More Linear Regression
Outliers, Influential Points, and
Confidence Interval Construction
Introduction
The following tutorial will show you how to:
• Make a scatterplot with confidence bands
• Find outliers and influential points in a data
set
• Conduct multiple linear regression,
including an interaction term
• Calculate confidence intervals for
parameter estimates, as well as individual
and mean prediction
Consider the following data set:
The file infant.txt contains data on the net food supply (#
calories per person per day) and the infant mortality rate
(# of infant deaths per 1000 live births) for 22 countries
before World War II. Copy and paste the data into SAS
using the following lines:
http://www.biostat.umn.edu/~susant/PH6415DATA/infant.txt
DATA infant;
INPUT country $ food mortality;
DATALINES;
[paste data lines here]
;
Plotting Confidence Bands
We want to determine whether there is a
relationship between infant mortality and
country’s net food supply.
We also want to construct confidence bands
around our regression line to visually
predict mortality rates given a certain level
of food supply.
SAS Code for Confidence Bands
Type the following code into SAS. This is similar to
previous regression analyses you have conducted, but
two options have been added. “pred” requests the
confidence bands for individual prediction, and “conf”
requests the confidence bands for mean prediction
levels.
PROC REG DATA = infant;
MODEL mortality = food;
PLOT mortality * food / pred conf;
RUN;
Plot of Confidence Bands
Interpreting the Plot
• “PRED” is the regression line
• “U95M” is the Upper 95% Confidence
Interval for mean prediction
• “L95M” is the Lower 95% Confidence
Interval for mean prediction
• “U95” is the Upper 95% Confidence Band
for individual prediction
• “L95” is the Lower 95% Confidence Band
for individual prediction
Notice that the individual prediction bands
are wider than the mean prediction bands.
Now that you have eye-balled the prediction levels,
there is a formal way to calculate mean and
individual prediction for a certain level of x
(food).
Suppose you wanted to know the mean and
individual prediciton mortality rates for a country
with a net food supply of 2900 calories. There is
a simple way to calculate this in SAS. Add
another line of data at the end of your data set
with a made-up country name, 2900 and ‘.’ for
the mortality value.
Remember, SAS sees periods (.) as missing data.
It will not take the missing value into
consideration when calculating the regression
line, but it will calculate prediction CIs for this
value.
SAS Code:
Add a new line of data to the datalines:
…
Uraguay 2380 94.1
Country 2900 .
;
Re-run the data set, so that “Country” has
been added to your data set, then type the
following code into SAS:
PROC REG DATA = infant;
MODEL mortality = food / clb clm cli;
RUN;
Explanation of SAS Code
• “clb” requests the 95% confidence
intervals for the parameter (β) estimates
• “clm” requests the 95% confidence interval
for mean prediction
• “cli” requests the 95% confidence interval
for individual prediction
SAS Output
Interpreting Output
• The Regression Line is:
Yhat = 311.45 – 0.08(food)
• The 95% CI for β1 is [-0.11,-0.05]
Notice that the CI for β1 does not contain 0,
indicating that we reject Ho: β1 = 0. There
is a linear relationship between food and
infant mortality.
The same conclusion is reached by looking
at the p-value for the test statistic (t* =
-5.68, p-value < 0.0001).
95% CI Prediction Output
Interpreting the CI Output
Notice that a new line has been added to your
output (Observation #23). This is the new
country you added with a calorie amount of
2900. The Yhat (Predicted Value) for this calorie
is 78.43, which you could also calculate from the
regression line: Yhat = 311.45 – 0.08(2900).
• The 95% CI for mean/average prediction is
found under “95% CL Mean”: [62.12,94.74]
• The 95% CI for individual/single prediction is
found under “95% CL Predict”: [2.81,154.05]
Notice that the CI for individual prediction is much
wider than that of mean prediction.
Outliers and Influential Points
To determine whether your data set contains
any outliers or points that are influencing
your model, use the options “r” to request
residuals and “influence” to request
measures of influence in your SAS output:
PROC REG DATA = infant;
MODEL mortality = food / r influence;
RUN;
QUIT;
Output from “r” and “influence”
Interpreting Output
• To determine if a point is an outlier, look
for a Student Residual with an absolute
value greater than 2.6.
Observation #7 (Chile) is an outlier, and
observation #16 (Japan) is close to being
an outlier.
• To determine if a point is influential, look
for a Cook’s D value greater than 1.
There appears to be no influential points.