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

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CHAPTER 7 ST 520, A. TSIATIS and D. Zhang Our primary focus will be on testing the null hypothesis H0 : µ1 = . . . = µK. Let us redefine our data so that (Yij, i = 1, . . .,nj, j = 1, . . .,K) denotes the response for the i-th individual within treatment j, and nj denotes the number of individuals in our sample assigned to treatment j (n = � K j=1 nj). From standard theory we know that the treatment-specific sample mean nj � ¯Yj = Yij/nj is an unbiased estimator for µj and that asymptotically i=1 ¯Yj ∼ N(µj, σ2 Y j nj ), j = 1, . . ., K. Remark: If the Y ’s are normally distributed, then the above result is exact. However, with the large sample sizes that are usually realized in phase III clinical trials, the asymptotic approxi- mation is generally very good. Also, we know that the treatment-specific sample variance s 2 Y j = �nj i=1(Yij − ¯ Yj) 2 nj − 1 is an unbiased estimator for σ 2 Y j, and that asymptotically ¯Yj ∼ N(µj, s2 Y j nj ), j = 1, . . .,K. Remark: If the treatment specific variances are all equal, then the common variance is often estimated using the pooled estimator s 2 Y = � �nj Kj=1 i=1(Yij − ¯ Yj) 2 . n − K Returning to the general results of section 7.1 of the notes, we let • ¯ Yj take the role of ˆ θj • µj take the role of θj PAGE 110
CHAPTER 7 ST 520, A. TSIATIS and D. Zhang • s2 Y j nj take the role of σ2 j; hence wj = nj s 2 Y j Using (7.1), we construct the test statistic where and wj = nj s2 . Y j K� Tn = wj( j=1 ¯ Yj − ¯ 2 Y ) , (7.9) �Kj=1 wj ¯Y = ¯ Yj , � Kj=1 wj For the case where we are willing to assume equal variances, we get that where �Kj=1 nj( Tn = ¯ Yj − ¯ 2 Y ) , (7.10) ¯Y = s 2 Y �Kj=1 nj ¯ Yj . n Under the null hypothesis H0, Tn is approximately distributed as a chi-square distribution with K − 1 degrees of freedom. Under the alternative hypothesis HA : µ1 = µ1A, . . .,µK = µKA where the µjA’s are not all equal, Tn is approximately distributed as a non-central chi-square distribution with K − 1 degrees of freedom and non-centrality parameter where and wj = nj σ2 . Y j φ 2 = K� wj(µjA − ¯µA) 2 , (7.11) j=1 ¯µA = � Kj=1 wjµjA � Kj=1 wj Assuming equal variances, we get the simplification where , φ 2 �Kj=1 nj(µjA − ¯µA) = 2 , (7.12) ¯µA = σ 2 Y � Kj=1 njµjA n PAGE 111 .
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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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