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Section 7.4: Other Nonlinear Models

Section 7.4: Other Nonlinear Models. STAT 992 Project by Kendra Schmid April 25, 2006. Outline. Introduction to Nonlinear Regression Nonlinear Regression and the Bootstrap Example 7.7 Extensions of Example 7.7. Nonlinear Regression.

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Section 7.4: Other Nonlinear Models

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  1. Section 7.4:Other Nonlinear Models STAT 992 Project by Kendra Schmid April 25, 2006

  2. Outline • Introduction to Nonlinear Regression • Nonlinear Regression and the Bootstrap • Example 7.7 • Extensions of Example 7.7

  3. Nonlinear Regression • A model is nonlinear if at least one of the derivatives of y with respect to the parameters is a function of at least one of the parameters • General Form • Example

  4. Nonlinear Regression • Intrinsically Linear: Can be linearized by a transformation • Usually not good to do • Intrinsically Nonlinear: Cannot be linearized by any transformation

  5. Nonlinear Regression 3 Main types of nonlinear models • Unbounded—exponential growth • Asymptotic—has lower or upper asymptote, but no inflection point • Sigmoidal—”S-shaped”, has inflection point where the rate of change is greatest

  6. Fitting Nonlinear Models • Recall: • Linear Regression results do not apply exactly • Least Squares Theory can be developed by Linear Approximation using a Taylor series expansion • Least Squares Estimates can often be computed by iterative linear fitting

  7. Fitting Nonlinear Models • Taylor Series Expansion • Where

  8. Distribution of • Based on the linear approximation • Where U is the derivative matrix • S2 is the MSE

  9. Curvature • Curvature is a property of nonlinear models that measures how good the linear approximation is • Parameter Effects—depends on data and parameterization • Intrinsic—can cause bias in fitted values and unreliable inference

  10. Why Bootstrap? • Regular assumptions on errors are not met • Linear approximation is not good due to high intrinsic curvature • Inference will be more reliable if either of the above is true

  11. Example 7.7 • Calcium uptake data • Response: Calcium uptake of cells • Explanatory: Time suspended in a solution of radioactive calcium • Fitted Model:

  12. Example 7.7 • Fit using Nonlinear Least Squares (nls) function in R • Used starting values of 5 and 0.2 • Convergence was reached in 3 iterations using the Gauss-Newton algorithm

  13. Example 7.7 Least Squares Estimates Where = 0.55 with df = 25

  14. Figure 7.10

  15. Example 7.7 3 Resampling Methods Tried • Resample cases by stratified sampling • BMA says they use this method • Case based resampling • Model based resampling

  16. Case Based Resampling • Sample pairs with replacement • Fit the model to each set of n resampled pairs and calculate the least squares parameter estimates • BMA says they resample using time as the strata variable

  17. Model Based Resampling • Sample from the mean adjusted modified residuals where • Calculate using the OLS estimates • Using these values, calculate parameter estimates for each resample

  18. Modified Residuals

  19. Modified Residuals

  20. Bootstrap Results

  21. Figure 7.11 Upper Normality of Parameter Estimates

  22. Figure 7.11 Lower Joint Distributions on original and log scales

  23. Proportion of Maximum • represents the maximum calcium that can be taken up by a cell • represents the proportion of the maximum that has been obtained by time x • Could use the delta method and bivariate normal approximation to calculate CI’s, or just use the bootstrap with the simulated parameter estimates

  24. Proportion of Maximum • Consider times x = 1, 5, 15 • 95% Basic Intervals

  25. Bootstrap Distributions X=15 X=1 X=5

  26. Bootstrap Distributions X=1 X=5 X=15

  27. Comparing 95% Basics

  28. Other Intervals • If the basic doesn’t work well, could do the transformations as illustrated • Why not just use one of the other bootstrap intervals? • Normal, Basic, Percentile, BCa • What is the difference for all of these between original and logit scales?

  29. Bootstrap Intervals

  30. Bootstrap Intervals

  31. Conclusion • Bootstrap is useful for nonlinear models if error assumptions aren’t met or high curvature • Same sampling methods as linear models • Some confidence intervals perform better when looking at aspects such as proportion of maximum • “Some simulation results about confidence intervals and bootstrap methods in nonlinear regression.” Huet, Jolivet, and Messean. 1990.

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