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Unformatted text preview: Logistic Regression Logistic Regression  Dichotomous Response variable and numeric and/or categorical explanatory variable(s) Goal: Model the probability of a particular as a function of the predictor variable(s) Problem: Probabilities are bounded between 0 and 1 Distribution of Responses: Binomial Link Function:  = 1 log ) ( g Logistic Regression with 1 Predictor Response  Presence/Absence of characteristic Predictor  Numeric variable observed for each case Model  ( x ) Probability of presence at predictor level x x x e e x + + + = 1 ) ( = 0 P(Presence) is the same at each level of x > 0 P(Presence) increases as x increases < 0 P(Presence) decreases as x increases Logistic Regression with 1 Predictor , are unknown parameters and must be estimated using statistical software such as SPSS, SAS, or STATA Primary interest in estimating and testing hypotheses regarding LargeSample test (Wald Test): H : = 0 H A : ) ( : : . . : . . 2 2 2 1 , 2 2 ^ ^ 2 ^ o b s o b s o b s X P v a l P X R R X S T  = Example  Rizatriptan for Migraine Response  Complete Pain Relief at 2 hours (Yes/No) Predictor  Dose ( mg ): Placebo (0),2.5,5,10 Dose # Patients # Relieved % Relieved 67 2 3.0 2.5 75 7 9.3 5 130 29 22.3 10 145 40 27.6 Example  Rizatriptan for Migraine (SPSS) Variables in the Equation .165 .037 19.819 1 .000 1.1802.490 .285 76.456 1 .000 .083 DOSE Constant Step 1 a B S.E. Wald df Sig. Exp(B) Variable(s) entered on step 1: DOSE. a. x x e e x 165 . 490 . 2 165 . 490 . 2 ^ 1 ) ( + + + = 000 . : 84 . 3 : 819 . 19 037 . 165 . : . . : : 2 1 , 05 . 2 2 2 val P X RR X S T H H obs obs A = = = = Odds Ratio Interpretation of Regression Coefficient ( ): In linear regression, the slope coefficient is the change in the mean response as x increases by 1 unit In logistic regression, we can show that:  = = + ) ( 1 ) ( ) ( ) ( ) 1 ( x x x odds e x odds x odds Thus e represents the change in the odds of the outcome (multiplicatively) by increasing x by 1 unit If = 0, the odds and probability are the same at all x levels ( e =1) If > 0 , the odds and probability increase as x increases ( e >1) 95% Confidence Interval for Odds Ratio Step 1: Construct a 95% CI for : + ^ ^ ^ ^ ^ ^ ^ ^ ^ 96 . 1 , 96 . 1 96 . 1 Step 2: Raise e = 2.718 to the lower and upper bounds of the CI: + ^ ^ ^ ^ ^ ^ 96 . 1 96 . 1 , e e If entire interval is above 1, conclude positive association If entire interval is below 1, conclude negative association If interval contains 1, cannot conclude there is an association Example  Rizatriptan for Migraine...
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 Fall '08
 Mukherjee
 Probability

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