Table entries show the absolute difference (new - standard) in 5-year survival for the hazard reduction due to the new treatment on the left-hand column and the 5-year survival in the standard treatment group (middle columns, top), assuming exponential survival patterns. The right-most column shows the number of events required (Eq. 2) for 80% power at a two-sided a = 0.05. The number of patients required for the trial to obtain results in some specified time period will depend on the standard group failure rate and the rate at which patients can be accrued.

From the specification of difference of interest or effect size, the sample size in terms of number of events required to detect this difference with desired statistical power and significance level can be determined. Depending on the anticipated accrual rate and the prognosis (e.g., rapidity of failure events) in the control treatment group, the number of patients required can then be approximated. Typically, the required number of events is based on the normal theory approximation of the natural logarithm of the hazard ratio. For a two-arm trial, the total number of events is

where Z1-p and Za/2 are the values from the standard normal distribution associated with the power and significance level desired, and pA and pB are the proportions of total patients to be allocated to the two arms, respectively (e.g., 0.5 for equal allocation). One can see from this equation that the number of events required depends strongly on the HR, becoming dramatically larger as the HR approaches 1.0 (see Table 8.4). The number of patients required and total duration of the trial depend on the rate of patient accrual and the failure rate in the control group, both of which contribute to the determination of how rapidly the requisite events will be observed. The accrual rate is typically estimated from previous experience and may also involve querying potential investigators to project the accrual rate per unit of time. Similarly, the failure rate for patients under standard therapy is derived from past observations and available literature estimates. The computations are straightforward but generally require computer programs (see Shuster in Reference 1, or commercial programs) or under certain assumptions, tabled values.52 Sample size methods have been extended to take into account other factors that will influence power, such as patients withdrawing from the study while it is ongoing (dropout), switching from the assigned treatment to the other group (cross-over), or deviating from protocol treatment (noncompliance).53-56

Another important design aspect concerns the desire to ensure that factors associated with outcomes, called prognostic factors, are balanced between treatment arms. Although random allocation to treatments naturally provides equal distribution of characteristics, using key prognostic variables as stratification factors, and incorporating these into the randomization process (by randomizing within strata or other means discussed in the next section), imbalances that can arise by chance can essentially be prevented. The number of stratification factors needs to be limited to a few key factors, because the strata increase multiplicatively with the number of factors and factor levels. For example, the use of four factors [say, age groups (less than 50 years, 50-64 years, 65 years or older), lymph node status (positive, negative), performance status (0-1, more than 1), and surgical procedure (procedure A or B)] produces 3 x 2 x 2 x 2 = 24 strata in which to balance treatment assignments. As the number of strata becomes large relative to the number of patients to be entered, the efficiency of stratification as a means to balance treatment arms diminishes.57

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