Logistic Regression and Odds Ratio：Logistic回归分析和比值比.pdf

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Logistic Regression and Odds Ratio：Logistic回归分析和比值比.pdf

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Logistic Regression and Odds Ratio Odds Ratio Review Risk Factor (Benzene) Yes No Total Risk Factor Yes (Exposed) No (Unexposed) Total Outcome (Brain Tumor) Yes No (Case) (Control) 50 20 100 130 150 150 Outcome Yes No (Disease) (NoDisease) ab c a+c d b+d Total 70 230 300 Total a+b c+d n Risk factor can always be arranged as either column or row variable, but the definition of odds ratio is still the same. Let p1 be the probability of success in row 1 (probability of Brain Tumor in row 1) 1 ? p1 is the probability of not success in row 1 (probability of no Brain Tumor in row 1) Odd of getting disease for the people who were exposed to the risk factor: ( p?1 is an estimate of a p1) O+ = P[disease | 1 - P[disease expo sed ] | expo sed ] = P[disease | expo sed ] P[no disease | expo sed ] = 1 p1 ? p1 ≈ p?1 1 ? p?1 = a +b b = a b = 50 20 = 2.5 a+b Let p0 be the probability of success in row 2 (probability of Brain Tumor in row 2) 1 ? p0 is the probability of not success in row 2 (probability of no Brain Tumor in row 2) Odd of getting disease for the people who were not exposed to the risk factor: ( p? 0 is an estimate of p0) O? = P[disease | un expo sed ] 1 - P[disease | un expo sed ] = P[disease | un expo sed ] P[no disease | un expo sed ] c = 1 p0 ? p0 ≈ 1 p? 0 ? p? 0 = c+d d = c d = 100 130 = .77 c+d The Odds Ratio of having brain tumor for people who were exposed to the risk factor versus not exposed: OR = θ = O+ O? = p1 1? p0 1? p1 p0 ≈ θ? = p? 1 1? p? 0 1? p? 1 p? 0 = [a /(a + b)] /[b [c /(c + d )] /[d /(a /(c + + b)] d )] = a b c d = ad bc = 50 ×130 20 ×100 = 3.25 Interpretation: The odds of having brain tumor are 3.25 times higher for those who exposed to benzene than those who were not exposed to benzene. If θ 1, then the odds of success are higher for column 1(risk factor present) than column 2(risk factor not present). If θ 1, then the odds of success are lower for column 1(risk factor present) than column 2(risk factor not present). If θ = 1, then the odds o