chapter15 - Logistic Regression Logistic Regression -...

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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 Large-Sample 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 1.180-2.490 .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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chapter15 - Logistic Regression Logistic Regression -...

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