Biplots: Do Not Stretch Them!

Two-way tables of data, either observed or standardized in some way, are commonly analyzed by spectral decomposition or singular value decomposition, providing scores for both rows and columns of the two-way classification. Two of the most common examples in plant and crop research are sample × variable data (principal component analysis) and genotype × environment data, the latter either centered for environment only (genotype main effect plus genotype × environment interaction biplots) or doubly centered for both genotype and environment (genotype × environment interaction biplots based on the additive main effects and multiplicative interaction [AMMI] model). Results are often displayed by plotting the row scores, column scores, or both to visually study the structure of the data. Usually, arrows or lines are drawn from the origin to facilitate interpretation. Graphical features such as angles between arrows and distances between points, as well as graphical operations such as orthogonal projections, allow a number of useful interpretations. For the validity of such properties and operations, it is imperative that the two axes of a plot or biplot be equally scaled exactly (i.e., 1 cm on the vertical axis must represent the same number of units as 1 cm on the horizontal axis). Unfortunately, this important fact is often neglected by users when preparing such plots or integrating them into a text document for publication, rendering all of these features of a plot essentially meaningless. The purpose of the present note, therefore, is to highlight the importance of equal scaling using pertinent examples.