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

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CHAPTER 4 ST 520, A. TSIATIS and D. Zhang = � (µ + αSi + βXi + ǫi)/nA Xi=1 where ¯ǫA = � Xi=1 ǫi/nA. Similarly, where ¯ǫB = � Xi=0 ǫi/nB. Therefore Stratified randomization = (nAµ + α � Xi=1 Si + β � Xi=1 = (nAµ + αnA1 + βnA + � = µ + α nA1 + β + ¯ǫA, nA Xi=1 ¯YB = µ + α nB1 + ¯ǫB, ¯YA − ¯ � nA1 YB = β + α nA nB Xi + � Xi=1 ǫi)/nA ǫi)/nA − nB1 � + (¯ǫA − ¯ǫB). (4.2) nB Let us first consider the statistical properties of the estimator for treatment difference if we used permuted block randomization within strata with equal allocation. Roughly speaking, the number assigned to the two treatments, by strata, would be nA = nB = n/2 nA1 = nB1 = n1/2 nA0 = nB0 = n0/2. Remark: The counts above might be off by b/2, where b denotes the block size, but when n is large this difference is inconsequential. Substituting these counts into formula (4.2), we get Note: The coefficient for α cancelled out. Thus the mean of our estimator is given by ¯YA − ¯ YB = β + (¯ǫA − ¯ǫB). E( ¯ YA − ¯ YB) = E{β + (¯ǫA − ¯ǫB)} = β + E(¯ǫA) − E(¯ǫB) = β, PAGE 62
CHAPTER 4 ST 520, A. TSIATIS and D. Zhang which implies that the estimator is unbiased. The variance of the estimator is given by Simple randomization var( ¯ YA − ¯ YB) = var(¯ǫA) + var(¯ǫB) = σ 2 � 2 n � 2 + n 4σ2 . (4.3) n With simple randomization the counts nA1 conditional on nA and nB1 conditional on nB follow a binomial distribution. Specifically, and nA1|nA, nB ∼ b(nA, θ) (4.4) nB1|nA, nB ∼ b(nB, θ), (4.5) where θ denotes the proportion of the population in stratum 1. In addition, conditional on nA, nB, the binomial variables nA1 and nB1 are independent of each other. The estimator given by (4.2) has expectation equal to Because of (4.4) Similarly Hence, E( ¯ YA − ¯ � YB) = β + α E E � � nA1 nA = E � � nA1 nA − E � �� nB1 nB � � �� nA1 E |nA = E nA E � � nB1 nB = θ. E( ¯ YA − ¯ YB) = β; + E(¯ǫA − ¯ǫB). (4.6) � � nAθ = θ. that is, with simple randomization, the estimator ¯ YA − ¯ YB is an unbiased estimator of the treatment difference β. In computing the variance, we use the formula for iterated conditional variance; namely var( ¯ YA − ¯ YB) = E{var( ¯ YA − ¯ YB|nA, nB)} + var{E( ¯ YA − ¯ YB|nA, nB)}. PAGE 63 nA
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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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CHAPTER 1 ST 520, A. TSIATIS and D.
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