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Residuals

A continuation ofregression analysis

- Continue to build on regression analysis.
- Learn how residual plotshelp identify problems with the analysis.

CaseXY

1 73 175

2 68 158

3 67 140

4 72 207

5 62 115

^

Wt = – 332.73 + 7.189 Ht

Example 1: Sample of n = 5 students,Y = Weight in pounds,X = Height in inches.

continued …

Prediction equation:

To be foundlater.

r-square = ?

Std. error = ?

Y = – 332.7 + 7.189X

Example 1, continued

220

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200

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160

WEIGHT

Residuals = distance from point to line, measuredparallel to Y- axis.

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140

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HEIGHT

Compute the fitted value and residual for the 4th person in the sample; i.e., X = 72 inches, Y = 207 lbs.

^

y =

fitted value =

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^

y4 - y4

Example 1, continued

-332.73 + 7.189()

= _________

residual = e4 =

=

= __________

Y = – 332.7 + 7.189X

Example 1, continued

e4 = +22.12.

220

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160

WEIGHT

Residuals = distance from point to line, measuredparallel to Y- axis.

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HEIGHT

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e4 is theresidual for the 4th case,= +22.12.

Residual Plot

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Residuals

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Regression line from previous plot is rotated to horizontal.

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HEIGHT

Residual Plot

Scatterplot of residuals versus the predicted means of Y, Y; or an X-variable, or Time.

^

Expect random dispersion around a horizontal line at zero.

Problems occur if: • Unusual patterns • Unusual cases

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Residuals

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Next step: ________ to determineif a recording error has occurred.

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Outliers?

X, or time

Next step: Add a “quadratic term,”or use “______.”

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Residuals

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Nonlinear relationship

X, or time

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Next step: Stabilize variance by using “________.”

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Variance is increasing

X, or time

Unusual patterns:

qPossible curvature in the data.

qVariances that are not constant as X changes.

Unusual cases:

qOutliers

q High leverage cases

q Influential cases

Residual Plots help identifyThree properties of Residuals

illustrated with somecomputations.

73 175

68 158

67 140

72 207

62 115

^

^

e = Y – Y

Y

.01

Property 1.

Y = Weight

X = Height

^

Y = – 332.73 + 7.189 X

Residuals

XY

–17.07

192.07

Find the sum of the

residuals.

156.12

1.88

. . .

round-off error

73 175

68 158

67 140

72 207

62 115

^

^

e = Y – Y

Y

867.98

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Property 2.

Y = Weight

X = Height

^

Y = – 332.73 + 7.189 X

e2

XY

192.07

156.12

148.93184.88112.99

–17.07

1.88

–8.93 22.12 2.01

291.38

3.53

79.74489.29 4.04

Find the sum of squaresof the residuals.

1.Residuals always sum to zero.

“SSE for any other line”.

Sei2= SSE = 867.98 <

Properties of Least Squares Line2. This “least squares” line produces a smaller “Sum of squared residuals” than any other straight line can.

1. Residuals always sum to zero.

2. This “least squares” line produces a smaller “Sum of squared residuals” than any other straight line can.

3. Line always passes through the point ( x, y ).

Properties of Least Squares LineIllustration of unusual cases:

- Outliers
- Leverage
- Influential

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outlier

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“Unusual point” does not follow pattern. It’s near the X-mean; the entire line pulled toward it.

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“Unusual point” does not follow pattern. The line is pulled down and twistedslightly.

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outlier

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“Unusual point” is farfrom the X-mean, but still follows the pattern.

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Highleverage

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“Unusual point” is far from the X-mean, but does not follow the pattern.Line really twists!

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leverage

& outlier,

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High Leverage Case:

An extreme X value relative to the other X values.

Definitions:

Outlier:

An unusual y-value relative to the pattern of the other cases.Usually has a large residual.

has an unusually largeeffecton the slope of the least squares line.

Definitions: continued

Influential CaseHigh leverage

Definitions: continued

Conclusion:

potentially influential.

High leverage & Outlier

influential!!

- Learn two ways to use Minitab to runa regression analysis.
- Learn how to read output from Minitab.

Male

Example 3, continued …

Can height be predicted using shoe size?

Graph

Scatterplot

Plot …

“Jitter” added in X-direction.

The scatter for eachsubpopulation is about the same; i.e., there is“constant variance.”

Copied from “Session Window.”

Can height be predicted using shoe size?

Regression Analysis: Height versus Shoe Size

The regression equation is

Height = 50.5 + 1.87 Shoe Size

Predictor Coef SE Coef T P

Constant 50.5230 0.5912 85.45 0.000

Shoe Siz 1.87241 0.06033 31.04 0.000

S = 1.947 R-Sq = 79.1% R-Sq(adj) = 79.0%

Analysis of Variance

Source DF SS MS F P

Regression 1 3650.0 3650.0 963.26 0.000

Error 255 966.3 3.8

Total 256 4616.3

Least squares estimated coefficients.

Example 3, continued …

Can height be predicted using shoe size?

