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# Notes 10 - \$ ST3241 Categorical Data Analysis I Models For...

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& \$ % ST3241 Categorical Data Analysis I Models For Matched Pairs Dependent Proportions and Conditional Models 1

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& \$ % Example: Rating of Performance For a poll of a random sample of 1600 votingage British citizens, 944 indicated approval of the Prime minister’s performance in the office. Six months later, of these same 1600 people, 800 indicated approval. 2
& \$ % Example Second Survey First Survey Approve Disapprove Total Approve 794 150 944 Disapprove 86 570 656 Total 880 720 1600 3

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& \$ % Dependent Categorical Data To compare categorical responses for two samples when each sample has the same subjects or when a natural pairing exists between each subject in one sample and a subject from the other sample. The responses in the two samples are then statistically dependent . The pairs of observations are called matched pairs . A two-way table having the same categories for both classifications summarizes such data. 4
& \$ % Comparing Dependent Proportions Let n ij = the number of subjects making response i at the first survey and response j at the second. In the example, the sample proportions approving are 944 / 1600 = 0 . 59 and 880 / 1600 = 0 . 55 . These marginal proportions are correlated, and statistical analyses must recognize this. Let π ij = probability that a subject makes response i at survey 1 and response j at survey 2. 5

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& \$ % Dependent Proportions The probabilities of approval at the two surveys are π 1+ and π +1 , the first row and first column totals. When these are identical, the probabilities of disapproval are also identical, and there is marginal homogeneity . Note that, π 1+ - π +1 = ( π 11 + π 12 ) - ( π 11 + π 21 ) = π 12 - π 21 Marginal homogeneity is equivalent to equality of off-maindiagonal probabilities; that is π 12 = π 21 . The table shows symmetry across the main diagonal. 6
& \$ % Inference For Dependent Proportions Use δ = π +1 - π 1+ Let d = p +1 - p 1+ = p 2+ - p +2 . From the results on multinomial distributions, cov ( p +1 , p 1+ ) = ( π 11 π 22 - π 12 π 21 ) /n Thus, var ( nd ) = π 1+ (1 - π 1+ ) + π +1 (1 - π +1 ) - 2( π 11 π 22 - π 12 π 21 ) For large samples, d has approximately a normal distribution. A confidence interval for δ is then d ± z α/ 2 ˆ δ ( d ) 7

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& \$ % Inference For Dependent Proportions Here ˆ δ 2 ( d ) = [ p 1+ (1 - p 1+ ) + p +1 (1 - p +1 ) - 2( p 11 p 22 - p 12 p 21 )] /n = [( p 12 + p 21 ) - ( p 12 - p 21 ) 2 ] /n (1) The hypothesis of marginal homogeneity is H 0 : pi 1+ = pi +1 (i.e. δ = 0 ). Wald test statistic is: z = d/ ˆ σ ( d ) 8
& \$ % McNemar’s Test Under H 0 , an alternative estimated variance is ˆ σ 2 0 ( d ) = p 12 + p 21 n = n 12 + n 21 n 2 The score test statistic is z 0 = d ˆ σ 0 ( d ) = n 21 - n 12 ( n 21 + n 12 ) 1 / 2 The square of z 0 is a chi-squared distrbution with df = 1 . The test using it called McNemar 0 s test . It depends only on cases classified in different categories for the two observations.

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Notes 10 - \$ ST3241 Categorical Data Analysis I Models For...

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