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Nonlinear Regression

Nonlinear Regression. KNNL – Chapter 13. Nonlinear Relations wrt X – Linear wrt b s. Nonlinear Regression Models. Data Description - Orlistat. 163 Patients assigned to one of the following doses (mg/day) of orlistat: 0, 60,120,150,240,300,480,600,1200

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Nonlinear Regression

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  1. Nonlinear Regression KNNL – Chapter 13

  2. Nonlinear Relations wrt X – Linear wrtbs

  3. Nonlinear Regression Models

  4. Data Description - Orlistat • 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

  5. Nonlinear Regression Model - Example • Simple Maximum Effect (Emax) model: • g0≡ Mean Response at Dose 0 • g1≡ Maximal Effect of Orlistat (g0+ g1 = Maximum Mean Response) • g2≡ Dose providing 50% of maximal effect (ED50)

  6. Nonlinear Least Squares

  7. Nonlinear Least Squares

  8. Estimated Variance-Covariance Matrix

  9. Orlistat Example • Reasonable Starting Values: • g0: Mean of 0 Dose Group: 5 • g1: Difference between highest mean and dose 0 mean: 33-5=28 • g2: Dose with mean halfway between 5 and 33: 160 • Create Vectors Y and f (g0) • Generate matrix F(g0) • Obtain first “new” estimate of g • Continue to Convergence

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

  11. Variance Estimates/Confidence Intervals

  12. Notes on Nonlinear Least Squares • For small samples: • When errors are normal, independent, with constant variance, we can often use the t-distribution for tests and confidence intervals (software packages do this implicitly) • When the extent of nonlinearity is extreme, or normality assumptions do not hold, should use bootstrap to estimate standard errors of regression coefficients

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