Chapter 5Residuals, Residual Plots, &

Influential pointsBy Doug Fritz

Residuals (error) -Residuals (error) -

• The vertical deviation between the observations & the LSRL

• the sum of the residuals is alwaysalways zero zero

• error = observed - expected

yy ˆresidual

Residual plotResidual plot

• A scatterplot of the (x, residual) pairs.

• Residuals can be graphed against other statistics besides x

• Purpose is to tell if a linear associationlinear association exist between the x & y variables

• If no patternno pattern exists between the points in the residual plot, then the association is linearlinear.

Residuals

x

Residuals

x

LinearLinear Not linearNot linear

Age Range of Motion

35 154

24 142

40 137

31 133

28 122

25 126

26 135

16 135

14 108

20 120

21 127

30 122

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.

Since there is no pattern in the residual plot, there is a linear relationship between age and range of motion

x

Residuals

Age Range of Motion

35 154

24 142

40 137

31 133

28 122

25 126

26 135

16 135

14 108

20 120

21 127

30 122

Plot the residuals against the y-hats. How does this residual plot compare to the previous one?

Residualsy

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

x

Residuals Residuals

y

Coefficient of determination-Coefficient of determination-• r2

• gives the proportion of variationvariation in yy that can be attributed to an approximate linear relationship between x & y

• remains the same no matter which variable is labeled x

Age Range of Motion

35 154

24 142

40 137

31 133

28 122

25 126

26 135

16 135

14 108

20 120

21 127

30 122

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.

SSy = 1564.917

Sum of the squared residuals (errors) using

the mean of y.

308.130y

Age Range of Motion

35 154

24 142

40 137

31 133

28 122

25 126

26 135

16 135

14 108

20 120

21 127

30 122

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 = 1085.735

583.107871.ˆ xy

Age Range of Motion

35 154

24 142

40 137

31 133

28 122

25 126

26 135

16 135

14 108

20 120

21 127

30 122

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?

SSy = 1085.735

SSy = 1564.917

3062916671564

7351085916671564

SSE

SSESSE

y

yy

..

..

ˆ

This is r2 – the amount of the

variation in the y-values that is explained by the x-values.

Age Range of Motion

35 154

24 142

40 137

31 133

28 122

25 126

26 135

16 135

14 108

20 120

21 127

30 122

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.

Interpretation of r2

Approximately rr22%% of the variation in yy can be explained by the LSRL of xx & yy.

Computer-generated regression analysis of knee surgery data:

Predictor Coef Stdev T P

Constant 107.58 11.12 9.67 0.000

Age 0.8710 0.4146 2.10 0.062

s = 10.42 R-sq = 30.6% R-sq(adj) = 23.7%

x . . y 871058107ˆ 5532.r

What is the equation of the LSRL?

Find the slope & y-intercept.

NEVER use adjusted r2!

Be sure to convert r2 to decimal beforebefore taking the square

root!

What are the correlation coefficient and the coefficient of

determination?

Outlier –Outlier –

• In a regression setting, an outlier is a data point with a largelarge residual

Influential point-Influential point-

• A point that influences where the LSRL is located

• If removed, it will significantly change the slope of the LSRL

Racket Resonance Acceleration

(Hz) (m/sec/sec)

1 105 36.0

2 106 35.0

3 110 34.5

4 111 36.8

5 112 37.0

6 113 34.0

7 113 34.2

8 114 33.8

9 114 35.0

10 119 35.0

11 120 33.6

12 121 34.2

13 126 36.2

14 189 30.0

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.

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

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

Which of these measures are Which of these measures are resistant?resistant?

• LSRL

• Correlation coefficient

• Coefficient of determination

NONENONE – all are affected by outliers