Detrended correspondence analysis

The main idea...

A common issue with correspondence analysis (CA) is the production of "arch-shaped" ordinations in which the ends of gradients are compressed and objects at the ends of gradients (the 'tips' at the base of the arch) tend to be closer than expected relative to those toward the arch's apex. The causality of this effect is unclear, however, it is often considered undesirable (see Jackson and Somers, 1991). A popular approach in addressing the arch effect is known as detrended correspondence analysis (DCA) and was developed and implemented by Hill and Gaugh (1980). The algorithm splits the first CA axis into a number of segments (which may be defined) and rescales values along each segment such that they have a mean of zero along the second CA axis. This flattens out the arch-like structure and renders the scores along CA2 meaningless. Additional rescaling decompresses the ordination at the ends of gradients relative to the middle of a gradient (Figure 1). 

DCA may improve the dispersion (the multivariate spread) of points in the ordination and remove the arch effect; however, the detrending and rescaling algorithms have been criticised for being somewhat arbitrary (see Jackson and Somers, 1991; Legendre and Legendre, 1998; Ejrnæs 2000). The introduction of canonical correspondence analysis (CCA) which allows direct gradient analysis offers a widely-used alternative which should be considered prior to conducting DCA for gradient analysis, particularly if pertinent environmental data is at hand

Additional properties

Legendre and Legendre (1998) note that DCA performed using segments and rescaling can help determine the length of ecological gradients when analysing datasets comparable to a "sites × species" table. Assuming a true unimodal distribution of species abundance along the gradient analysed, the units of a DCA axis correspond to the average standard deviation of species turnover. Species turnover (i.e. the appearance, increment in abundance to a maximum, and decrement to absence) can be used to assess whether the gradient sampled was of appropriate length. For example, if a DCA ordination shows that species have not 'turned over' and been replaced over a gradient, extending the sampling campaign may be in order. Thus, DCA may have some utility in the analysis of pilot studies in ecology.
 
Figure 1: Detrending a CA ordination. The original CA solution (a) shows an arch-like effect. Hill and Gauch's (1980) detrending algorithm "flattens" the arch by first segmenting the first axis (CA1) and, for all objects within a given segment, setting the average object scores along CA2 to zero. Additional steps use non-linear rescaling to decompress the ends of the flattened arch, relative to its middle. 

Warnings 
  • Scores along a detrended axis are rendered meaningless.
  • It is not useful to apply DCA to response data which are influenced by only one environmental gradient. In this case, only the first CA axis provides meaningful information (Legendre and Legendre, 1998).
  • Proximity among points should not be considered meaningful; if they (arbitrarily) belonged to different segments during detrending, they will be rescaled differently. This can strongly effect interpretation (Jackson and Somers, 1991).
  • It is vital to report the number of segments used in DCA.
  • Defining different numbers of segments will lead to different ordinations, particularly for datasets with lower sample numbers. Some ordinations with similar numbers of segments may seem similar; however, this is not a true indicator of the stability of the solution.
  • If the eigenvalues of CA axes are similar, performing DCA with differing segment numbers may lead to reordering of axes. 

Implementations
  • R 
    • The decorana() function in vegan is based on Hill and Gaugh's (1980) FORTRAN code
References
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