Prediction Interval Ellipse
In the Ellipse dialog box, you can select
to compute for the 2D scatterplot
the prediction interval ellipse
for a given value of alpha. This interval describes the area in which
a single new observation can be expected to fall with a certain probability
(alpha), given that the new observation comes from a bivariate
normal distribution with the parameters (means,
standard deviations, covariance)
as estimated from the observed points shown in the plot.
The coordinates for the ellipse are computed so that:
[(n-p)*n]/[p*(n-1)*(n+1)]*(X-Xm)' S-1 (X-Xm)~ F(alpha,p,n-p)
where
n |
number
of cases |
p |
number
of variables; i.e., p=2 in the case of the bivariate scatterplot |
X |
vector
of coordinates (pair of coordinates, since p=2) |
Xm |
vector
of means for the p dimensions (variables) in the plot |
S-1 |
inverse
of the variance covariance matrix for the p variables |
F(alpha,p,
n-p) |
the value
of F, given alpha,
p, and n-p |
Note that if the number of observations in
the scatterplot is small, then the prediction
interval may be very large, exceeding the area shown in the graph
for the default scaling of the axes. Thus, in some cases (with small n) you may not see the prediction interval
ellipse on the default graph (change the scaling to show larger intervals
for the two variables in the plot). For additional information see, for
example, Tracy, Young, and Mason (1992), or Montgomery 1996).