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: