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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:  = μ μ μ 1 log ) ( g Logistic Regression with 1 Predictor • Response  Presence/Absence of characteristic • Predictor  Numeric variable observed for each case • Model  π ( x ) ≡ Probability of presence at predictor level x x x e e x β α β α π + + + = 1 ) ( • β = 0 ⇒ P(Presence) is the same at each level of x • β > 0 ⇒ P(Presence) increases as x increases • β < 0 ⇒ P(Presence) decreases as x increases Logistic Regression with 1 Predictor • α, β are unknown parameters and must be estimated using statistical software such as SPSS, SAS, or STATA · Primary interest in estimating and testing hypotheses regarding β · LargeSample test (Wald Test): · H : β = 0 H A : β ≠ ) ( : : . . : . . 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 ≥ ≥ = χ χ σ β α β 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 ) ( + + + = π 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 = ≥ = = ≠ = χ β β Odds Ratio • Interpretation of Regression Coefficient ( β ): – 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 π π β • Thus e β represents the change in the odds of the outcome (multiplicatively) by increasing x by 1 unit • If β = 0, the odds and probability are the same at all x levels ( e β =1) • If β > 0 , the odds and probability increase as x increases ( e β >1) • β 95% Confidence Interval for Odds Ratio • Step 1: Construct a 95% CI for β : + ≡ ± ^ ^ ^ ^ ^ ^ ^ ^ ^ 96 . 1 , 96 . 1 96 . 1 β β β σ β σ β σ β • Step 2: Raise e = 2.718 to the lower and upper bounds of the CI: + ^ ^ ^ ^ ^ ^ 96 . 1 96 . 1 , β β σ β σ β 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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This note was uploaded on 07/08/2011 for the course STA 6127 taught by Professor Mukherjee during the Fall '08 term at University of Florida.
 Fall '08
 Mukherjee
 Probability

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