Orlistat for Fat Absorption 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

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

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)

Nonlinear Least Squares

Nonlinear Least Squares

Nonlinear Least Squares

Estimated Variance-Covariance Matrix

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

Orlistat Example – Iteration History (Tolerance = .0001)

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)

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;