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

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

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

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)

Example - Rizatriptan for Migraine (SPSS)

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)

95% Confidence Interval for Odds Ratio Step 1: Construct a 95% CI for  : Step 2: Raise e = 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

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

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

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)

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

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

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)

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”

Example - P24 Antigens and AZT

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)

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)

Example - MK639 in HIV + Patients