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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 follow-up 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