1. Chapter 5 Residuals, Residual Plots, & Influential points

2. Residuals (error) - The vertical deviation between the observations & the LSRL always zerothe sum of the residuals is always zero error = observed - expected

3. Residual plot A scatterplot of the (x, residual) pairs. Residuals can be graphed against other statistics besides x linear associationPurpose is to tell if a linear association exist between the x & y variables

4. Consider a population of adult women. Let’s examine the relationship between their height and weight. Height Weight 606468

5. Suppose we now take a random sample from our population of women. Height Weight 606468 Residuals

6. Residual plot A scatterplot of the (x, residual) pairs. Residuals can be graphed against other statistics besides x model is an appropriate fitPurpose is to tell if the model is an appropriate fit between the x & y variables no pattern model is appropriate.If no pattern exists between the points in the residual plot, then the model is appropriate.

7. Model is appropriate Model is NOT appropriate

8. AgeRange of Motion 35154 24142 40137 31133 28122 25126 26135 16135 14108 20120 21127 30122 One measure of the success of knee surgery is post-surgical range of motion for the knee joint following a knee dislocation. Is there a linear relationship between age & range of motion? Sketch a residual plot. Since there is no pattern in the residual plot, there is a linear relationship between age and range of motion x Residuals

9. AgeRange of Motion 35154 24142 40137 31133 28122 25126 26135 16135 14108 20120 21127 30122 Plot the residuals against the y- hats. How does this residual plot compare to the previous one? Residuals

10. Residual plots are the same no matter if plotted against x or y-hat. x Residuals

11. Coefficient of determination- r 2 variationythe proportion of variation in y that can be attributed to a approximate linear relationship between x & y remains the same no matter which variable is labeled x

12. AgeRange of Motion 35154 24142 40137 31133 28122 25126 26135 16135 14108 20120 21127 30122 Let’s examine r 2. Suppose you were going to predict a future y but you didn’t know the x-value. Your best guess would be the overall mean of the existing y’s. SS total = 1564.917 Total sum of the squared deviations - Total variation

13. AgeRange of Motion 35154 24142 40137 31133 28122 25126 26135 16135 14108 20120 21127 30122 Now suppose you were going to predict a future y but you DO know the x-value. Your best guess would be the point on the LSRL for that x-value (y-hat). Sum of the squared residuals using the LSRL. SS Resid = 1085.735

14. AgeRange of Motion 35154 24142 40137 31133 28122 25126 26135 16135 14108 20120 21127 30122 By what percent did the sum of the squared error go down when you went from just an “overall mean” model to the “regression on x” model? SS Resid = 1085.735 SS Total = 1564.917 This is r 2 – the amount of the variation in the y-values that is explained by the x-values.

15. AgeRange of Motion 35154 24142 40137 31133 28122 25126 26135 16135 14108 20120 21127 30122 How well does age predict the range of motion after knee surgery? 30.6% of the variation in range of motion after knee surgery can be explained by the approximate linear regression of age and range of motion.

16. Interpretation of r 2 r 2 %y xy r 2 % of the variation in y can be explained by the approximate linear regression of x & y.

17. Outlier – largeIn a regression setting, an outlier is a data point with a large residual

18. Influential point- A point that influences where the LSRL is located If removed, it will significantly change the slope of the LSRL

19. RacketResonance Acceleration (Hz) (m/sec/sec) 110536.0 210635.0 311034.5 411136.8 511237.0 611334.0 711334.2 811433.8 911435.0 1011935.0 1112033.6 1212134.2 1312636.2 1418930.0 One factor in the development of tennis elbow is the impact-induced vibration of the racket and arm at ball contact. Sketch a scatterplot of these data. Calculate the LSRL & correlation coefficient. Does there appear to be an influential point? If so, remove it and then calculate the new LSRL & correlation coefficient.

20. (189,30) could be influential. Remove & recalculate LSRL

21. (189,30) was influential since it moved the LSRL

22. Which of these measures are resistant? LSRL Correlation coefficient Coefficient of determination NONE NONE – all are affected by outliers