Notes 10 - ’ & ST3241 Categorical Data...

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Unformatted text preview: ’ & ST3241 Categorical Data Analysis I Models For Matched Pairs Dependent Proportions and Conditional Models 1 ’ & 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 ’ & 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 ’ & 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 ’ & 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 : pi 1+ = pi +1 (i.e. δ = 0 ). • Wald test statistic is: z = d/ ˆ σ ( d ) 8 ’ & McNemar’s Test • Under H , an alternative estimated variance is ˆ σ 2 ( d ) = p 12 + p 21 n = n 12 + n 21 n 2 • The score test statistic is z = d ˆ σ ( d ) = n 21- n 12 ( n 21 + n 12 ) 1 / 2 • The square of z is a chi-squared distrbution with df = 1 ....
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This note was uploaded on 11/20/2011 for the course STATISTICS ST3241 taught by Professor Manwai's during the Spring '11 term at National University of Singapore.

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Notes 10 - ’ & ST3241 Categorical Data...

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