Statistical models for categorical responses

1. Statistical models for categorical responses Logistic Regression Analysis • 2 by 2 Contingency Tables • Odd Ratios • Risk Ratios • Mantel-Haenszel Test Survival Analysis

2. 2 by 2 Contingency Tables • Risk of disease: p1 or p2 • Est: a/(a+b) and c/(c+d) • Risk Ratios: p1/p2 • Est: a/(a+b) • c/(c+d) • Odd Ratios: • a/b or ad • c/d cb

3. 2 Mantel-Haenszel Tests Suppose we have k strata producing k tables such as: • Mantel-Haenszel Test for association in stratified 2x2 tables • O = a1+…+ak • E = (a1+b1) (a1+c1)/n1 +…+ (ak+bk) (ak+ck)/nk • M-H = (|O-E|-0.5)2/V • Mantel-Haenszel Test for common OR in stratified 2x2 tables

4. Dataset EAR Column Variable Format or Code ---------------------------------------------------- 1-3 ID 5 Clearance by 14 days 1=yes/0=no 7 Antibiotic 1=CEF/2=AMO 9 Age 1=<2 yrs/2=2-5 yrs/3=6+ yrs 11 Ear 1=1st ear/2=2nd ear ----------------------------------------------------

5. Logistic Regression Analysis • When to Use it: • Response is categorical • Several Confounding factors or • Numeric Covariates • Logistic Regression model: • Logit(p) = log(p/(1-p))= +1x1+…+ kxk

6. Logistic curves Slope=2 Slope=1 Slope=0.5 Slope=0.1

7. Odds Ratio: • Assume that x1 is a binary factor. • B is the group x1=0 • A is the group x1=1 • Then the odds ratio • OR = pA(1- pA)/(pB(1- pB)) = exp{1}