The presence–absence matrix reloaded: the use and interpretation of range–diversity plots

Aim A great deal of information on distribution and diversity can be extracted from presence–absence matrices (PAMs), the basic analytical tool of many biogeographic studies. This paper presents numerical procedures that allow the analysis of such information by taking advantage of mathematical relationships within PAMs. In particular, we show how range–diversity (RD) plots summarize much of the information contained in the matrices by the simultaneous depiction of data on distribution and diversity. Innovation We use matrix algebra to extract and process data from PAMs. Information on the distribution of species and on species richness of sites is computed using the traditional R (by rows) and Q (by columns) procedures, as well as the new Rq (by rows, considering the structure of columns) and Qr (by columns, considering the structure by rows) methods. Matrix notation is particularly suitable for summarizing complex calculations using PAMs, and the associated algebra allows the implementation of efficient computational programs. We show how information on distribution and species richness can be depicted simultaneously in RD plots, allowing a direct examination of the relationship between those two aspects of diversity. We explore the properties of RD plots with a simple example, and use null models to show that while parameters of central tendency are not affected by randomization, the dispersion of points in RD plots does change, showing the significance of patterns of co‐occurrence of species and of similarity among sites. Main conclusion Species richness and range size are both valid measures of diversity that can be analysed simultaneously with RD plots. A full analysis of a system requires measures of central tendency and dispersion for both distribution and species richness.