Nonlinear Regression KNNL – Chapter 13.

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Presentation transcript:

Nonlinear Regression KNNL – Chapter 13

Nonlinear Relations wrt X – Linear wrt bs

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 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 - 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)

Nonlinear Least Squares

Nonlinear Least Squares

Estimated Variance-Covariance Matrix

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

Orlistat Example – Iteration History (Tolerance = .0001)

Variance Estimates/Confidence Intervals Parameter Estimate Std. Error 95% CI g0 6.12 1.08 (3.96 , 8.28) g1 27.62 3.48 (20.66 , 34.58) g2 124.7 47.31 (30.08 , 219.32)

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