DEMONSTRATION I
Consider a 5-year clinical trial with uniform accrual in the
first 3 years. The interim analyses are scheduled at 1, 1.5,…, 4.5, 5 years for
a total of 9 interim analyses. The logrank statistic is used and the Haybittle
method is selected with b=3. The baseline survival function is the exponential
survival function with 50% failure in 5 years. A two-sided logrank test with an
overall Type I error is performed at the interim analyses. For the
alternative survival function, the failure rate for the first year is 4%, for
the second and third year is 8% and for the fourth and fifth year is 10%.
We assume that there is 7.7 % of censoring (loss to follow-up) for the control
group and 7.3 % censoring for the treatment group in the 5-year period.
In addition, there is a 3% drop-in rate per year and a 10% drop-out rate per
year, and these cross-overs occur at the beginning of the year. We
demonstrate how the program dsss1.exe can be used to determine the sample size
of the trial with 90% power at the specified alternative.
The first step is to modify the sample input file dsss1.inp. The modified
sample input file exam1.inp can be found on this Web page. For example,
since we have selected the Haybittle method, on line 11 of the input file, we
change the number to 3. On line 20, we change the number to 9 since we are
planning 9 interim analyses. We specify on line 26 that we are going to
simulate 1 statistic. On line 30, we specify the rho of the statistic, and
input 0, which is the rho of the logrank statistics. On line 33, we input
the tau of the statistic, and also input 0. Another important input parameter
is the number of simulations used in each iteration. This parameter is located
on line 37. For the first run of the program, we use a moderate number,
500, so that the program will finish rather quickly and tell us roughly what
the accrual rate should be. On lines 74, 78 and 83, we specify the
baseline survival function. On line 74, we use 1 to indicate to the program
that the baseline survival function will be specified by 1 intervals. On line
78, one right endpoints, 5 of the interval is specified. On line 83, the
survival probabilities at 5 is specified, it is .5. Since the survival
probabilities between the endpoints are interpolated by using the exponential
distribution, the survival distribution between 0 and 5 is the exponential
distribution with median 5. Another important input is at line 88, where we
specify the accrual length and the lower and upper limits on the accrual rate.
For the first run, we use 3 300 500, indicating an accrual length of 3 years, a
lower bound of 300 patients and an upper bound of 500 subjects per year on the
accrual rate. At line 95, we input .90 for the desired power. At
lines 100, 103, 108 and 113 we specify the alternative distribution function.
At line 100, we use 5 to indicate that five intervals will be used to specify
the alternative distribution function. At line 103, the five right endpoints of
the intervals are specified. They are 1 2 3 4 5. The input on line 108
indicates whether the numbers on line 113 are the survival probabilities at the
points used in line 103 or they are the hazard ratios on the six intervals. The
number 1 indicates to the program that they are the survival
probabilities. We input the six survival probabilities in line 113 as
.96, .88, .80, .70, .60, according to our alternative failure rates specified
in last paragraph. The inputs starting at line 118 are for the baseline and
alternative survival probabilities for the censoring distribution and the
drop-in and drop-out rates. The logic of these inputs is the same as that of
the survival probabilities for the baseline distribution.
Once we finish editing the input file, we can run the program with the input
file. It took a PC with Pentium II 300 processor less than 1 minute to get the
result. The program prints out a message stating that the lower bound on the
accrual rate is too large for the desired power. Since we have specified
exam1.log on line 14 of the input file, we can use an editor to open the file
exam1.log to see the actual powers simulated by our specification. At iteration
1, the accrual rate is 500, the simulated power for the log-rank statistic is
1. At iteration 4 with accrual rate 300, the simulated power is about .94. Since
300 is the lower bound on the accrual rate we specified on line 88 of the input
file, the program stops without finding the accrual rate for the desired power.
In the second run of the program, we change the input on line 88 of the input
file to 3 200 400. The input file is called exam2.inp. This time the program
stops at accrual rate 252.0 with a simulated power .892 for the
log-rank statistic. Since the standard deviation of this simulated power is
.0139, the program stops because it has reached within 2 standard errors of the
desired power. In order to get a more accurate accrual rate, we can take a
third run of the program with a larger number of simulations. In the input
file, we first change on line 14 so that the output file name this time is
exam3.log . We then change on line 37 the number of simulations to 5000. We
also change on line 88 the lower and upper bounds on the accrual rate to 200
300. This time, after about 5 minutes run on the PC, the program finds the
accrual rate to be 262.1 with a simulated power .904 for the log-rank
statistic. The input file is called exam3.inp, which can also be found on
this Web page.
Once the user becomes familiar with the program, he/she may use the program to
exploit many other aspects of the design of a trial. For example, an
investigator may not be sure what the cross-over rates are and may try the
program to find the sample size with different configurations of the cross-over
rate. Another example concerns the sample size determination for the Peto-Prentice
statistic in the setting just described above. Under our specification of the
alternative, the Peto-Prentice statistic should be more powerful than the
log-rank statistic. We have used the program to find that the accrual rate
should be 228 over a 3-year period to have 90% power for the Peto-Prentice
statistic. (This can be done by changing the parameter on line 30 of the
input file from 0 to 1. The resulting input file is called exam4.inp, which can
be found on this Web page.)