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

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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.

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Categorical Data Analysis - Lei Sun 2 Review of 2 × 2 contingency tables. Birth Smoking Status Weight No Yes Total Normal n 11 ( π 11 ) n 12 ( π 12 ) n 1 . ( π 1 . ) Low n 21 ( π 21 ) n 22 ( π 22 ) n 2 . ( π 2 . ) Total n . 1 ( π . 1 ) n . 2 ( n . 2 ) n .. ( π .. = 1) Joint, marginal and conditional distributions. Comparing two binomial proportions: normal approximation. Z = ˆ π 1 - ˆ π 2 r ˆ π (1 - ˆ π ) n 1 . + ˆ π (1 - ˆ π ) n 2 . N (0 , 1) . Comparing two binomial proportions: likelihood ratio test. X = 2 lnλ = 2( l ( ˆ θ ) - l ( ˜ θ )) χ 2 r , 2 X observed × log ( observed expected ) , where expected count is under the null hypothesis, e.g. E [ n 11 ] = n 1 . ˜ π 1 = n 1 . n . 1 n .. . Comparing two binomial proportions: Pearson χ 2 test of homogeneity. X = X (observed - expected) 2 expected χ 2 r ,
Categorical Data Analysis - Lei Sun 3 where the expected count of the ij th cell under the null hypothesis of homogeneity: ˜ π .j = n .j n .. , and E ij = n i. ˜ π .j = n i. n .j n .. . Pearson χ 2 test of independence. X = X (observed - expected) 2 expected χ 2 r , where the expected count of the ij th cell under the null hypothesis of independence: ˜ π ij = ˜ π i. ˜ π .j = n i. n .. n .j n .. , and E ij = n .. ˜ π 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.

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Categorical Data Analysis - Lei Sun 4 Odds, Log-Odds, Odds Ratio, Log-Odds Ratio. For the two parameters, π 1 and π 2 , there are many ways to measure possible differences, e.g. π 1 - π 2 , π 1 2 , π 1 1 - π 1 / π 2 1 - π 2 , or simple function of of them such as log, log ( π 1 1 - π 1 / π 2 1 - π 2 ). Odds: π 1 - π . Logistic/Logit/Log-Odds: ψ = logit ( π ) = log ( π 1 - π ) . Odds Ratio: π 1 1 - π 1 / π 2 1 - π 2 = π 1 (1 - π 2 ) π 2 (1 - π 1 ) . Log-Odds Ratio (difference in logit) Δ = log ( π 1 1 - π 1 / π 2 1 - π 2 ) = log ( π 1 1 - π 1 ) - log ( π 2 1 - π 2 ) = logit ( π 1 ) - logit ( π 2 ) = ψ 1 - ψ 2 .
Categorical Data Analysis - Lei Sun 5 Relative risk: π 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 . and n 2 . , are fixed. * We can estimate P (smoking | case) and P (smoking | control). * However, the more interested quantities P (cases | smoking) and P (cases | non-smoking) are not estimable. (They can be estimated from a prospective study where the num- bers of smokers and non-smokers are fixed.) If we measure the difference between the two probabilities by odds ratio, a simple exercise in conditional probability shows: odds(cases | smoking) odds(cases | non-smoking) = odds(smoking | cases) odds(smoking | controls) .

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

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