Contribution Biplots

Abstract

To interpret the biplot, it is necessary to know which points—usually variables—are the ones that are important contributors to the solution, especially when there are many variables involved. This information can be calculated separately as part of the biplot's numerical results, but this means that a table has to be consulted along with the graphical display. We propose a new scaling of the display, called the contribution biplot, which incorporates this diagnostic information directly into the display itself, showing visually the important contributors and thus facilitating the biplot interpretation and often simplifying the graphical representation considerably. The contribution biplot can be applied to a wide variety of analyses, such as correspondence analysis, principal component analysis, log-ratio analysis, and various forms of discriminant analysis, and, in fact, to any method based on dimension reduction through the singular value decomposition. In the contribution biplot, one set of points, usually the rows of a data matrix, optimally represents the spatial positions of the cases or sample units, according to an appropriate distance measure. The other set of points, usually the columns of the data matrix, is represented by vectors that are related to their contributions to the low-dimensional solution. A fringe benefit is that often only one common scale for the row and column points is needed on the principal axes, thus avoiding the problem of enlarging or contracting the scale of one set of points to make the biplot legible. Furthermore, the contribution biplot also solves the problem in correspondence analysis and log-ratio analysis of low-frequency categories that are located on the periphery of the map, giving the false impression that they are important, when they are in fact contributing minimally to the solution. This article has supplementary materials online.

Key Words:

• Correspondence analysis
• Discriminant analysis, Log-ratio analysis
• Principal component analysis
• Singular value decomposition
• Weighting

ACKNOWLEDGMENTS

The research support of the BBVA Foundation in Madrid, Spain, is gratefully acknowledged, as well as partial support from the Spanish Ministry of Science and Innovation grants MTM2008-00642 and MTM2009-09063. The associate editor's constructive comments on the earlier versions of this article led to major improvements, as did the comprehensive reports of two reviewers. The Calanus glacialis fatty acid data were kindly provided by Janne Søreide of UNIS (The University Centre in Svalbard).

Additional information

Notes on contributors

Michael Greenacre

Michael Greenacre, Professor of Statistics, Department of Economics and Business, Universitat Pompeu Fabra, Barcelona, Spain, and Barcelona Graduate School of Economics, Barcelona, Spain (E-mail: [email protected]).