Confidence Ellipse

A confidence ellipse is the bivariate equivalent of a confidence interval. It shows the region containing 95% of the bivariate distribution for a group, assuming the data are bivariate-normally distributed. It is the correct tool for visualizing group overlap in two-dimensional scatter plots —more informative than drawing group hulls or circles, which have no statistical meaning.

STEP 1 —When to Use

Use a confidence ellipse when: you want to show group separation or overlap in a two-variable space; you have multiple groups on a scatter plot and need to communicate their distributional boundaries; the data are approximately bivariate normally distributed (check: are x and y individually approximately normal within each group?).

Do not use when: the data are clearly non-normal or multimodal (the ellipse will misrepresent the group boundary); you want to show the distribution of individual PCA scores (use a KDE contour instead for large n); you have only 2— data points per group (the ellipse will be meaningless).

[Image placeholder: Scatter plot with 3 groups (n=25 each). Each group shown with: individual points (colored by group) and a 95% confidence ellipse in matching color. One group's ellipse overlaps significantly with another. The overlap region is visually highlighted. Center of each ellipse marked with a cross (group centroid).]

95% confidence ellipses per group: individual points + ellipse boundary + centroid cross. Overlap region between groups 1 and 2 is visible.

STEP 2 —Journal Requirement

The legend must state: "Ellipses represent 95% bivariate confidence regions assuming bivariate normality."
n per group must be stated.
If the data are not bivariate-normal, a KDE contour should be used instead (disclosed if ellipse is used despite non-normality).
For PCA score plots, the ellipse represents the 95% confidence region for the group centroid (not the data distribution) —this must be clearly stated.

STEP 3 —Licklider Implementation

Input

Two continuous variables (or PCA/UMAP scores)
Categorical grouping variable
Confidence level: 95% (default), 90%, or 99%
Bivariate normality check: auto-run using Mardia's test (if n >= 20 per group)

Output

Individual scatter points per group (colored by group)
95% confidence ellipse per group (matching color, semi-transparent fill)
Group centroid marked (cross or filled circle)
Bivariate normality test result reported if check was run
Overlap annotation optional: area of intersection as a percentage

[Image placeholder: Licklider confidence ellipse plot. 3 groups (n=25 each), individual points, 95% ellipses with semi-transparent fill, centroid crosses, legend: "Ellipses: 95% bivariate confidence regions (Mardia test: normality acceptable for all groups, all p > 0.15)."]

Licklider confidence ellipses: bivariate normality verified, semi-transparent fills, group centroids, normality check result in legend.

STEP 4 —Draft Output (Draft / Needs review)

Scatter plot of [Y] versus [X] for three groups (n = 25 per group).
Ellipses represent 95% bivariate confidence regions assuming bivariate
normality. Bivariate normality was assessed using Mardia's multivariate
normality test; all groups showed acceptable normality (all p > 0.15).
Group centroids are indicated by crosses. The overlap between Group A
and Group B ellipses indicates partial but not complete group separation
in this two-dimensional space.