exam1 - 3. Cr epeau et al. (1985) describe a study of the...

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STAT 8630 Exam 1 – Due Tuesday, March 24 SHOW ALL WORK Name: Do not discuss this exam with anyone other than your instructor. You may use any notes, books, journal articles etc., but you are not to collaborate! 1. Consider the random intercept and slope model from p.131 of the class notes: y ij = β 0 + b 0 i + ( β 1 + b 1 i ) x ij + ε ij , i = 1 ,...,n , j = 1 ,...,t i , where D = var( b i ) = ± σ 2 0 σ 01 σ 01 σ 2 1 , R i = var( ε i ) = σ 2 I , i. a. Show that the variance-covariance matrix V i for y i in this model is given by V i = σ 2 0 j n j T n + σ 01 ( x i j T n + j n x T i ) + σ 2 1 x i x T i + σ 2 I n . (The elements V ijj and V ijk of this matrix are given on p.131 of the notes.) Here y i = y i 1 ,...,y it i and x i = x i 1 ,...,x it i . b. Show that corr( y ij ,y ik ) is monotonically decreasing in x ik for x ik > x ij . c. Show one of the following two results (your choice): i. For fixed x ij , lim x ik →∞ corr( y ij ,y ik ) = σ 01 + σ 2 1 x ij q σ 2 1 ( σ 2 0 + 2 σ 01 x ij + σ 2 1 x 2 ij + σ 2 ) . ii. Suppose that δ = x ik - x ij is fixed. Then lim x ij →∞ corr( y ij ,y ik ) = 1 . 2. Do problem 8.2 in the FLW text (all parts).
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Unformatted text preview: 3. Cr epeau et al. (1985) describe a study of the eect of halothane on responses to irreversible myocardial ischemia and subsequent infarction. The experiment consisted of inducing a heart attack in rats exposed to dierent concerntrations of halothane (group1: 0%, group 2: 0.25%, group 3: 0.5%, group4: 1%). The table below (see also exam1-3data.dat on the course web site) displays blood pressure 1 measurements (mm Hg) at up to nine time points ranging from one minute to 240 minutes after heart attack induction. Investigate the eect of halothane dose on blood pressure using an appropriate model and appropriate methods of inference. Explain and justify your analysis thoroughly. 2...
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exam1 - 3. Cr epeau et al. (1985) describe a study of the...

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