1. Adequacy of Linear Regression Models
Transforming Numerical Methods Education for STEM Undergraduates
5/3/2019 1

2. Data

3. Therm exp coeff vs temperatureα 60
6.36 40
6.24 20
6.12
6.00
-20
5.86
-40
5.72
-60
5.58
-80
5.43 T α
-100
5.28
-120
5.09
-140
4.91
-160
4.72
-180
4.52
-200
4.30
-220
4.08
-240
3.83 T α
-280
3.33
-300
3.07
-320
2.76
-340
2.45

4. Is this adequate?
Straight Line Model

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

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

7. Four checks Does the model look like it explains the data?
Do 95%of the residuals fall with ±2 standard error of estimate?
Is the coefficient of determination acceptable?
Does the model meet the assumption of random errors?

8. Check 1: Does the model look like it explains the data?

9. Data and Model
-340
-260
-180
-100
-20 60
2.45
3.58
4.52
5.28
5.86
6.36

10. Check 2. Do 95%of the residuals fall with ±2 standard error of estimate?

11. Standard error of estimate

12. Standard Error of Estimate
-340
-260
-180
-100
-20 60
2.45
3.58
4.52
5.28
5.86
6.36
2.7357
3.5114
4.2871
5.0629
5.8386
6.6143

13. Standard Error of Estimate

14. Standard Error of Estimate

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

16. Scaled Residuals Ti αi
Residual
Scaled Residual
-340
-260
-180
-100
-20 60
2.45
3.58
4.52
5.28
5.86
6.36

17. 3. Find the coefficient of determination

18. Coefficient of determination

19. Sum of square of residuals between data and meany x

20. Sum of square of residuals between observed and predictedy x

21. Calculation of St
-340
-260
-180
-100
-20 60
2.45
3.58
4.52
5.28
5.86
6.36
1.1850
1.6850

22. Calculation of Sr -340 -260 -180 -100 -20 60 2.45 3.58 4.52 5.28 5.86
6.36
2.7357
3.5114
4.2871
5.0629
5.8386
6.6143

23. Coefficient of determination

24. Limits of Coefficient of Determination

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

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

27. Final Exam Grade

28. Final Exam Grade vs Pre-Req GPA

29. Redoing Check 1, 2 and 3 with 20 data points

30. Check 1:Plot Model and Dataα 60
6.36 40
6.24 20
6.12
6.00
-20
5.86
-40
5.2
-60
5.58
-80
5.43 T α
-100
5.28
-120
5.09
-140
4.91
-160
4.72
-180
4.52
-200
4.30
-220
4.08
-240
3.83
-280
3.33
-300
3.07
-320
2.76
-340
2.45

31. Check 2: Using Standard Error of Estimate

32. Check 3: Using Coefficient of Determination

33. Check 4. Does the model meet assumption of random errors?

34. 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.

35. Are residuals negative and positive?

36. Is variation of residuals as a function of independent variable random?

37. Do the residuals follow normal distribution?

38. END

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

40. Which model to choose?

41. Optimum Polynomial: Wrong Criterion

42. Optimum Polynomial of Regression

43. END

44. Effect of an Outlier

45. Effect of Outlier

46. Effect of Outlier