can be obtained easily.Utilizing the written SAS macro program %power,the empirical trial powers under calculated sample sizes of different accrual models were obtained and they were all near the expected power,i.e.,90%.2.With the truncated GMD patient accrual model,the larger the mean of each Gaussian component,the larger the sample size required in survival trials.The larger the standard deviation of the Gaussian component that was more close to the left truncation point and the smaller the standard deviation of the Gaussian component that was more close to the right truncation point,the larger the sample size.The smaller the weight of the former Gaussian component,the larger the sample size.With the increase of standard deviation of each Gaussian component,the sample size of truncated GMD accrual was more and more close to that of the uniform accrual.3.For ease of use,the EM algorithm that estimated the truncated GMD parameters has been compiled as the SAS macro program %EM_trunGMD.Based on the simulated data of different sample sizes and distributions,the truncated GMD estimated by EM algorithm fit the data well.Under the same trial background,the difference between the sample sizes calculated with the real patient accrual and the fitting truncated GMD accrual was within 3%.Based on the truncated GMD that fit the historical accrual data,we proposed four strategies,i.e.,direct adoption strategy,normalization strategy,interception strategy,and custom strategy to determine the parameters of truncated GMD accrual for new designed trials.4.In survival trials that patients entered into the trial late while decreasing accrual model was adopted in the design stage,the sample size was seriously underestimated and the power of the trial was undermined.Patients entering into the trial early while increasing accrual model adopted in the design stage led to overestimated sample size and mild increase in trial power.In the scenario that patients entered into the trial relatively evenly,both of the increasing and decreasing accrual models resulted in sample size estimation bias,but the bias was smaller than that caused by the case that the actual accrual rate and the anticipant accrual rate were completely opposite in the change trend.When the accrual duration fixed,the bias of sample size estimation and the change of power caused by misjudged accrual model shrunk with the extension of the total trial length.ConclusionTruncated GMD accrual and the corresponding formula derived in this paper can be used to accurately estimate the sample size when the patient accrual rate fluctuates in survival trials with fixed trial length.In the design phase of survival trials,it is necessary to estimate the changing trend of the patient accrual rate correctly to decrease the sample size estimation biases caused by misjudged accrual models.The truncated GMD estimated by EM algorithm based on historical accrual data,along with the impact of truncated GMD parameters on sample size can provide references for parameter determining of accrual models in new trials.