Better biplots

The elements of a biplot are (i) a set of axes representing variables, usually concurrent at the centroid of (ii) a set of points representing samples or cases. The axes are (approximations to) conventional coordinate axes, and therefore may be labelled and calibrated. Especially when there are many points (perhaps several thousand) the whole effect can be very confusing but this may be mitigated by:1. Giving a density representation of the points.2. While respecting the calibrations, moving the axes to new positions more remote from the points, and possibly jointly rotating axes and points.3. The use of colour — when permissible.4. Choosing more than one centre of concurrency.The principles are quite general but we illustrate them by examples of the Categorical Principal Component Analysis of the responses to questions concerning migration in Germany. This application introduces the additional interest of representing ordered categorical variables by irregularly calibrated axes.