Regression Analysis: Height versus Shoe Size

The regression equation is

Height = 50.5 + 1.87 Shoe Size

Predictor Coef SE Coef T P

Constant 50.5230 0.5912 85.45 0.000

Shoe Siz 1.87241 0.06033 31.04 0.000

S = 1.947 R-Sq = 79.1% R-Sq(adj) = 79.0%

Analysis of Variance

Source DF SS MS F P

Regression 1 3650.0 3650.0 963.26 0.000

Error 255 966.3 3.8

Total 256 4616.3

Total “Degrees of Freedom”= Number of cases - 1

3650.04616.3

R-Sq =

=

Example 3, continued …

Can height be predicted using shoe size?

Regression Analysis: Height versus Shoe Size

The regression equation is

Height = 50.5 + 1.87 Shoe Size

Predictor Coef SE Coef T P

Constant 50.5230 0.5912 85.45 0.000

Shoe Siz 1.87241 0.06033 31.04 0.000

S = 1.947 R-Sq = 79.1% R-Sq(adj) = 79.0%

Analysis of Variance

Source DF SS MS F P

Regression 1 3650.0 3650.0 963.26 0.000

Error 255 966.3 3.8

Total 256 4616.3

S = MSE

=

Example 3, continued …

Can height be predicted using shoe size?

Regression Analysis: Height versus Shoe Size

The regression equation is

Height = 50.5 + 1.87 Shoe Size

Predictor Coef SE Coef T P

Constant 50.5230 0.5912 85.45 0.000

Shoe Siz 1.87241 0.06033 31.04 0.000

S = 1.947 R-Sq = 79.1% R-Sq(adj) = 79.0%

Analysis of Variance

Source DF SS MS F P

Regression 1 3650.0 3650.0 963.26 0.000

Error 255 966.3 3.8

Total 256 4616.3

Standard Error of Regression.Measure of variation around the regression line.

Sum of squared residuals

Mean Squared ErrorMSE

Can height be predicted using shoe size?

Are there anyproblems visiblein this plot?

___________

No “Jitter” added.

Example 3, continued …

Can height be predicted using shoe size?

Least squares regression equation:

Std. error = 1.947 inches

r-square = 79.1%,

The two summary measuresthat should always begiven with the equation.

Can height be predicted using shoe size?

Stat

Method 2

This program gives a scatterplot with the regression superimposed on it.

Regression

Fitted Line Plot …

Y = a + bX

Can height be predicted using shoe size?

Regression Analysis: Height versus Shoe Size

The regression equation is

Height = 50.5 + 1.87 Shoe Size

Predictor Coef SE Coef T P

Constant 50.5230 0.5912 85.45 0.000

Shoe Siz 1.87241 0.06033 31.04 0.000

S = 1.947 R-Sq = 79.1% R-Sq(adj) = 79.0%

Analysis of Variance

Source DF SS MS F P

Regression 1 3650.0 3650.0 963.26 0.000

Error 255 966.3 3.8

Total 256 4616.3

What information do these values provide?

How do you determine if theX-variable is a useful predictor?

Use the“t-statistic”or the F-stat.

“t” measures how many standard errors the estimated coefficient is from “zero.”

“F” = t2 for simple regression.

How do you determine if theX-variable is a useful predictor?

A “P-value” is associated with “t” and “F”.

The further “t” and “F” are from zero,in either direction, the smaller the corresponding P-value will be.

P-value: a measure of the “likelihoodthat the true coefficient IS ZERO.”

If the P-value IS SMALL (typically “<0.10”),

3

then conclude:

1. It is unlikely that the true coefficient is really zero, and therefore,

2. The X variable IS a useful predictor for the Y variable. Keep the variable!

If the P-value is NOT SMALL (i.e., “> 0.10”), then conclude:

1. For all practical purposes the true coefficient MAY BE ZERO; therefore

2. The X variable IS NOT a useful predictor of the Y variable. Don’t use it.

Can height be predicted using shoe size?

Could “shoe size”have a truecoefficient thatis actually “zero”?

Regression Analysis: Height versus Shoe Size

The regression equation is

Height = 50.5 + 1.87 Shoe Size

Predictor Coef SE Coef T P

Constant 50.5230 0.5912 85.45 0.000

Shoe Siz 1.87241 0.06033 31.04 0.000

S = 1.947 R-Sq = 79.1% R-Sq(adj) = 79.0%

Analysis of Variance

Source DF SS MS F P

Regression 1 3650.0 3650.0 963.26 0.000

Error 255 966.3 3.8

Total 256 4616.3

“t” measures how many standard errors the estimated coefficient is from “zero.”

P-value: a measure of the likelihoodthat the true coefficient is “zero.”

The P-value for Shoe Size IS SMALL (< 0.10).

Conclusion:

The “shoe size” coefficient is NOT zero!“Shoe size” IS a useful predictor of the mean of “height”.

is statistical inference.

This will be covered in more detail during the last three weeks of the course.

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