statistics 350 lecture 22. today last day: multicollinearity today: example

Statistics 350 Lecture 22

Today

• Last Day: Multicollinearity

• Today: Example

Example

• Investigators studied physical characteristics and ability in 13 football punters

• Each volunteer punted a football ten times

• The investigators recorded the average distance for the ten punts, in feet

• In addition, the investigators recorded five measures of strength and flexibility for each punter: right leg strength (pounds), left leg strength (pounds), right hamstring muscle flexibility (degrees), left hamstring muscle flexibility (degrees), and overall leg strength (foot-pounds)

• From the study "The relationship between selected physical performance variables and football punting ability" by the Department of Health, Physical Education and Recreation at the Virginia Polytechnic Institute and State University, 1983

Example

• Variables:

• Y: Distance traveled in feet

• X1: Right leg strength in pounds

• X2: Left leg strength in pounds

• X3: Right leg flexibility in degrees

• X4: Left leg flexibility in degrees

• X5: Overall leg strength in pounds

Example

Example

• What do you notice from this output?

• Hypothesis Test:

Model Summary

.902a .814 .682 14.64982Model1

R R SquareAdjustedR Square

Std. Error ofthe Estimate

Predictors: (Constant), X5, X4, X2, X1, X3a.

ANOVAb

6590.987 5 1318.197 6.142 .017a

1502.321 7 214.617

8093.308 12

Regression

Residual

Total

Model1

Sum ofSquares df Mean Square F Sig.

Predictors: (Constant), X5, X4, X2, X1, X3a.

Dependent Variable: Yb.

Coefficientsa

-29.580 65.700 -.450 .666

.279 .456 .245 .611 .561

.070 .484 .062 .144 .890

1.241 1.449 .373 .857 .420

-.395 .745 -.131 -.531 .612

.224 .131 .412 1.714 .130

(Constant)

X1

X2

X3

X4

X5

Model1

B Std. Error

UnstandardizedCoefficients

Beta

StandardizedCoefficients

t Sig.

Dependent Variable: Ya.

Example

• Why do you suppose that this phenomenon has occurred?

Coefficient Correlationsa

1.000 .104 .309 -.298 -.456

.104 1.000 .105 .187 -.646

.309 .105 1.000 -.712 -.450

-.298 .187 -.712 1.000 -.103

-.456 -.646 -.450 -.103 1.000

.017 .010 .019 -.018 -.086

.010 .555 .038 .064 -.698

.019 .038 .234 -.157 -.316

-.018 .064 -.157 .208 -.068

-.086 -.698 -.316 -.068 2.100

X5

X4

X2

X1

X3

X5

X4

X2

X1

X3

Correlations

Covariances

Model1

X5 X4 X2 X1 X3

Dependent Variable: Ya.

Example

• What do you notice from this output?

• Hypothesis Test:

Model Summary

.886a .786 .714 13.88586Model1

R R SquareAdjustedR Square

Std. Error ofthe Estimate

Predictors: (Constant), X5, X4, X1a.

ANOVAb

6357.954 3 2119.318 10.991 .002a

1735.355 9 192.817

8093.308 12

Regression

Residual

Total

Model1

Sum ofSquares df Mean Square F Sig.

Predictors: (Constant), X5, X4, X1a.

Dependent Variable: Yb.

Coefficientsa

5.012 45.178 .111 .914

.549 .224 .482 2.454 .037

.109 .518 .036 .211 .838

.266 .109 .489 2.434 .038

(Constant)

X1

X4

X5

Model1

B Std. Error

UnstandardizedCoefficients

Beta

StandardizedCoefficients

t Sig.

Dependent Variable: Ya.

Example

Coefficient Correlationsa

1.000 -.258 -.540

-.258 1.000 -.151

-.540 -.151 1.000

.012 -.015 -.013

-.015 .268 -.017

-.013 -.017 .050

X5

X4

X1

X5

X4

X1

Correlations

Covariances

Model1

X5 X4 X1

Dependent Variable: Ya.

Example

• What do you notice from this output?

• Hypothesis Test:

Model Summary

.886a .785 .741 13.20584Model1

R R SquareAdjustedR Square

Std. Error ofthe Estimate

Predictors: (Constant), X5, X1a.

ANOVAb

6349.366 2 3174.683 18.204 .000a

1743.943 10 174.394

8093.308 12

Regression

Residual

Total

Model1

Sum ofSquares df Mean Square F Sig.

Predictors: (Constant), X5, X1a.

Dependent Variable: Yb.

Coefficientsa

12.768 24.993 .511 .621

.556 .210 .488 2.644 .025

.272 .100 .500 2.709 .022

(Constant)

X1

X5

Model1

B Std. Error

UnstandardizedCoefficients

Beta

StandardizedCoefficients

t Sig.

Dependent Variable: Ya.

Example

• What should we have done before hypothesis tests?

Example

Example

Example

• Other plots:

Example

• Could also look at extra sums of squares:

Example

Coefficientsa

12.768 24.993 .511 .621 -42.919 68.455

.556 .210 .488 2.644 .025 .087 1.025

.272 .100 .500 2.709 .022 .048 .495

(Constant)

X1

X5

Model1

B Std. Error

UnstandardizedCoefficients

Beta

StandardizedCoefficients

t Sig. Lower Bound Upper Bound

95% Confidence Interval for B

Dependent Variable: Ya.

Example

• 95% Confidence interval for 1:

• 95% confidence region for the estimated coefficients:

Example

• Other stuff of possible interest: