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

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CHAPTER 10 ST 520, A. TSIATIS and D. Zhang information necessary to achieve a certain level of significance and power for a fixed sample design and multiplying by an inflation factor. For designs with a maximum of K analyses after equal increments of information, the inflation factor is a function of α (the significance level), β (the type II error or one minus power), K, and Φ (the shape parameter of the boundary). We denote this inflation factor by IF(α, K, Φ, β). Let V denote the number of interim analyses conducted before a study is stopped. V is a discrete integer-valued random variable that can take on values from 1, . . ., K. Specifically, for a K-look group-sequential test with boundaries b1, . . .,bK, the event V = j (i.e. stopping after the j-th interim analysis) corresponds to (V = j) = {|T(t1)| < b1, . . ., |T(tj−1)| < bj−1, |T(tj)| ≥ bj}, j = 1, . . ., K. The expected number of interim analyses for such a group-sequential test, assuming ∆ = ∆ ∗ is given by E∆∗(V ) = K� j=1 j × P∆∗(V = j). Since each interim analysis is conducted after increments MI/K of information, this implies that the average information before a study is stopped is given by Since MI = I FS × IF(α, K, Φ, β), then AI(α, K, Φ, β, ∆ ∗ ) = I FS AI(∆ ∗ ) = MI E∆∗(V ). K �� IF(α, K, Φ, β) K E∆∗(V ) Note: We use the notation AI(α, K, Φ, β, ∆ ∗ ) to emphasize the fact that the average information depends on the level, power, maximum number of analyses, boundary shape, and alternative of interest. For the most part we will consider the average information at the null hypothesis ∆ ∗ = 0 and the clinically important alternative ∆ ∗ = ∆A. However, other values of the parameter may also be considered. Using recursive numerical integration, the E∆∗(V ) can be computed for different sequential designs at the null hypothesis, at the clinically important alternative ∆A, as well as other values for the treatment difference. For instance, if we take K = 5, α = .05, power equal to 90%, then under HA : ∆ = ∆A, the expected number of interim analyses for a Pocock design is PAGE 190 � .
CHAPTER 10 ST 520, A. TSIATIS and D. Zhang equal to E∆A (V ) = 2.83. Consequently, the average information necessary to stop a trial, if the alternative HA were true would be I FS �� IF(.05, 5, .5, .10) × 2.83 5 = I FS � 1.21 5 = I FS × .68. � × 2.83 Therefore, on average, we would reject the null hypothesis using 68% of the information necessary for a fixed-sample design with the same level (.05) and power (.90) as the 5-look Pocock design, if indeed, the clinically important alternative hypothesis were true. This is why sequential designs are sometimes preferred over fixed-sample designs. Remark: If the null hypothesis were true, then it is unlikely (< .05) that the study would be stopped early with the sequential designs we have been discussing. Consequently, the average information necessary to stop a study early if the null hypothesis were true would be close to the maximum information (i.e. for the 5-look Pocock design discussed above we would need almost 21% more information than the corresponding fixed-sample design). In contrast, if we use the 5-look O-F design with α = .05 and power of 90%, then the expected number of interim analyses equals E∆A (V ) = 3.65 under the alternative hypothesis HA. Thus, the average information is I FS �� IF(.05, 5, 0.0, .10) × 3.65 5 = I FS � 1.03 5 = I FS × .75. PAGE 191 � × 3.65
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ST 520 Statistical Principles of Cl
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