1. Outliers and influential data points

2. No outliers?

3. An outlier? Influential?

9. Impact on regression analyses Not every outlier strongly influences the estimated regression function. Always determine if estimated regression function is unduly influenced by one or a few cases. Simple plots for simple linear regression. Summary measures for multiple linear regression.

10. The hat matrix H

11. Least squares estimates The regression model Fitted values

14. Identifying outlying Y values

15. Residuals Standardized residuals –also called internally studentized residuals Deleted residuals Deleted t residuals –also called studentized deleted residuals –also called externally studentized residuals

16. Residuals Ordinary residuals defined for each observation, i = 1, …, n: Using matrix notation:

17. Variance of the residuals Residual vector Variance matrix Variance of the i th residual Estimated variance of the i th residual

18. Standardized residuals Standardized residuals defined for each observation, i = 1, …, n: Standardized residuals quantify how large the residuals are in standard deviation units. Standardized residuals larger than 2 or smaller than -2 suggest that the y values are unusual.

19. An outlying y value?

20. x y FITS1 HI1 s(e) RESI1 SRES1 0.10000 -0.0716 3.4614 0.176297 4.27561 -3.5330 -0.82635 0.45401 4.1673 5.2446 0.157454 4.32424 -1.0774 -0.24916 1.09765 6.5703 8.4869 0.127014 4.40166 -1.9166 -0.43544 1.27936 13.8150 9.4022 0.119313 4.42103 4.4128 0.99818 2.20611 11.4501 14.0706 0.086145 4.50352 -2.6205 -0.58191... 8.70156 46.5475 46.7904 0.140453 4.36765 -0.2429 -0.05561 9.16463 45.7762 49.1230 0.163492 4.30872 -3.3468 -0.77679 4.00000 40.0000 23.1070 0.050974 4.58936 16.8930 3.68110 S = 4.711 Unusual Observations Obs x y Fit SE Fit Residual St Resid 21 4.00 40.00 23.11 1.06 16.89 3.68R R denotes an observation with a large standardized residual

21. Deleted residuals If observed y i is extreme, it may “pull” the fitted equation towards itself, thereby yielding a small ordinary residual. Delete the i th case, estimate the regression function using remaining n-1 cases, and use the x values to predict the response for the i th case. Deleted residual

22. Deleted t residuals A deleted t residual is just a standardized deleted residual: The deleted t residuals follow a t distribution with ((n-1)-p) degrees of freedom.

23. x y RESI1 TRES1 1 2.1 -1.59 -1.7431 2 3.8 0.24 0.1217 3 5.2 1.77 1.6361 10 2.1 -0.42 -19.7990

24. Row x y RESI1 SRES1 TRES1 1 0.10000 -0.0716 -3.5330 -0.82635 -0.81916 2 0.45401 4.1673 -1.0774 -0.24916 -0.24291 3 1.09765 6.5703 -1.9166 -0.43544 -0.42596... 19 8.70156 46.5475 -0.2429 -0.05561 -0.05413 20 9.16463 45.7762 -3.3468 -0.77679 -0.76837 21 4.00000 40.0000 16.8930 3.68110 6.69012

25. Identifying outlying X values

26. Use the diagonal elements, h ii, of the hat matrix H to identify outlying X values. The h ii are called leverages.

27. Properties of the leverages (h ii ) The h ii is a measure of the distance between the X values for the i th case and the means of the X values for all n cases. The h ii is a number between 0 and 1, inclusive. The sum of the h ii equals p, the number of parameters.

28. HI1 0.176297 0.157454 0.127014 0.119313 0.086145 0.077744 0.065028 0.061276 0.048147 0.049628 0.049313 0.051829 0.055760 0.069311 0.072580 0.109616 0.127489 0.141136 0.140453 0.163492 0.050974 Sum of HI1 = 2.0000

29. Properties of the leverages (h ii ) If the i th case is outlying in terms of its X values, it has a large leverage value h ii, and therefore exercises substantial leverage in determining the fitted value.

30. Using leverages to identify outlying X values Minitab flags any observations whose leverage value, h ii, is more than 3 times larger than the mean leverage value…. …or if it’s greater than 0.99.

31. Unusual Observations Obs x y Fit SE Fit Residual St Resid 21 14.0 68.00 71.449 1.620 -3.449 -1.59 X X denotes an observation whose X value gives it large influence. x y HI1 14.00 68.00 0.357535

32. x y HI2 13.00 15.00 0.311532 Unusual Observations Obs x y Fit SE Fit Residual St Resid 21 13.0 15.00 51.66 5.83 -36.66 -4.23RX R denotes an observation with a large standardized residual. X denotes an observation whose X value gives it large influence.

33. Identifying influential cases

34. Influence A case is influential if its exclusion causes major changes in the estimated regression function.

35. Identifying influential cases Difference in fits, DFITS Cook’s distance measure

36. DFITS The difference in fits … … represent the number of standard deviations that the fitted value increases or decreases when the i th case is included.

37. DFITS A case is influential if the absolute value of its DFIT value is … … greater than 1 for small to medium data sets …greater than for large data sets

38. x y DFIT1 14.00 68.00 -1.23841

39. x y DFIT2 13.00 15.00 -11.4670

40. Cook’s distance Cook’s distance measure … … considers the influence of the i th case on all n fitted values.

41. Cook’s distance Relate D i to the F(p, n-p) distribution. If D i is greater than the 50th percentile, F(0.50, p, n-p), then the i th case has lots of influence.

42. x y COOK1 14.00 68.00 0.701960

43. x y COOK2 13.00 15.00 4.04801