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Logistic Regression

Logistic Regression. Logistic Regression - Binary Response variable and numeric and/or categorical explanatory variable(s) Goal: Model the probability of a particular outcome as a function of the predictor variable(s) Problem: Probabilities are bounded between 0 and 1.

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Logistic Regression

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  1. Logistic Regression • Logistic Regression - Binary Response variable and numeric and/or categorical explanatory variable(s) • Goal: Model the probability of a particular outcome as a function of the predictor variable(s) • Problem: Probabilities are bounded between 0 and 1

  2. Logistic Regression with 1 Predictor • Response - Presence/Absence of characteristic • Predictor - Numeric variable observed for each case • Model - p(x)  Probability of presence at predictor level x • b = 0  P(Presence) is the same at each level of x • b > 0  P(Presence) increases as x increases • b < 0  P(Presence) decreases as x increases

  3. Logistic Regression with 1 Predictor • b0, b1 areunknown parameters and must be estimated using statistical software such as SPSS, SAS, or STATA • Primary interest in estimating and testing hypotheses regarding b1 • Large-Sample test (Wald Test): • H0: b1= 0 HA: b1 0

  4. Example - Rizatriptan for Migraine • Response - Complete Pain Relief at 2 hours (Yes/No) • Predictor - Dose (mg): Placebo (0),2.5,5,10 Source: Gijsmant, et al (1997)

  5. Example - Rizatriptan for Migraine (SPSS)

  6. Odds Ratio • Interpretation of Regression Coefficient (b1): • 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: • Thus eb1represents the change in the odds of the outcome (multiplicatively) by increasing x by 1 unit • If b1=0, the odds (and probability) are equal at all x levels (eb1=1) • If b1>0 , the odds (and probability) increase as x increases (eb1>1) • If b1< 0 , the odds (and probability) decrease as x increases (eb1<1)

  7. 95% Confidence Interval for Odds Ratio • Step 1: Construct a 95% CI for b : • Step 2: Raise e = 2.718 to the lower and upper bounds of the CI: • 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

  8. Example - Rizatriptan for Migraine • 95% CI for b1: • 95% CI for population odds ratio: • Conclude positive association between dose and probability of complete relief

  9. Multiple Logistic Regression • Extension to more than one predictor variable (either numeric or dummy variables). • With p predictors, the model is written: • Adjusted Odds ratio for raising xi by 1 unit, holding all other predictors constant: • Inferences on bi and ORi are conducted as was described above for the case with a single predictor

  10. Example - ED in Older Dutch Men • Response: Presence/Absence of ED (n=1688) • Predictors: (p=12) • Age stratum (50-54*, 55-59, 60-64, 65-69, 70-78) • Smoking status (Nonsmoker*, Smoker) • BMI stratum (<25*, 25-30, >30) • Lower urinary tract symptoms (None*, Mild, Moderate, Severe) • Under treatment for cardiac symptoms (No*, Yes) • Under treatment for COPD (No*, Yes) * Baseline group for dummy variables Source: Blanker, et al (2001)

  11. Example - ED in Older Dutch Men • Interpretations: Risk of ED appears to be: • Increasing with age, BMI, and LUTS strata • Higher among smokers • Higher among men being treated for cardiac or COPD

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