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Unformatted text preview: Addendum to Lecture 10 Application of the cumulative odds and baselinecategory logit model with random effects to the hypnotic drug experiment Laurent Briollais Samuel Lunenfeld Research Institute, Mount Sinai Hospital Dalla Lana School of Public Health, UofT November 28, 2011 1 CHL 5210 Fall 2011 Categorical data analysis Lecture #11 2 1 Example: Hypnotic drug experiment Randomized, doubleblind clinical trial comparing active hypnotic drug with placebo in insomnia patients. Response is time in minutes to fall asleep be fore going to bed. Each person has ( Y i 1 , Y i 2 , x i ) where Y i 1 = 1, 2, 3, 4 denotes time to fall asleep at baseline and Y i 2 = 1, 2, 3, 4 is time to fall asleep after two weeks of treatment on one of x i = 0 placebo or x i = 1 hypnotic. Table 1: Time to Falling Asleep by Treatment and Occasion Time to Falling Asleep Followup Treatment Occasion < 20 20 to 30 30 to 60 > 60 Active < 20 7 4 1 20 to 30 11 5 2 2 30 to 60 13 23 3 1 > 60 9 17 13 1 Placebo < 20 7 4 2 1 20 to 30 14 5 1 30 to 60 6 9 18 2 > 60 4 11 14 22 Table 2: Sample Marginal Distributions of Table 1 Response Treatment Occasion < 20 20 to 30 30 to 60 > 60 Active Initial 0.101 0.168 0.336 0.395 Follow up 0.336 0.412 0.160 0.092 Placebo Initial 0.117 0.167 0.292 0.425 Follow up 0.258 0.242 0.292 0.208 This is repeated measures data on an individual with ordinal outcomes. A natural model to consider is an extension of the proportional odds model with a random effect that accounts for an individuals predisposition toward insomnia CHL 5210 Fall 2011 Categorical data analysis Lecture #11 3 2 The proportional cumulative odds logit model with random effects logit[ P ( Y t ≤ j  u i )] = log P ( Y t ≤ j  u i ) P ( Y t > j  u i ) = u i + α j + β 1 t + β 2 x + β 3 ( tx ) , j = 1 , 2 , 3 (1) We are primarily interested in how the odds of taking less time to get to sleep changes from drug to placebo after being treated for two weeks (so j = 2). For x i = 1 (treatment) logit[ P ( Y 2 ≤ j  u i )] = u i + α j + β 1 + β 2 + β 3 , For x i = 0 (placebo) logit[ P ( Y 2 ≤ j  u i )] = u i + α j + β 2 . The difference of these, expressed in log odds, is constant over the different categories j of the response variable with log P ( Y 2 ≤ j  x i = 1 , u i ) /P ( Y 2 > j  x i = 1 , u i ) P ( Y 2 ≤ j  x i = 0 , u i ) /P ( Y 2 > j  x i = 0 , u i ) = β 1 + β 3 The likelihood, conditional on the u i is built from the multinomial proba bilities CHL 5210 Fall 2011 Categorical data analysis Lecture #11 4 P ( Y t = 1  u i ) = P ( Y t ≤ 1  u i ) P ( Y t = 2  u i ) = P ( Y t ≤ 2  u i ) P ( Y t ≤ 1  u i ) P ( Y t = 3  u i ) = P ( Y t ≤ 3  u i ) P ( Y t ≤ 2  u i ) P ( Y t = 4  u i ) = P ( Y t ≤ 4  u i ) P ( Y t ≤ 3  u i ) (2) where P ( Y t ≤ j  u i ) = exp( u i + α j + β 1 t + β 2 x + β 3 ( tx )) 1 + exp( u i + α j + β 1 t + β 2 x + β 3 ( tx )) The CI for e β 1 + β 3 is (0.9, 5.4). We estimate the odds of falling asleep moreis (0....
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This note was uploaded on 02/23/2012 for the course CHL 5210H taught by Professor Leisun during the Fall '11 term at University of Toronto.
 Fall '11
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