## Parametric Curves: Confidence Ellipses

SAS/INSIGHT software provides two types of confidence
ellipses for each pair of **X** and **Y** variables
assuming a bivariate normal distribution.
One is a confidence ellipse for the population mean,
and the other is a confidence ellipse for prediction.
Let and **S** be the sample mean and the
unbiased estimate of the covariance matrix of a random
sample of size *n* from a bivariate normal distribution
with mean and covariance matrix .

The variable is distributed
as a bivariate normal variate with mean 0 and covariance
,and it is independent of **S**.
The confidence ellipse for is based on
Hotelling's *T*^{2} statistic:

A confidence ellipse
for is defined by the equation

where is the critical value of an *F*
variate with degrees of freedom 2 and *n*-2.
A confidence ellipse for prediction is a confidence region
for predicting a new observation in the population.
It also approximates a region containing
a specified percentage of the population.
Consider **Z** as a bivariate random
variable for a new observation.
The variable
is distributed
as a bivariate normal variate with mean 0 and
covariance ,and it is independent of **S**.

A confidence ellipse
for prediction is then given by the equation

The family of ellipses generated by different
*F* critical values has a common center (the
sample mean) and common major and minor axes.
The ellipses graphically indicate the
correlation between two variables.
When the variable axes are standardized (by dividing the
variables by their respective standard deviations), the ratio
of the two axis lengths (in Euclidean distances) reflects the
magnitude of the correlation between the two variables.
A ratio of 1 between the major and minor axes
corresponds to a circular confidence contour
and indicates that the variables are uncorrelated.
A larger value of the ratio indicates a larger positive
or negative correlation between the variables.

You can choose the level of the confidence region
from the **Confidence Ellipse** menus, as illustrated by
Figure 39.34.

**Figure 39.34:** Confidence Ellipse Menu

A confidence ellipse for the population mean is
displayed with dashed lines, and a confidence ellipse
for prediction is displayed with dotted lines.
Figure 39.35 displays a scatter plot with
50% and 80% confidence ellipses for prediction.
Use the sliders in the **Confidence Ellipses** table
to change the coefficient of the confidence ellipses.

**Figure 39.35:** Confidence Ellipses for Prediction

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.