Statistical models for categorical responsesStatistical models for categorical responses

2 by 2 Contingency Tables2 by 2 Contingency Tables•Odd Ratios•Risk Ratios•Mantel-Haenszel Test Survival AnalysisSurvival Analysis

Logistic Regression AnalysisLogistic Regression Analysis

2 by 2 Contingency Tables2 by 2 Contingency Tables

Disease

Yes No Yes a b a+b=r1

No c d c+d=r2

Exposure

a+c=m1 b+d=m2 n

•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

2 Mantel-Haenszel Tests2 Mantel-Haenszel Tests

•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

Disease Yes No Yes ai bi ai+bi=r1i

No ci di ci+di=r2i

Exposure

ai+ci=m1i bi+di=m2i ni

Suppose we have k strata producing k tables such as:

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----------------------------------------------------

Logistic Regression AnalysisLogistic 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

kk11

kk11

kk11

xx

xx

xx

k1

e11

e1e

)x,...,x|1Y(Pp

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

0.6

0.8

1.0

Dose

P(Y=1)

Slope=0.1

Slope=0.5Slope=1Slope=2

Logistic curvesLogistic curves

Odds Ratio:

Assume that x1 is a binary factor.B is the group x1=0

A is the group x1=1

Then the odds ratioOR = pA(1- pA)/(pB(1- pB)) = exp{1}