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

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

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

Conditional and Marginal Odds Ratios

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)

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

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

CMH Test for Conditional Independence

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.

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

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;

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)

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

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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