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Inference About Conditional Associations In 2 x 2 x K Tables Demeke Kasaw Gary Gongwer

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An Example from §2.3 Death Penalties in Florida for Multiple Murders, Odds Ratio = 1.45 Defendants Race Death Penalty Yes No Percent Yes White Black

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Converting this to a 2 X 2 X 2 Table We now have 2 Partial Tables, by race of the victim Conditional Odds Ratios: Victims RaceDefendants Race Death Penalty Yes No Percent Yes White Black BlackWhite Black

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Conditional and Marginal Odds Ratios

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This can be generalized to K different levels To study whether an association exists between an explanatory and response variable after controlling for a possibly confounding variable Different medical centers Severity of Condition Age Different Studies of the same sort (Meta Analysis)

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CenterTreatmentResponse Success Failure Odds Ratio Drug Control Drug Control Drug Control Drug Control Drug Control Drug Control Drug Control Drug Control

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Using logit Models to Test Independence We wish to estimate the conditional probabilities If Y depends on X, then If Y and X are independent

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CMH Test for Conditional Independence

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Estimation of Common Odds Ratio When the association seems stable among the partial tables, it is helpful to combine the K odds ratios into a summary measure of conditional association.

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Testing Homogeneity of Odds Ratios Ha: At least one is different

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SAS CODES data cmh; input center $ treat response count ; datalines; a a a h ; /*Consider 2x2xk*/ proc freq data = cmh; weight count; tables center*treat*response / cmh chisq All; run; /*Consider 2x2*/ proc freq data = cmh; weight count; tables treat*response / cmh chisq All; run;

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Partial outputs Odds Ratio for calculated on each centers; for center 1 Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits Case-Control (Odds Ratio) Center 2 Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits Case-Control (Odds Ratio) Center 3 Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits Case-Control (Odds Ratio)

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Table 5 of treat by response Controlling for center=e treat response Frequency Percent Row Pct Col Pct 1 2 Total ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ Total Table 6 of treat by response Controlling for center=f treat response Frequency Percent Row Pct Col Pct 1 2 Total ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ Total

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Center 7 Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits Case-Control (Odds Ratio) Center 8 Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits Case-Control (Odds Ratio) Total Type of Study Method Value 95% Confidence Limits Case-Control Mantel-Haenszel (Odds Ratio) Logit ** Estimates of the Common Relative Risk (Row1/Row2) Type of Study Method Value 95% Confidence Limits Case-Control Mantel-Haenszel (Odds Ratio) Logit Homogeneity test: Breslow-Day Test for Homogeneity of the Odds Ratios ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square DF 7 Pr > ChiSq Total Sample Size = 273

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Thank you Good luck with Prof. Trumbos Exam

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