1. Residuals The residuals are estimate of the error
term, εi , in the model.
What to we know about the ei?
Further, since ei are linear function of the Yi they are random
variables with mean and variance….
Further, they have a Normal distribution but they are NOT
uncorrelated.
However, as n  ∞, with number of predictors stay constant, we
have that the correlation in ei’s goes to 0 and the variance become
constant.
So we will ignore these problems with using ei as estimates of εi.
week 5

2. Possible Departures from Model Assumptions
We will use the residuals to examine the following possible departures from the simple linear regression model with normal errors.
The regression function is not linear, i.e, the straight line model is not appropriate.
The error terms do not have constant variance.
The error terms are not normally distributed.
There are outliers and /or influential points.
week 5

3. Residual Plots Residual plots are used to check the model assumptions.
We look for evidence of any of the possible departure described above.
The recommended plots are: residuals versus the predicted values,
residuals versus the Xi and a normal quantile plot of the residuals.
week 5

4. Influential Points, Outliers and Leverage Points
Observations whose inclusion/exclusion result in substantial
changes in the fitted model are said to be influential.
A point can be outlying in any (or all) of the value of the
explanatory variable, the dependent variable or its residual.
Outlier with respect to the residual represents model failure, i.e., line
doesn’t fit this point adequately. These are typically outliers with
respect to the dependent variable.
Outlier with respect to the explanatory variable are called leverage
points. They may be influential.
Points can be any, none or all of the above.
week 5

5. Other Diagnostics tools
Univariate analysis of residuals such as stem-and-leaf plot, box-plot
and histogram is useful for examining departure from the normal
distribution.
Absolute value of residuals versus predicted (fitted) values is useful
in examining if the variance of the errors is constant. This plot show
non-constant variance more sharply.
Residuals versus time or other spatial sequence in observations helps
indicate correlation in observations.
Residuals versus potential other predictors. This plot helps us
determine whether we should include the other predictor in the
model.
week 5