Logistic and Nonlinear Regression

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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

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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 gt 0 ? P(Presence) increases as x increases
• b lt 0 ? P(Presence) decreases as x increases

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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)
• H0 b 0 HA b ? 0

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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)
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Example - Rizatriptan for Migraine (SPSS)
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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 (eb1)
• If b gt 0 , the odds and probability increase as
  x increases (ebgt1)
• If b lt 0 , the odds and probability decrease as
  x increases (eblt1)

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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

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Example - Rizatriptan for Migraine

• 95 CI for b

• 95 CI for population odds ratio

• Conclude positive association between dose and
  probability of complete relief

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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

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Example - ED in Older Dutch Men

• Response Presence/Absence of ED (n1688)
• Predictors (p12)
• Age stratum (50-54, 55-59, 60-64, 65-69, 70-78)
• Smoking status (Nonsmoker, Smoker)
• BMI stratum (lt25, 25-30, gt30)
• 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)
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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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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

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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
• kout is the rate constant of decrease of P24
  antigen

Source Sasomsin, et al (2002)
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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

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Example - P24 Antigens and AZT
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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)
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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)

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Example - MK639 in HIV Patients