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).