Adequacy of Linear Regression Models

1. Adequacy of Linear Regression Models http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates 8/6/2014 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

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9. 1. Plot the data and the model

10. Data and model

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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

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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. Correlation coefficient How do you know if r is positive or negative ?

29. What does a particular value of r mean? 0.8 to 1.0 - Very strong relationship 0.6 to 0.8 - Strong relationship 0.4 to 0.6 - Moderate relationship 0.2 to 0.4 - Weak relationship 0.0 to 0.2 - Weak or no relationship 

30. 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

31. Final Exam Grade

32. Final Exam Grade vs Pre-Req GPA

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34. 4. Model meets assumption of random errors

35. 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.

36. Therm exp coeff vs temperature

37. Data and model

38. Plot of Residuals

39. Histograms of Residuals

40. 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.

41. Is there autocorrelation?

42. 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!

43. 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!

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

46. First Order of Polynomial

47. Second Order Polynomial

48. Which model to choose?

49. Optimum Polynomial

50. THE END