CH10-4 - Outline 10.1: Comparing Dependent Proportions;...

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Chapter 10. Models for Matched Pairs Deyuan Li School of Management, Fudan University Feb. 28, 2011 1/1 Outline 10.1: Comparing Dependent Proportions; 10.2: Conditional Logistic Regression for Binary Matched Paries; 10.3: Marginal Models for Squared Contingency Tables; 10.4: Symmetry, Quasi-symmetry and Quasi-independence; 10.5: Measure Agreement Between Observers; 10.6: Bradley-Terry Models for Paired Preferences. 2/1 In this chapter, we introduce methods for comparing categorical responses for two samples when each observation in one sample pairs with an observation in the other. For easy understanding, we assume n independent subjects and let Y i =( Y i 1 , Y i 2 , ..., Y it i ) is the observation of subject i at di±erent time. In statistics, { Y 1 , Y 2 , ..., Y n } are called longitudinal data ( ± ± ± ± ); in econometrics it is called panel data ( ± ± ± ± ). For ²xed i , Y i is a time series; for ²xed time j , { Y 1 j , Y 2 j , ..., Y nj } is a sequence of independent random variables. If t i =2fora l l i , { Y 1 , Y 2 , ..., Y n } is called matched-pairs data ( ± ± ± ± ). Note that the two samples { Y 11 , Y 21 , ..., Y n 1 } and { Y 12 , Y 22 , ..., Y n 2 } are not independent . 3/1 Table 10.1 illustrates matched-pairs data. TABLE 10.1 Rating of Performance of Prime Minister Second Survey First Survey Approve Disapprove Total Approve 794 150 944 Disapprove 86 570 656 Total 880 720 1600 For a poll of a random sample of 1600 voting-age British citizens, 944 indicated approval of the Prime Minister’s performance in office. Six months later, of these same 1600 people, 880 indicated approval. For matched pairs with a categorical response, a two-way contingency table with the same row and column categories summarizes the data. The table is square . 4/1
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Let n = 1600 and Y i =( Y i 1 , Y i 2 ), where Y ij = 1 means approval of subject i in the j -th survey, Y ij = 0 means disapproval of subject i in the j -th survey, for i =1 , 2 , ..., n and j , 2. Then from Table 10.1, we have X i Y i 1 = 944 , X i Y i 2 = 880 , X i Y i 1 Y i 2 = 794 , X i (1 Y i 1 )(1 Yi 2) = 570 . A strong association exists between opinions six months apart, since the sample odds ratio is (794 × 570) / (150 × 86) = 35 . 1. 5/1 10.1 Comparing Dependent Proportions Fo reachmatchedpa ir Y Y 1 , Y 2 ), de±ne π ab = P ( Y 1 = a , Y 2 = b ) . Assume Y , Y 1 , Y 2 , ..., Y n are i.i.d. matched pairs. Let n ab count the number of such pairs, with p ab = n ab / n the sample proportion. We treat { n ab } as a sample from a multinomial ( n ; { π ab } ) distribution. Denote p a + is the proportion in category a for observation 1, and p + a is the corresponding proportion for observation 2. 6/1 We compare samples by comparing marginal proportions { p a + } with { p + a } . Methods for independent samples (Section 2) are inappropriate. In this section we consider binary outcomes (i.e. a , b , 2). When π 1+ = π +1 ,then π 2+ = π +2 also, and there is marginal homogeneity . Since π 1+ π +1 π 11 + π 12 ) ( π 11 + π 21 )= π 12 π 21 , marginal homogeneity in 2 × 2 tables is equivalent to π 12 = π 21 .
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CH10-4 - Outline 10.1: Comparing Dependent Proportions;...

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