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 -  (x)  Probability of presence at predictor level x  = 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

3. 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 :  = 0 H A :   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 (  ): –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 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) If  < 0, the odds and probability decrease as x increases (e  <1)

7. 95% Confidence Interval for Odds Ratio Step 1: Construct a 95% CI for  : 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  : 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 x i by 1 unit, holding all other predictors constant: Inferences on  i and OR i 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 ( 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-E max 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 E 0 is the initial level A is the coefficient of reduction of P24 antigen k out is 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 = log 10 (RNA change) Predictor: x = MK639 AUC 0-6h Model: Sigmoid-E max : where:  0 is the maximum effect (limit as x  )  1 is the x level producing 50% of maximum effect  2 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