Algorithms and biplots for double constrained correspondence analysis

Cajo J.F. Ter Braak*, Petr Šmilauer, Stéphane Dray

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

14 Citations (Scopus)


Correspondence analysis with linear external constraints on both the rows and the columns has been mentioned in the ecological literature, but lacks full mathematical treatment and easily available algorithms and software. This paper fills this gap by defining the method as maximizing the fourth-corner correlation between linear combinations, by providing novel algorithms, which demonstrate relationships with related methods, and by making a detailed study of possible biplots and associated approximations. The method is illustrated using ecological data on the abundances of species in sites and where the species are characterized by traits and sites by environmental variables. The trait data and environment data form the external constraints and the question is which traits and environmental variables are associated, how these associations drive species abundances and how they can be displayed in biplots. With microbiome data becoming widely available, these and related multivariate methods deserve more study as they might be routinely used in the future.
Original languageEnglish
Pages (from-to)171-197
JournalEnvironmental and Ecological Statistics
Issue number2
Early online date16 Jan 2018
Publication statusPublished - Jun 2018


  • Biplot
  • Canonical correlation analysis
  • Canonical correspondence analysis
  • Community ecology
  • Fourth-corner correlation
  • Multivariate analysis
  • Trait-environment relations


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