Influential Observations in Regression

1
Influential Observations in Regression

• Measurements on Heat Production as a Function of
  Body Mass and Work Effort.
• M. Greenwood (1918). On the Efficiency of
  Muscular Work, Proc. Roy. Soc. Of London, Series
  B, Vol. 90, 627, pp. 199-214

2
Data Description

• Study involved Algerians accustomed to heavy
  labor. Experiment consisted of several hours on
  stationary bicycle.
• Dependent (Response) Variable
• Heat Production (Calories)
• Independent (Explanatory/Predictor) Variables
• Work Effort (Calories)
• Body Mass (kg)
• Model
• H b0 b1W b2M e

3
Raw Data (Table III, p.203)
4
Estimated Regression Coefficients

• Note that that we can conclude, controlling for
  the other factor
• Work Effort increase ? Heat Production increases
  (p .0136)
• Body Mass increase does not ? Heat Production
  increases (p .1957)

5
Plot of Residuals versus Fitted Values
Huge, Positive, Residual
6
Influential Measures (I) Note n37, p3
Parameters
7
Standardized / Studentized Residuals
8
Influential Measures (II)
9
Influential Measures (III)
10
Diagnosing Influential Observations

• Clearly, Observation 19 exerts a huge influence
  (although it has a small hat or leverage value,
  so it must be near center of Mass/Work
  observations
• Upon further review to authors original
  calculations provided in paper, the mean and S.D.
  are much to high for H (but exactly the same for
  M and W).
• Could observation been a typo?
• Try replacing H193936 with H192936
• Note Do not do this arbitrarily, check your data
  sources in practice

11
Analysis with Corrected Data Point
Note that both factors are significant, and that
the intercept and body mass coefficients have
changed drastically
12
Plot of Residuals versus Predicted Values