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

2. Residuals (error) -
The vertical deviation between the observations & the LSRL
the 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
Purpose is to tell if a linear association exist between the x & y variables
If no pattern exists between the points in the residual plot, then the association is linear.

4. Linear
Not linear

5. Residuals x Age Range of Motion 35 154 24 142 40 137 31 133 28 122
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. x
Residuals
Since there is no pattern in the residual plot, there is a linear relationship between age and range of motion

6. Age Range of Motion
Plot the residuals against the y-hats. How does this residual plot compare to the previous one?
Residuals

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

8. Coefficient of determination-
gives the proportion of variation in y that can be attributed to an approximate linear relationship between x & y
remains the same no matter which variable is labeled x

9. Sum of the squared residuals (errors) using the mean of y.
Age Range of Motion
Let’s examine r2.
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.
Sum of the squared residuals (errors) using the mean of y.
SSy =

10. Sum of the squared residuals (errors) using the LSRL.
Age Range of Motion
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 (errors) using the LSRL.
SSy =

11. Age Range of Motion
SSy =
SSy =
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?
This is r2 – the amount of the variation in the y-values that is explained by the x-values.

12. How well does age predict the range of motion after knee surgery?
Age Range of Motion
How well does age predict the range of motion after knee surgery?
Approximately 30.6% of the variation in range of motion after knee surgery can be explained by the linear regression of age and range of motion.

13. Interpretation of r2
Approximately r2% of the variation in y can be explained by the LSRL of x & y.

14. Be sure to convert r2 to decimal before taking the square root!
Computer-generated regression analysis of knee surgery data:
Predictor Coef Stdev T P
Constant
Age
s = R-sq = 30.6% R-sq(adj) = 23.7%
Be sure to convert r2 to decimal before taking the square root!
NEVER use adjusted r2!
What is the equation of the LSRL?
Find the slope & y-intercept.
What are the correlation coefficient and the coefficient of determination?

15. Outlier –
In a regression setting, an outlier is a data point with a large residual

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

17. Racket Resonance Acceleration
(Hz) (m/sec/sec)
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.

18. (189,30) could be influential. Remove & recalculate LSRL

19. (189,30) was influential since it moved the LSRL

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