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# CH11-4 - Outline Chapter 11 Analyzing Repeated Categorical...

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Chapter 11. Analyzing Repeated Categorical Response Data Deyuan Li School of Management, Fudan University Feb. 28, 2011 1/32 Outline 11.1: Comparing marginal distributions: multiple responses; 11.2: Marginal modeling: maximum likelihood approach; 11.5: Markov chains: transitional modeling. 2/32 Repeated categorical response data: (1) response variable for each subject is measured repeatedly (at several times or under various conditions); dependence among the repeated responses. For example, blood measures of one person at several times. (2) responses of subjects in a set or cluster; dependence in the set or cluster. For example, survival response for each fetus in a litter for a sample of pregnant mice. 3/32 11.1 Comparing marginal distributions: multiple responses Usually, the multivariate dependence among repeated responses is of less interest than their marginal distributions. For instance, in treating a chronic condition with some treatment, the primary goal might be to study whether the probability of success increases over the T weeks of a treatment period. In Sections 10.2.1 and 10.3 we compared marginal distributions for matched pairs ( T = 2) using models that apply directly to the marginal distributions. In this section we extend this approach to T > 2. 4/32

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11.1.1 Binary marginal models and marginal homogeneity Denote T binary responses by ( Y 1 , Y 2 , ..., Y T ). The marginal logit model is logit[ P ( Y t =1)]= α + β t , t =1 , 2 , ..., T , (11 . 1) with β T =0 . Inpart icu lar , β 1 = β 2 = ... = β T implies marginal homogeneity (MH) . For the outcome i =( i 1 , ..., i T )w ith i t = 1 or 0, let the joint mass probability be π i = P ( Y 1 = i 1 , Y 2 = i 2 , ..., Y T = i T ) , and the set of mass probabilities be π = { π i } . Of course, # π =2 T . 5/32 Let n i denote the sample count of outcome i . The kernel of the log likelihood is L ( π )= ± i n i log π i . To test marginal homogeneity, the likelihood-ratio test uses 2[ L π MH ) L ( p )] = 2 ± i n i log( p i / ˆ π MH i ) χ 2 df , where df = T 1, ˆ π MH and p are the ML estimators under and without MH assumption, respectively. 6/32 11.1.2 Crossover drug comparison example Table 11.1 comes from a crossover study in which each subject used each of three drugs for treatment of a chronic condition at three times (±rst drug A, then B and C). Responses are binary (favorable or unfavorable), 2 3 = 8 di²erent outcomes in total. TABLE 11.1 Responses to Three Drugs in a Crossover Study Drug A Favorable Drug A Unfavorable B Favorable B Unfavorable B Favorable B Unfavorable C Favorable 6 2 2 6 C Unfavorable 16 4 4 6 Ž. Source: Reprinted with permission from the Biometric Society Grizzle et al. 1969 . The sample proportion favorable was (0 . 61 , 0 . 61 , 0 . 35) for drugs A, B, C. The likelihood-ratio statistic for testing marginal homogeneity is 5 . 95 ( df =2) . For more analysis details, see the book. 7/32 11.1.3 Modeling margins of a multicategory response Multinomial response . With baseline-category logits for I outcome categories, the saturated model is log[ P ( Y t = j ) / P ( Y t = I )] = β ij , t , ..., T , j , ..., I 1 .
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CH11-4 - Outline Chapter 11 Analyzing Repeated Categorical...

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