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Logistic and Nonlinear Regression

Logistic and Nonlinear 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

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Logistic and Nonlinear Regression

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  1. Logistic and Nonlinear 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 • Nonlinear Regression: Numeric response and explanatory variables, with non-straight line relationship • Biological (including PK/PD) models often based on known theoretical shape with unknown parameters

  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 • a, b areunknown parameters and must be estimated using statistical software such as SPSS, SAS, or STATA • Primary interest in estimating and testing hypotheses regarding b • Large-Sample test (Wald Test): • H0: b = 0 HA: b 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 (b): • 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 ebrepresents the change in the odds of the outcome (multiplicatively) by increasing x by 1 unit • If b = 0, the odds and probability are the same at all x levels (eb=1) • If b > 0 , the odds and probability increase as x increases (eb>1) • If b < 0 , the odds and probability decrease as x increases (eb<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 b : • 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

  12. Nonlinear Regression • Theory often leads to nonlinear relations between variables. Examples: • 1-compartment PK model with 1st-order absorption and elimination • Sigmoid-Emax S-shaped PD model

  13. Example - P24 Antigens and AZT • Goal: Model time course of P24 antigen levels after oral administration of zidovudine • Model fit individually in 40 HIV+ patients: • where: • E(t) is the antigen level at time t • E0 is the initial level • A is the coefficient of reduction of P24 antigen • koutis the rate constant of decrease of P24 antigen Source: Sasomsin, et al (2002)

  14. Example - P24 Antigens and AZT • Among the 40 individuals who the model was fit, the means and standard deviations of the PK “parameters” are given below: • Fitted Model for the “mean subject”

  15. Example - P24 Antigens and AZT

  16. Example - MK639 in HIV+ Patients • Response: Y = log10(RNA change) • Predictor: x = MK639 AUC0-6h • Model: Sigmoid-Emax: • where: • b0 is the maximum effect (limit as x) • b1 is the x level producing 50% of maximum effect • b2 is a parameter effecting the shape of the function Source: Stein, et al (1996)

  17. Example - MK639 in HIV+ Patients • Data on n = 5 subjects in a Phase 1 trial: • Model fit using SPSS (estimates slightly different from notes, which used SAS)

  18. Example - MK639 in HIV+ Patients

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