## Planning the size and duration of a clinical trial studying the time to some critical event |

** Author(s): ** ,

** Journal/Book: **J Chron Dis. 1974; 27: 15-24.

** Abstract: **This paper considers the dual problem of planning the size (i.e. the required number of patients) and the required duration of trial for a fixed sample size clinical trial comparing the length of time to some critical event (such as death or relapse) in two treatment regimens (´treatment' and ´control'). The required number of patients is derived from the exact distribution of the usual test statistic and a comparison is made with two normal approximations. The required duration of the trial is derived assuming that the patient entry into the study is a process in continuous time, and that the time to failure is exponential. This approach may be considered to be a generalization of the approach in [4]. The principal results may be summarized as follows: (1) The required number of patients (2d), based an the exact distribution of the test statistic, is uniformly lower than the required number, based an the approach which assumes that the difference in means is approximately normal. However, the approximation d=[2a(k+kß)I/(1n?)2], based an the logarithmic transformation is, found to be quite accurate.(2) A solution to the problem of finding the minimum required duration of study is found by expressing the accumulated number of patient-years in the time interval (0, t) as a filtered Poisson process and may be considered to be the analogue of the discrete patient entry model in [4]. For a given formulation, the required duration of study is found by solving a non-linear equation by iterative techniques. However, a very good lower bound is T~2d/a, the expected number of years necessary to enter the required 2d patients.(3) The implicit assumption in [4] that the optimal duration of study requires no follow-up time is given a theoretical justification by proving that it follows directly from the approach adopted in this paper.(4) One tables given from which the required number of patients may be read and another table is given, which illustrates the results obtained for the required duration of study. Computer programs that compute these quantities for any given specification are available.(5) Although perhaps preferable an theoretical grounds, the required duration of study T, as determined by the methods of this paper, is quite similar to the T determined by the methods in [4]. However, in several instances, the T determined in [4] is even less than the expected time necessary to enter the required 2d patients. Also, every T in [4) that satisfied the lower bound for the required T derived in this paper was larger than necessary to achieve the desired power.

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