Confidence Ellipse

Figure purpose

Confidence ellipse is best read as a grouped scatter overlay. It can help show the broad spread and orientation of groups in a two-dimensional view without forcing the reader to inspect every point equally.

The key limit is that "confidence" here should not be read as a guarantee. This page should stay focused on visual context, not on overclaiming formal certainty.

This page is about a visual summary layer on top of two-dimensional grouped points. It does not produce a p-value, effect size, or formal separation test by itself. Its role is to help readers see one compact summary of center, spread, and orientation around the displayed points.

When to use or avoid

Use confidence ellipse when a grouped scatter already exists and you want a compact visual summary of spread and orientation around the observed points.

Avoid reading ellipse overlap or separation as proof of real group difference. Small samples, unstable covariance estimates, and outliers can change the shape substantially.

This figure works best as an overlay that complements point-level reading rather than replacing it.

That caution matters because an ellipse can look clean and stable even when the underlying point cloud is sparse, skewed, or driven by a few influential observations.

Required columns

One numeric x variable
One numeric y variable
A grouping variable if multiple ellipses are being compared

The current implementation supports grouped two-dimensional scatter overlays. Docs should keep the meaning at that level rather than broadening it into a general multivariate-inference workflow.

At a high level, the ellipse shape is driven by the location and covariance structure of the displayed points within each group. That makes it a useful visual summary of one grouped scatter view, but it does not turn the figure into a general-purpose test of clustering, discrimination, or classification.

Confidence ellipse can help orient the eye, but it does not prove separation, significance, or stable boundary structure. Readers should keep sample size and outlier sensitivity in mind when interpreting the plotted shape.

Licklider does not determine automatically from ellipse overlap or non-overlap whether groups differ significantly, whether one group is "truly" separate, or whether the plotted boundary is stable to small data changes. The current figure is an overlay, not a standalone inference system.

This limitation matters because two groups can show visually separated ellipses without supporting a robust inferential claim, and two groups can show overlapping ellipses even when a formal analysis would still detect a meaningful difference.

Design Rationale & References

This page follows a simple rule: use the ellipse to summarize one scatter view, but do not ask it to carry more inferential meaning than the plotted points can support.

That is why the page stays conservative. The ellipse is useful for quick visual orientation, especially when many points make group shape harder to read at a glance. But the same compactness that makes it useful also hides details about outliers, skew, multimodality, and sample-size fragility.

Friendly, M., Monette, G., & Fox, J. (2013). Elliptical Insights: Understanding Statistical Methods through Elliptical Geometry. Statistical Science, 28(1), 1-39. https://doi.org/10.1214/12-STS402
Fox, J., & Weisberg, S. (2019). An R Companion to Applied Regression (3rd ed.). Sage.

These references support the use of elliptical geometry as a visual summary of multivariate spread and orientation. In Licklider, the confidence ellipse uses that idea as an exploratory overlay; it does not automatically certify separation, significance, or one correct group boundary.

Alternative figures

Use Scatter Plot when point-level visibility is more important than a summary overlay.
Use PCA Biplot when you need a reduced two-axis multivariate view rather than a direct grouped scatter overlay.
Use Convex Hull when a geometric outer boundary is easier to read than an ellipse summary.

TODO (Phase02+)