Diagnostics

1
Diagnostics

• Checking Assumptions and Bad Data

2
Questions

• What is the linearity assumption? How can you
  tell if it seems met?
• What is homoscedasticity (heteroscedasticity)?
  How can you tell if its a problem?
• What is an outlier?
• What is leverage?

• What is a residual?
• How can you use residuals in assuring that the
  regression model is a good representation of the
  data?
• Why consider a standardized residual?
• What is a studentized residual?

3
Linear Model

• Linear relations b/t X and Y
• Normal distribution of error of prediction
• Homoscedasticity (homogeneity of error in Y
  across levels of X)

4
Good-Looking Graph
No apparent departures from line.
5
Same Data, Different Graph
No systematic relations between X and residuals.
6
Problem with Linearity
7
Problem with Heteroscedasticity
Common problem when Y
8
Outliers
Outlier pathological point
9
Review

• What is the linearity assumption? How can you
  tell if it seems met?
• What is homoscedasticity (heteroscedasticity)?
  How can you tell if its a problem?
• What is an outlier?

10
Residuals

• Zresid
• Look for large values (some say zgt2)
• Studentized residual (Student Residual)

The studentized residual considers the distance
of the point from the mean. The farther X is
from the mean, the smaller the standard error and
the larger the residual. Look for large values.
Also, studentized deleted residual (RStudent).
11
Influence Analysis

• Leverage
• Leverage is an index of the importance of an
  observation to a regression analysis.
• Function of X only
• Large deviations from mean are influential
• Maximum is 1 min is 1/N
• Average value is (k1)/N, where k is the number
  of IVs

12
Influence Analysis (2)

• DFBETA and standardized DFBETA
• Change in slope or intercept resulting when you
  delete the ith person.
• Allow for influence of both X and Y

13
Example
r .82 r2 .67 p lt .05.
X Y
2 2
3 3
3 1
4 1
4 3
5 2
8 8
4.14 2.86
SX 1.95, SY 2.41
b1.01, a-1.34
M
14
Example (2)
Y Pred Resid Student Residual Rstudent DFBETA a DFBETAb
2 .6875 1.3125 1.072 1.0923 .7577 -.6044
3 1.7 1.3 .962 .9526 .3943 -.2546
1 1.7 -.7 -.518 -.476 -.1970 .1272
1 2.7125 -1.7125 -1.224 -1.3086 -.2524 .0423
3 2.7125 .2875 .206 .1846 .0356 -.006
2 3.725 -1.725 -1.256 -1.3584 .0198 -.2681
8 6.7625 1.2375 1.803 2.7249 -3.5303 4.4807
15
Remedies

• Fit Curves if needed.
• Note heteroscedasticity for applied problems.
• Investigate all outliers. May delete them or
  not, depending. Report your actions.

16
Review

• What is leverage?
• What is a residual?
• How can you use residuals in assuring that the
  regression model is a good representation of the
  data?
• Why consider a standardized residual?
• What is a studentized residual?