Determination of Power and Sample Size in the Design of

Clinical Trials with Failure-Time Endpoints and Interim Analyses

There are two programs, dssp2.exe (Design of Survival Studies: Power, Version 2), which is used to simulate power and expected duration of the trial, and dess1.exe (Design of Survival Studies: Sample Size, Version 1), which finds the sample size of a test having the pre-specified power at given baseline and alternative distributions. Both programs require the following input information on the clinical trial and the test statistics.

1. Interim analysis information

(a) Number of interim analyses: an integer from 1 to 12. If 1 is entered, this means that there is only one analysis at the prescheduled end of the trial (i.e., trial cannot terminate prior to that date).

(b) Times of interim analysis: if k is entered in (1), then enter k positive numbers (separated by spaces), in increasing order of magnitude, to denote the time (in years) of the first, second, ..., kth interim analysis.

2. Basic information on the test and simulations

(a) type I-error: Input a number between 0 and 0.1.

(b) One-sided or two-sided test: Input 1 or 2.

(c) Number of simulations: We recommend 1000 for the power simulation program, and 500 for the sample size program.

(d) Seed for random number generation: The built-in seed is -123457. The user can input 0 for this built-in seed, or specify and a larger negative integer as the seed used.

3. Accrual period and accrual rate

The accrual period is the length of time in years (starting from the beginning of the trial) during which accrual occurs. The accrual rate is the number of subjects accrued per year. Their product, therefore, is the total sample size, which should not exceed 7000 (which is the value m set in the parameter line of the FORTRAN source code. For those who want to conduct simulation with accrual period * accrual rate > 7000, you can download the FORTRAN source code dssp2.f, change the parameter value m, compile the program and run the program with the newly generated executable file).

For the power simulation program, the user inputs the accrual rate so that the program simulates the power corresponding to that rate. However, for the sample size determination program, the user inputs lower and upper bounds on the accrual rate. There are usually practical constraints on the sample size in clinical trials. For example, budget considerations would lead one to a natural

upper bound on the sample size, while publication and administrative considerations would lead one to a natural lower bound on the sample size. After dividing by the accrual period, one therefore obtains upper and lower bounds on the accrual rate. The product of the upper bound and the accrual length should not exceed 7000 in our executable version of the program because of the pre-specified dimension of the data vectors. (For those who want to conduct simulation with accrual period * upper bound > 7000, you can download the FORTRAN source code dsss1.f, change the parameter value m, compile the program and run the program with the newly generated executable file.)

4. Group sequential boundary

There are 4 methods of choosing the boundary. The choice is made by typing 1, 2, 3, or 4.

Method 1 is that of Slud and Wei, for which the user has to input the values e(1), ..., e(k), where k is the number of interim analyses and e(i) is the error to be spent at the ith interim analysis. The e(i) are positive numbers whose sum must equal to the specified Type I error.

Method 2 is the ``use function'' method of Lan and DeMets. The user has to input the use function choice by typing 1, 2, or 3, in which 1 denotes the O'Brien-Fleming use function, 2 denotes the Pocock use function and 3 denotes the linear use function.

Method 3 is the Haybittle-type method, in which the user has to input a common boundary value b, which is a positive number, for the first k - 1 interim analyses. The user who is not sure how to choose b can simply type in 0, and each of the two programs will automatically reset. Moreover, either program will reset b suitably if the user has chosen too low a value of b.

Method 4 is the user-specified boundary. Here the user has to input the boundary values b(1), ..., b(k) for the k interim analyses. Each b(i) is a positive number representing the threshold whose exceedance by the normalized statistic (or by its absolute value for a two-sided test) signals stopping.

5. Choice of test statistics

(a) The statistics are to be chosen from the Beta family, which has two parameters, rho and tau. For the logrank statistic, rho = 0 = tau. For the Peto-Prentice generalized Wilcoxon statistic, rho = 1 and tau = 0. More generally, the Harrington-Fleming family of statistics has tau = 0.

(b) For the sample size determination program, the user chooses one statistic and the program determines the sample size of the test using the statistic chosen. For the power simulation program, the user can simultaneously simulate the power associated with no more than five statistics chosen. The user has to input (i) the number of statistics used, (ii) the rho's of these statistics (separated by spaces), and (iii) the tau's of these statistics.

6. Baseline survival function

The baseline (or control-group) survival function S(t) is specified by a non-increasing piecewise log-linear function as follows.

(a) The user has to input first the number, J, of intervals on which log S(t) is determined by linear interpolation from the values at the endpoints. This number can be any integer between 1 and 20.

(b) The left endpoint of the first interval is 0, with S(0) = 1. The user has to input the right endpoints of the J intervals. These are J positive numbers (separated by blank spaces) in increasing order of magnitude, with the Jth number at least as large as the time of the last interim analysis (see 6(b) above).

(c) The user then inputs the survival probabilities at the J right endpoints in the input (b). These are J numbers between 0 and 1, in decreasing order of magnitude.

7. Censoring distributions

The survival functions of the censoring (due to loss to follow-up) distributions for the control and treatment groups are specified by piecewise log-linear functions in the same way as the baseline survival function. If there is no loss in follow-up, the user can set S(T) = 1, where T is the prescheduled end of the trial, after setting number of intervals = 1 and right endpoint of the interval = T.

8. Alternative survival function

There are two ways to specify alternative (or treatment-group under the alternative hypothesis) survival function. One is to specify it as a piecewise log-linear function in the same way as the baseline survival function. The user inputs 1 if this way is chosen. The second way, for which the user inputs 2, is to specify it via the hazard ratio, which is the ratio of the alternative hazard function to the baseline survival function.

(a) In either way, the user has to input first the number, I, of intervals, followed by the right endpoints of these intervals.

(b) For the first way, the user inputs the survival probabilities at these right endpoints.

(c) For the second way, the user inputs for each interval the hazard ratio (i.e., hazard function of the alternative distribution divided by that of the baseline distribution), which is assumed to be constant over each interval. Thus, the user inputs I positive numbers (separated by blank spaces) for the hazard ratios of the successive (from left to right) intervals.

9. Noncompliance rates

The noncompliance (or crossover) rates are specified by the drop-out rate (from treatment group to control group) and drop-in rate (from the control group to treatment group). The specification of the drop-out rate is similar to that of the drop-in rate. It assumes that crossovers can only occur at specified time points.

(a) To specify the drop-in rate, the user first specifies the number of time point(s) at which drop-in occurs. This number, K, is an integer ranging from 1 to 20.

(b) The user then inputs the K time-points at which drop-in occurs, separated by blank spaces and in increasing order of magnitude.

(c) This is followed by inputting the drop-in rates at these times. The drop-in rate, which is the proportion of subjects that switch from the control group to the treatment group, is a number between 0 and 1.

(d) If there is no drop-in at all, the user can set K = 1, drop-in time = 0, and drop-in rate at that time = 0.

10. Desired power

For the sample size determination program dsss1.exe, the user has to input the desired power so that the program will find the appropriate accrual rate for the desired power. Please read the last few lines of the output file to see if the desired power has been reached. This information can be used to modify either the bounds on the accrual rate or the desired power in a rerun of the program.