Example Program and Statement Details

Example Graph

The following graph was generated by the Example Program:

Example Program

This example overlays two ELLIPSE statements on a SCATTERPLOT of the same data.

Both ELLIPSE statements use TYPE=PREDICTED.
One ELLIPSE statement uses ALPHA=.2 and the other uses ALPHA=.05.

proc template;
  define statgraph ellipse;
    begingraph;    
      entrytitle "Prediction Ellipses";
      layout overlayequated / equatetype=equate;
        scatterplot x=petallength y=petalwidth /
          datatransparency=.5;
        ellipse x=petallength y=petalwidth /
          type=predicted  alpha=.2
          name="p80" legendlabel="80%"
          outlineattrs=graphconfidence;
        ellipse x=petallength y=petalwidth /
          type=predicted  alpha=.05
          name="p95" legendlabel="95%"
          outlineattrs=graphconfidence2;
        discretelegend "p80" "p95" /
          location=inside autoalign=(topleft);
      endlayout;	
      entryfootnote halign=left "Fisher's Iris Data";
    endgraph;
  end;
run;
proc sgrender data=sashelp.iris template=ellipse; 
run;

Statement Summary

The ELLIPSE statement can be used only within 2-D overlay-type layouts. It computes an ellipse for a set of points specified by the X and Y columns and a confidence level specified by the ALPHA= option. Use the TYPE= option to control whether a predicted or confidence ellipse is generated.

Confidence and Prediction Ellipses

Two types of ellipses can be computed for the input data (where observations correspond to points in a scatter plot). One is a confidence ellipse for the population mean (TYPE=MEAN), and the other is a prediction ellipse for a new observation (TYPE=PREDICT). Both assume a bivariate normal distribution.

Let

and

be the sample mean and sample 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 zero and covariance

, and it is independent of

. Using Hotelling’s

statistic, which is defined as

confidence ellipse for

is computed from the equation

where

is the

critical value of an

distribution with degrees of freedom 2 and

A prediction ellipse is a region for predicting a new observation in the population. It also approximates a region containing a specified percentage of the population.

Denote a new observation as the bivariate random variable

. The variable

is distributed as a bivariate normal variate with mean zero (the zero vector) and covariance

, and it is independent of

. A

prediction ellipse is then given by the equation

The family of ellipses generated by different critical values of the

distribution has a common center (the sample mean) and common major and minor axis directions.

The shape of an ellipse depends on the aspect ratio of the plot. The ellipse indicates the correlation between the two variables if the variables are standardized (by dividing the variables by their respective standard deviations). In this situation, the ratio between the major and minor axis lengths is

In particular, if

, the ratio is 1, which 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.

Required Arguments

X=numeric-column | expression

specifies the numeric column for the X values.

Y=numeric-column | expression

specifies the numeric column for the Y values.

Options

Statement Option	Description
ALPHA	Sets a significance value for the confidence level to compute for the ellipse.
CLIP	Specifies whether the data for the ellipse are considered when determining the data ranges for the axes.
DATATRANSPARENCY	Specifies the degree of the transparency of the ellipse fill color and outline.
DISPLAY	Specifies whether to display an outlined ellipse, a filled ellipse, or an outlined and filled ellipse.
FILLATTRS	Specifies the appearance of the interior fill of the ellipse.
FREQ	Specifies a numeric column that provides frequencies for each observation read.
LEGENDLABEL	Specifies the legend label.
NAME	Assigns a name to a plot statement for reference in other template statements.
OUTLINEATTRS	Specifies the properties of the ellipse outline.
TYPE	Specifies the type of ellipse.
XAXIS	Specifies whether data are mapped to the primary X (bottom) axis or the secondary X2 (top) axis.
YAXIS	Specifies whether data are mapped to the primary Y (left) axis or the secondary Y2 (right) axis.

ALPHA=positive-number

sets a significance value for the confidence level to compute for the ellipse.

Default: .05

Range: 0 < number < 1

ALPHA=.05 represents a 95% confidence level.