Regression Diagnostics
Using Residual Plots in SAS
to Determine the
Appropriateness of the Model
Introduction
When conducting linear regression, it is
important to make sure the assumptions
(L.I.N.E.) behind the model are met. It is
also important to verify that the estimated
linear regression model is a good fit for the
data (often a linear regression line can be
estimated by SAS, even if it’s not
appropriate—in this case it is up to you to
judge whether the model is a good one).
Consider the Following Data Set:
Peak blood level data (in mg/ml) were obtained for 20 patients for a
single dose of a drug. In addition to the blood level, the patient’s
weight (in lbs) and the amount of drug (in mg) were recorded. The
data can be found in the file blood.txt with level (column 1), dose
(column 2), and weight (column 3). Use the INFILE statement to read
this data set into SAS:
http://www.biostat.umn.edu/~susant/PH6415DATA/blood.txt
DATA blood;
INFILE 'C:\Documents and Settings\My
Documents\blood.txt' dlm = ' ' firstobs = 1;
INPUT level dose weight;
RUN;
PROC PRINT DATA = blood;
VAR level dose weight;
RUN;
After you have checked your Log for any errors
and the data set has been printed in the Output,
so you can see there is no missing data, create
a plot of the data to determine whether a linear
relationship between level and weight seems
justified:
PROC GPLOT DATA = blood;
PLOT level * weight;
RUN;
Plot of Level * Weight
It appears from the plot that a linear
relationship between blood level and
weight may not be justified. There may be
a slightly negative relationship between
the two variables, but in general there
does not appear to be a linear relationship.
However, we will continue with linear
regression (knowing that it may be
inappropriate), in order to explore
regression diagnostics.
PROG REG
Submit the following program in SAS. In addition to the
first two statements with which you are familiar, the third
statement requests a plot of the residuals by weight and
the fourth statement requests a plot of the studentized
(standardized) residuals by weight:
PROC REG DATA = blood;
MODEL level = weight;
PLOT level * weight;
PLOT residual. * weight;
PLOT student. * weight;
RUN;
Regression Output
Interpreting Output
Notice that the overall F-test has a p-value
of 0.2160, which is greater than 0.05.
Therefore, we would conclude that blood
level and weight are independent (fail to
reject Ho: β1 = 0).
Now look at the following plots:
Plot of Regression Line: Notice it is the same plot as the
one you created from PROC GPLOT, except the fitted
regression line has been added to it.
Plot of residuals * weight: you want an even spread of
points above and below the dashed line. This is a good
way to eyeball the data for potential outliers.
Plot of studentized residuals * weight: look for
values with an absolute value larger than 2.6 to
determine if there are any outliers.
You can see from the plot that the
observation with weight = 128 (observation
#4) is an outlier.
The residual plots also help you determine
whether the assumption of constant
variance is met. Because the residuals
appear to be randomly scattered without
any definite pattern, this suggests that the
data are independent with constant
variance.
The Normality Assumption
A convenient way to test for normality is by
constructing a “Normal Quantile Quantile”
plot. This plots the residuals you would
see under normality versus the residuals
that are actually observed. If the data are
completely normal, the residuals will follow
a 45° line. Use the following code in SAS
to make the NQQ plot:
PLOT residual. * nqq.;
RUN;
Residual vs. NQQ Plot
Interpreting the NQQ Plot
The residuals do not clearly follow a 45° line.
Because the tails of this line seem curved,
this suggests that the data may be
skewed, not normally distributed.
Conclusions
When conducting linear regression, it is
important to verify whether the
assumptions under which the model is
created (L.I.N.E.) are met. This tutorial
has given you an introduction to ways of
assessing whether your data meets the
criteria.