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

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CHAPTER 6 ST 520, A. TSIATIS and D. Zhang power of a test and to compute sample sizes, we need to not only specify the clinically important difference ∆A, but also plausible values of the nuisance parameters. 2. It is often the case that under the alternative hypothesis the standard deviation σ∗(∆A, θ) will be equal to (or approximately equal) to one. If this is the case, then the mean under the alternative φ(n, ∆A, θ) is referred to as the non-centrality parameter. For example, when testing the equality in mean response between two treatments with normally distributed continuous data, we often use the t-test Tn = sY ¯Y1 − ¯ Y2 � ≈ 1/2 1 + n1 n2 � 1 σY ¯Y1 − ¯ Y2 � , 1/2 1 + n1 n2 which is approximately distributed as a standard normal under the null hypothesis. Under the alternative hypothesis HA : µ1 −µ2 = ∆ = ∆A, the distribution of Tn will also be approximately normally distributed with mean and variance Hence Thus and ⎧ ⎪⎨ Hence ⎪⎩ EHA (Tn) ⎧ ⎪⎨ ≈ E ⎪⎩ ∆A � 1 σY + n1 1 n2 σY ¯Y1 − ¯ Y2 � 1 n1 + 1 n2 � 1/2 ⎫ ⎪⎬ ⎪⎭ = σY � 1 µ1 − µ2 � 1 varHA (Tn) = {var(¯ Y1) + var( ¯ Y2)} σ2 � � = 1 1 Y + n1 n2 σ2 Y σ2 Y ⎛ n1 � = 1/2 1 + n2 � 1 n1 � 1 ⎞ n1 HA ⎜ ∆A ⎟ Tn ∼ N ⎝ � � , 1 1/2 ⎠. 1 1 σY + n1 n2 φ(n, ∆A, θ) = σY � 1 n1 ∆A σY � 1 n1 ∆A � 1 + n2� = 1. 1 + n2 � , 1/2 1 + n2 � , (6.1) 1/2 1 + n2 ⎫ ⎪⎬ σ∗(∆A, θ) = 1. (6.2) �1/2 is the non-centrality parameter. ⎪⎭ Note: In actuality, the distribution of Tn is ⎧a non-central t distribution with n1 +n2 −2 degrees ⎪⎨ ∆A of freedom and non-centrality parameter � ⎪⎩ 1 σY + n1 1 ⎫ ⎪⎬ �1/2 . However, with large n this is well ⎪⎭ n2 approximated by the normal distribution given above. PAGE 86
CHAPTER 6 ST 520, A. TSIATIS and D. Zhang 6.2 Deriving sample size to achieve desired power We are now in a position to derive the sample size necessary to detect a clinically important difference with some desired power. Suppose we want a level-α test (one or two-sided) to have power at least equal to 1 −β to detect a clinically important difference ∆ = ∆A. Then how large a sample size is necessary? For a one-sided test consider the figure below. Density 0.0 0.1 0.2 0.3 0.4 0.5 It is clear from this figure that Therefore, if we specify Figure 6.1: Distributions of T under H0 and HA • the level of significance (type I error) “α” • the power (1 - type II error) “1 − β” • the clinically important difference “∆A” • the nuisance parameters “θ” −2 0 2 4 6 8 x φ(n, ∆A, θ) = {Zα + Zβσ∗(∆A, θ)}. (6.3) PAGE 87
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ST 520 Statistical Principles of Cl
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