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Chapter 5

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Chapter 5

Residuals, Residual Plots, & Influential points

- The vertical deviation between the observations & the LSRL
- the sum of the residuals is always zero
- error = observed - expected

- 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

60

64

68

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

Weight

Height

Residuals

Weight

60

64

68

Height

Suppose we now take a random sample from our population of women.

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

Model is appropriate

Model is NOT appropriate

Residuals

x

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

Residuals

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

Residuals

x

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

- r2
- the 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

AgeRange of Motion

35154

24142

40137

31133

28122

25126

26135

16135

14108

20120

21127

30122

Let’s examine r2.

Suppose you were going to predict a future y but you didn’tknow the x-value. Your best guess would be the overall mean of the existing y’s.

Total sum of the squared deviations -

Total variation

SStotal = 1564.917

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.

SSResid = 1085.735

SSTotal = 1564.917

AgeRange of Motion

35154

24142

40137

31133

28122

25126

26135

16135

14108

20120

21127

30122

SSResid = 1085.735

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.

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.

Interpretation of r2

r2% of the variation in y can be explained by the approximate linear regression of x & y.

Computer-generated regression analysis of knee surgery data:

PredictorCoefStdevTP

Constant107.5811.129.670.000

Age0.87100.41462.100.062

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

Be sure to convert r2 to decimal beforetaking 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?

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

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

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.

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

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

- LSRL
- Correlation coefficient
- Coefficient of determination

NONE – all are affected by outliers