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Nonlinear Regression

Nonlinear Regression. Orlistat for Fat Absorption Zhi, J., Melia, A.T., Guericiolini, R. et al. (1994) “Retrospective Population-Based Analysis of the Dose-Response (Fecal Fat Excretion) Relationship of Orlistat in Normal and Obese Volunteers,” Clinical Pharmacology and Therapeutics , 56:82-85.

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Nonlinear Regression

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  1. Nonlinear Regression Orlistat for Fat Absorption Zhi, J., Melia, A.T., Guericiolini, R. et al. (1994) “Retrospective Population-Based Analysis of the Dose-Response (Fecal Fat Excretion) Relationship of Orlistat in Normal and Obese Volunteers,” Clinical Pharmacology and Therapeutics, 56:82-85

  2. Data Description • 163 Patients assigned to one of the following doses (mg/day) of orlistat: 0, 60,120,150,240,300,480,600,1200 • Response measured was fecal fat excretion (purpose is to inhibit fat absorption, so higher levels of response are considered favorable) • Plot of raw data displays a generally increasing but nonlinear pattern and large amount of variation across subjects

  3. Nonlinear Regression Model • Simple Maximum Effect (Emax) model: • b0≡ Mean Response at Dose 0 • b1≡ Maximal Effect of Orlistat (b0+ b1 = Maximum Mean Response) • b2≡ Dose providing 50% of maximal effect (ED50)

  4. Nonlinear Least Squares

  5. Nonlinear Least Squares

  6. Nonlinear Least Squares

  7. Estimated Variance-Covariance Matrix

  8. Orlistat Example • Reasonable Starting Values: • b0: Mean of 0 Dose Group: 5 • b1: Difference between highest mean and dose 0 mean: 33-5=28 • b2: Dose with mean halfway between 5 and 33: 160 • Create Vectors Y and f (b0) • Generate matrix F (b0) • Obtain first “new” estimate of b • Continue to Convergence

  9. Orlistat Example – Iteration History (Tolerance = .0001)

  10. Variance Estimates/Confidence Intervals

  11. SAS Code Proc nlin; Parms b0=5 b1=28 b2=160; Model y = b0 + ((b1*x)/(b2+x)); Der.b0 = 1; Der.b1 = x/(b2+x); Der.b2 = -((b1*x)/((b2+x)**2)); Run;

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