Logistic and Nonlinear Regression Logistic Regression Dichotomous Response

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 - 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

Logistic Regression with 1 Predictor · a, b are unknown 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): · H 0: b = 0 H A: b 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 (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 eb represents 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)

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

Example - Rizatriptan for Migraine • 95% CI for b : • 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 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

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)

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 1 st-order absorption and elimination – Sigmoid-Emax S-shaped PD model

Example - P 24 Antigens and AZT • Goal: Model time course of P 24 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 P 24 antigen • kout is the rate constant of decrease of P 24 antigen Source: Sasomsin, et al (2002)

Example - P 24 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 - P 24 Antigens and AZT

Example - MK 639 in HIV+ Patients • Response: Y = log 10(RNA change) • Predictor: x = MK 639 AUC 0 -6 h • Model: Sigmoid-Emax: • where: • b 0 is the maximum effect (limit as x ) • b 1 is the x level producing 50% of maximum effect • b 2 is a parameter effecting the shape of the function Source: Stein, et al (1996)

Example - MK 639 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 - MK 639 in HIV+ Patients