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

2. An Example from §2.3 Death Penalties in Florida for Multiple Murders, 1976-1987 Odds Ratio = 1.45 Defendants Race Death Penalty Yes No Percent Yes White Black 53 430 15 176 11.0 7.9

3. 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 53 414 11 37 11.3 22.9 BlackWhite Black 0 16 4 139 0.0 2.8

4. Conditional and Marginal Odds Ratios

5. 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)

6. CenterTreatmentResponse Success Failure Odds Ratio 1234567812345678 Drug Control Drug Control Drug Control Drug Control Drug Control Drug Control Drug Control Drug Control 11 25 10 27 16 4 22 10 14 5 7 12 2 14 1 16 6 11 0 12 1 10 0 10 1 4 1 8 4 2 6 1 1.19 1.82 4.80 2.29 2.0 0.33

7. 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

8. CMH Test for Conditional Independence

9. 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.

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

11. SAS CODES data cmh; input center $ treat response count ; datalines; a 1 1 11 a 1 2 25 a 2 1 10 h 2 2 1 ; /*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;

12. 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) 1.1880 0.4307 3.2766 Center 2 Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits Case-Control (Odds Ratio) 1.8182 0.4826 6.8496 Center 3 Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits Case-Control (Odds Ratio) 4.8000 1.2044 19.1292

13. Table 5 of treat by response Controlling for center=e treat response Frequency Percent Row Pct Col Pct 1 2 Total ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ 1 6 11 17 20.69 37.93 58.62 35.29 64.71 100.00 47.83 ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ 2 0 12 12 0.00 41.38 41.38 0.00 100.00 0.00 52.17 ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ Total 6 23 29 20.69 79.31 100.00 Table 6 of treat by response Controlling for center=f treat response Frequency Percent Row Pct Col Pct 1 2 Total ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ 1 1 10 11 4.76 47.62 52.38 9.09 90.91 100.00 50.00 ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ 2 0 10 10 0.00 47.62 47.62 0.00 100.00 0.00 50.00 ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ Total 1 20 21 4.76 95.24 100.00

14. Center 7 Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits ---------------------------------------------------------------------------------------------------------------------- Case-Control (Odds Ratio) 2.0000 0.0976 41.0034 Center 8 Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits ---------------------------------------------------------------------------------------------------------------------- Case-Control (Odds Ratio) 0.3333 0.0221 5.0271 Total Type of Study Method Value 95% Confidence Limits --------------------------------------------------------------------------------------------------------------------------- Case-Control Mantel-Haenszel 2.1345 1.1776 3.8692 (Odds Ratio) Logit ** 1.9497 1.0574 3.5949 Estimates of the Common Relative Risk (Row1/Row2) Type of Study Method Value 95% Confidence Limits --------------------------------------------------------------------------------------------------------------------------- Case-Control Mantel-Haenszel 1.4979 0.9151 2.4518 (Odds Ratio) Logit 1.4979 0.9151 2.4518 Homogeneity test: Breslow-Day Test for Homogeneity of the Odds Ratios ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 7.9955 DF 7 Pr > ChiSq 0.3330 Total Sample Size = 273

15. Thank you Good luck with Prof. Trumbos Exam