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Unformatted text preview: PubH 8452 Fall 2011 Homework #1 Due October 7, 2011 1. In a randomized clincal trial, subjects receive either the active treatment or placebo after recording a baseline measurement, Y i . At the end of trial, the outcome is again recorded for each participant, Y i 1 . The goal of the study is to assess whether there is an impact on Y due to treatment. There are a number of potential analyses that can be proposed for this study design. Let TX denote the treatment assignment and let post be an indicator for the followup time. Assume that we have m subjects in each group. Assume that σ 2 = Var( Y i ) = Var( Y i 1 ). (a) Consider the regression model: E( Y i j ) = β + β 1 TX i + β 2 post i j + γ TX i post i j . Interpret the parameter γ . (b) If we use a repeated measures model with Y i = ( Y i , Y i 1 ) T , assume an exchangeable covariance matrix, Cov( Y i ) = σ 2 1 ρ ρ 1 ¶ and assume normality, show that the MLE for γ is given by the difference of the differ ences: ˆ γ (1) equals the means of Y i 1 Y i for the treatment group minus the mean of Y i 1 Y i for the placebo group. (c) Calculate the variance of ˆ γ (1) . (d) Another model uses the fact that groups were randomized to constrain the means at baseline, E( Y i  TX i = 0) = E( Y i  TX i = 1). This model can be written as: E( Y i j ) = β + β 2 post i j + γ TX i post i j . Derive the MLE for γ in the constrained model. Denote this estimator as ˆ γ ( ρ ) . (e) Calculate the variance of ˆ γ ( ρ ) . (f) The estimators ˆ γ ( k ) are special cases of the general estimator: ˆ γ ( α ) = Y i 1 ( TX = 1) α Y i ( TX = 1) Y i 1 ( TX = 0) α Y i ( TX = 0)....
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 Fall '11
 XianghuaLuo
 Variance, Yi

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