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Nonlinear Regression Techniques for Orlistat Data Analysis

Explore nonlinear regression models for Orlistat data with varying doses to predict fecal fat excretion. Learn about Emax model and nonlinear least squares estimation. Understand the iterative process and variance-covariance matrix calculation.

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Nonlinear Regression Techniques for Orlistat Data Analysis

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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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