ST 520 Statistical Principles of Clinical Trials - NCSU Statistics ...

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CHAPTER 9 ST 520, A. TSIATIS and D. Zhang • number of patients • accrual period • follow-up time It was shown by Schoenfeld that to obtain reasonable approximations for the power, we need the expected number of events (deaths), computed under the alternative, to equal the number given by equation (9.3). Specifically, the expected number of deaths need to be computed separately for each treatment arm, under the assumption that the alternative hypothesis were true, and the sum from both treatments should equal (9.3) to achieve the desired power. In order to compute the expected number of deaths for each treatment during the design stage, we must consider the following. Also, see figure below. 0 Figure 9.6: Illustration of accrual and follow-up Ei A L = A + F (Accrual) (Follow-up) Remark: We will assume that any censoring that may occur is due to end of study censoring resulting from staggered entry. We will sometimes refer to this as “administrative censoring”. If, in addition, we wanted to consider other forms of censoring, such as lost to follow-up due to withdrawal or censoring due to death from other causes, then the formulas given below would have to be modified. We define the following notation: • A denotes the accrual period; that is, the calendar period of time that patients are entering the study • F denotes the calendar period of time after accrual has ended before the final analysis is conducted • L = A + F denotes the total calendar time of the study from the time the study opens until the final analysis PAGE 154 ✲
CHAPTER 9 ST 520, A. TSIATIS and D. Zhang • Denote the accrual rate at calendar time u by a(u); more precisely as � � Expected no. of patients entering between [u, u + h) a(u) = lim . h→0 h The total expected number of patients in the clinical trial is given by � A a(u)du. 0 If we have a constant accrual rate (this is the most common assumption made during the design stage), then a(u) = a and the expected number of patients is aA. In a randomized study where patients are assigned with equal probability to each of two treatments the accrual rate to each treatment would be a(u) . If we denote the distribution function 2 of the survival time for each treatment (0 or 1) by Fj(u) = P(T ≤ u|A = j) = 1 − Sj(u), j = 0, 1, then the expected number of observed deaths from treatment j is � A dj = 0 a(u) 2 Fj(L − u)du, j = 0, 1. (9.5) In words, we expect a(u) du patients to enter between times [u, u + du), of which the proportion 2 Fj(L − u) are expected to die by the end of the study (at time L). This number summed (integrated) over u, for values of u during the accrual period [0, A], will yield the expected number of deaths on treatment j = 0, 1. To get the desired power we want d1 + d0 = 4(Zα/2 + Zβ) 2 γ2 , A Note: The number of failures can be affected by • the accrual rate • accrual period (sample size) • follow-up period • the failure (hazard) rate (survival distribution) PAGE 155
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ST 520 Statistical Principles of Cl
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TABLE OF CONTENTS ST 520, A. Tsiati
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ST 520 Statistical Principles of Clinical Trials - NCSU Statistics ...

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