Download - Statistical models for categorical responses
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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
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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
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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:
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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----------------------------------------------------
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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
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-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
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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}