Consider a design with 10 stages, involving an interim analysis after every 60 deaths, at the ith of which the standardized log-rank statistic Z i would be computed and compared with the upper limit, u i , and the lower limit, i , as described above. This primary interim analysis requires: patient code number, recruitment date, date of death or date of last known status, and treatment arm.
The trial would recruit 16 patients per month for 5 years, and follow them up until a stopping boundary is reached. The study follows the triangular design,
6,10 an asymmetric design which stops for futility if no evidence that the novel treatment is superior is apparent. Following,
10 it can be established that the boundary points are given by the following equations:
These boundary points are listed in
Table 2, together with the number of deaths (d
i ) that determine the timing of the
ith interim analysis. Also shown are the probabilities of stopping on the upper boundary at or before each interim analysis, for the null hazard ratio of 1, the alternative hazard ratio of 0.737, and the intermediate value of 0.858. Corresponding probabilities for the lower boundary are also shown. A plot of
i √(15
i) and u
i √(15
i) against the number of deaths (not shown) reveals the triangular shape that gives the test its name.
Table 3 shows the number of patients recruited at each interim analysis and the number of months that will have elapsed by then. Additional patients will be recruited while the interim analysis is being conducted and any decision to stop is being considered and confirmed. Stopping would have to occur by the third or fourth interim analysis to reduce sample size.
Table 4 presents the expected sample size at study termination and the expected duration of the study in months (neglecting patients recruited and time elapsed during the conduct of the interim analysis and subsequent discussions about stopping). Reductions in sample size and study duration would be substantial if either the null or alternative values of hazard ratio were true, but less marked for the intermediate value. For all possible hazard ratios, these expected values improve on both the single-stage and two-stage designs. To achieve these savings, there must be a commitment to continue to the maximum trial duration of 97 months, should interim analyses prove inconclusive.
Figure 2 plots the expected sample sizes, for all three designs considered here against the value of θ = −ln(hazard ratio) (a scale that produces a symmetric plot). Note that −ln(1) = 0; −ln(0.737) = 0.30; and −ln(0.858) = 0.15. The maximum sample sizes occur when the hazard ratio is slightly smaller than 0.858. Longer recruitment might allow the sample size to be reduced because a greater proportion of patients would die during the trial.