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

2. Residuals Vertical distance between data & LSRL
Sum of residuals is always zero
Residual = observed – expected

3. Residual Plot Scatterplot of (x, residual) points
Purpose: See if x and y have a linear relationship
No pattern in the residual plot  linear relationship

4. Linear
Not Linear

5. Residuals
Get the regression equation on the home screen (Stat  Calc  8)
Stat  Edit
Highlight "L3"
List (2nd  Stat) , choose 7: RESID
Bottom of screen says "L3 = LRESID"  Hit Enter 
Residual Plot:
With residuals in L3, make a scatterplottwith:   Xlist: L1  Ylist: L3 

6. Age Range of Motion
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
No pattern in the residual plot, so there is a linear relationship between age and range of motion.

7. Sketch a residual plot using the y-hat values on the horizontal axis.
Age Range of Motion
Sketch a residual plot using the y-hat values on the horizontal axis.
Residuals

8. Residual plots are the same plotted against x or y-hat.
Residuals
Residuals
Residual plots are the same plotted against x or y-hat.

9. Coefficient of Determination
Proportion of variation in y that can be explained by the linear relationship between x & y
Remains the same no matter which variable is called x

10. Sum of the squared residuals (errors) using the mean of y
Age Range of Motion
Let’s examine r2.
Suppose we want to predict a y-value, but we don't have an x-value. Best guess: the mean of the y’s we have.
Sum of the squared residuals (errors) using the mean of y
SSEy =

11. Sum of the squared residuals (errors) using the LSRL
Age Range of Motion
Now suppose we want to predict a y-value and we DO have an x-value  we can use y-hat on the LSRL for that x-value.
Sum of the squared residuals (errors) using the LSRL
SSEy =

12. Age Range of Motion
SSEy =
SSEy =
How much does the sum of the squared error go down when we go from the "mean model" to the "regression model"?
This is r2 – the amount of the variation in the y-values that is explained by the x-values

13. 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 LSRL of age and range of motion.

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

15. What is the equation of the LSRL? Find the slope & y-intercept.
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%
NEVER use adjusted r2!
What are the correlation coefficient and the coefficient of determination?
What is the equation of the LSRL?
Find the slope & y-intercept.

16. Outlier
In regression, an outlier is a point with a large residual

17. Influential Point Influences where the LSRL is located
If removed, slope of LSRL changes significantly

18. 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 calculate the new LSRL & correlation coefficient.

19. (189, 30) could be influential. Remove & recalculate LSRL.

20. (189, 30) was influential since it moved the LSRL

21. Which of these measures are resistant?
LSRL
Correlation coefficient (r)
Coefficient of determination (r2)
NONE – all are affected by outliers