lect03 - Categorical Data Analysis - Lei Sun 1 CHL 5210 -...

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Unformatted text preview: Categorical Data Analysis - Lei Sun 1 CHL 5210 - Statistical Analysis of Qualitative Data Topic: Contingency Tables ctd. Outline Review of 2 2 contingency tables. Odds ratio. Small sample inference. Some relevant SAS codes and outputs. Categorical Data Analysis - Lei Sun 2 Review of 2 2 contingency tables. Birth Smoking Status Weight No Yes Total Normal n 11 ( 11 ) n 12 ( 12 ) n 1 . ( 1 . ) Low n 21 ( 21 ) n 22 ( 22 ) n 2 . ( 2 . ) Total n . 1 ( . 1 ) n . 2 ( n . 2 ) n .. ( .. = 1) Joint, marginal and conditional distributions. Comparing two binomial proportions: normal approximation. Z = 1- 2 r (1- ) n 1 . + (1- ) n 2 . N (0 , 1) . Comparing two binomial proportions: likelihood ratio test. X = 2 ln = 2( l ( )- l ( )) 2 r , 2 X observed log ( observed expected ) , where expected count is under the null hypothesis, e.g. E [ n 11 ] = n 1 . 1 = n 1 . n . 1 n .. . Comparing two binomial proportions: Pearson 2 test of homogeneity. X = X (observed- expected) 2 expected 2 r , Categorical Data Analysis - Lei Sun 3 where the expected count of the ij th cell under the null hypothesis of homogeneity: .j = n .j n .. , and E ij = n i. .j = n i. n .j n .. . Pearson 2 test of independence. X = X (observed- expected) 2 expected 2 r , where the expected count of the ij th cell under the null hypothesis of independence: ij = i. .j = n i. n .. n .j n .. , and E ij = n .. ij = n i. n .j n .. . Test of homogeneity: row (or column) margins are fixed. Test of independence: grand total is fixed. However, they have identical test statistic and d.f. Categorical Data Analysis - Lei Sun 4 Odds, Log-Odds, Odds Ratio, Log-Odds Ratio. For the two parameters, 1 and 2 , there are many ways to measure possible differences, e.g. 1- 2 , 1 / 2 , 1 1- 1 / 2 1- 2 , or simple function of of them such as log, log ( 1 1- 1 / 2 1- 2 ). Odds: 1- . Logistic/Logit/Log-Odds: = logit ( ) = log ( 1- ) . Odds Ratio: 1 1- 1 / 2 1- 2 = 1 (1- 2 ) 2 (1- 1 ) . Log-Odds Ratio (difference in logit) = log ( 1 1- 1 / 2 1- 2 ) = log ( 1 1- 1 )- log ( 2 1- 2 ) = logit ( 1 )- logit ( 2 ) = 1- 2 . Categorical Data Analysis - Lei Sun 5 Relative risk: 1 / 2 . Properties of (log-)odds or (log-)odds ratio. Most importantly: the (log-)odds ratio can be estimated from ei- ther retrospective or prospective studies, e.g. the LBW example is a retrospective study: * We assume the numbers of cases (low) and controls (normal), n 1 ....
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This note was uploaded on 02/22/2012 for the course CHL 5210H taught by Professor Leisun during the Fall '11 term at University of Toronto- Toronto.

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lect03 - Categorical Data Analysis - Lei Sun 1 CHL 5210 -...

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