Nonlinear Regression

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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, 5682-85

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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

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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)

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Nonlinear Least Squares
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Nonlinear Least Squares
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Nonlinear Least Squares
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Estimated Variance-Covariance Matrix
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Orlistat Example

• Reasonable Starting Values
• b0 Mean of 0 Dose Group 5
• b1 Difference between highest mean and dose 0
  mean 33-528
• 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

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Orlistat Example Iteration History (Tolerance
.0001)
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Variance Estimates/Confidence Intervals
Parameter Estimate Std. Error 95 CI
b0 6.12 1.08 (3.96 , 8.28)
b1 27.62 3.48 (20.66 , 34.58)
b2 124.7 47.31 (30.08 , 219.32)
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SAS Code

• Proc nlin
• Parms b05 b128 b2160
• Model y b0 ((b1x)/(b2x))
• Der.b0 1
• Der.b1 x/(b2x)
• Der.b2 -((b1x)/((b2x)2))
• Run