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Adequacy of Regression Models

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  1. Adequacy of Regression Models http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates 10/23/2014 http://numericalmethods.eng.usf.edu 1

  2. Data

  3. Is this adequate? Straight Line Model

  4. Quality of Fitted Data • Does the model describe the data adequately? • How well does the model predict the response variable predictably?

  5. Linear Regression Models • Limit our discussion to adequacy of straight-line regression models

  6. Four checks • Plot the data and the model. • Find standard error of estimate. • Calculate the coefficient of determination. • Check if the model meets the assumption of random errors.

  7. Example: Check the adequacy of the straight line model for given data

  8. END

  9. 1. Plot the data and the model

  10. Data and model

  11. END

  12. 2. Find the standard error of estimate

  13. Standard error of estimate

  14. Standard Error of Estimate

  15. Standard Error of Estimate

  16. Standard Error of Estimate

  17. Scaled Residuals 95% of the scaled residuals need to be in [-2,2]

  18. Scaled Residuals

  19. END

  20. 3. Find the coefficient of determination

  21. Coefficient of determination

  22. y x Sum of square of residuals between data and mean

  23. y x Sum of square of residuals between observed and predicted

  24. Limits of Coefficient of Determination

  25. Calculation of St

  26. Calculation of Sr

  27. Coefficient of determination

  28. Caution in use of r2 • Increase in spread of regressor variable (x) in y vs. x increases r2 • Large regression slope artificially yields high r2 • Large r2 does not measure appropriateness of the linear model • Large r2 does not imply regression model will predict accurately

  29. Final Exam Grade

  30. Final Exam Grade vs Pre-Req GPA

  31. END

  32. 4. Model meets assumption of random errors

  33. Model meets assumption of random errors • Residuals are negative as well as positive • Variation of residuals as a function of the independent variable is random • Residuals follow a normal distribution • There is no autocorrelation between the data points.

  34. Therm exp coeff vs temperature

  35. Data and model

  36. Plot of Residuals

  37. Histograms of Residuals

  38. Check for Autocorrelation • Find the number of times, qthe sign of the residual changes for the n data points. • If (n-1)/2-√(n-1) ≤q≤ (n-1)/2+√(n-1), you most likely do not have an autocorrelation.

  39. Is there autocorrelation?

  40. y vs x fit and residuals n=40 (n-1)/2-√(n-1) ≤p≤ (n-1)/2+√(n-1) Is 13.3≤21≤ 25.7? Yes!

  41. y vs x fit and residuals n=40 (n-1)/2-√(n-1) ≤p≤ (n-1)/2+√(n-1) Is 13.3≤2≤ 25.7? No!

  42. END

  43. What polynomial model to choose if one needs to be chosen?

  44. First Order of Polynomial

  45. Second Order Polynomial

  46. Which model to choose?

  47. Optimum Polynomial

  48. THE END

  49. Effect of an Outlier

  50. Effect of Outlier