Introduction

An ordination is a “graphical representation of the similarity of sampling units and/or attributes in resemblance space” (Wildi 2010, p. 35).  The ‘attributes’ in this quote are the suite of response variables – species, traits, chemical concentrations, etc.  The ‘resemblance space’ is the distance or dissimilarity measure used to synthesize the attributes on each sampling unit.

Ordination techniques summarize the data in a reduced number of dimensions while accounting for as much of the variability in the original data set as possible.  As a result, they enable you to visualize relationships among sample units or with respect to individual variables.

Characteristics of Ordination

One author (http://ordination.okstate.edu/ideal.htm) suggests the following desirable qualities for an ordination:

• It recovers gradients without distortion.
• If clusters exist in nature, they should be obvious in the ordination.
• It does not produce clusters which do not exist in nature.
• It gives the same result every time for a given data set.
• There is a unique solution.
• Ecological similarity is related to proximity in ordination space.
• Scaling of axes is related to beta diversity.
• Not sensitive to noise.
• “Signal” and “Noise” are easily separated.
• You do not need to pre-specify the number of axes.
• The solution is the same, no matter how many dimensions one chooses to look at.
• Unless by choice, all sample units are treated equally.
• The solution does not take much computer time.
• The method is robust – it works well for:
  • short and long gradients
  • low and high noise
  • sparse and full matrices
  • big and small data sets
  • species-rich and species-poor systems
• For the mathematician: elegant.
• For the ecologist: available, inexpensive, and easy to understand.

The items on this ‘wish list’ are not equal, and items that are important to one person may not be important to another.  For example, we have enough computing power these days that computer time is no longer an issue for most of these techniques.

There is no “ideal” ordination that possesses all of these qualities.  Different techniques emphasize different qualities.

Types of Ordination

The following distinctions (from http://ordination.okstate.edu/) may be helpful in categorizing different types of ordinations.

Indirect Gradient Analysis

Indirect gradient analysis utilizes a single matrix containing the response variables (e.g., sample units x species).  These methods are also described as being unconstrained because they do not consider other data (environmental characteristics, grouping variables, spatial location, etc.).  Those other data may, however, be used in a post-hoc fashion to interpret the results of an indirect gradient analysis.

Indirect gradient analysis can be conducted via eigenanalysis or on the basis of a distance matrix.

Eigenanalysis-based approaches produce a set of eigenvalues, each of which has an associated eigenvectors.  These approaches are further distinguished by whether they assume linear or unimodal relationships among variables:

• Principal Component Analysis (PCA) assumes a linear relationships among variables
• Correspondence Analysis (CA) and Detrended Correspondence Analysis (DCA) assume a unimodal relationship among variables

Distance-based approaches do not make assumptions about how the variables relate to one another, and instead focus on the patterns contained within the distance matrix.  I use ‘distance’ here as a general term that encompasses both strict distance matrices (produced by metric measures) and dissimilaritiy matrices (produced by semi-metric measures) – see the chapter about distance measures for a refresher on this distinction.  Examples include:

• Non-metric MultiDimensional Scaling (NMDS)
• Principal Coordinates Analysis (PCoA; aka metric multidimensional scaling (MDS))
• Polar ordination (aka Bray-Curtis ordination)

Direct Gradient Analysis

Direct gradient analysis methods relate a matrix of response variables to one or more explanatory variables.  As a result, this type of analysis is constrained.  Given this focus, it should be apparent that these analyses require two objects – a matrix of response variables (e.g., sample units × species) and a matrix or dataframe containing one or more explanatory variables associated with each sample unit.  These types of ordinations are conceptually similar to regression, where the objective is to relate sample units on the basis of their values in the two matrices. 

Direct gradient analysis can be conducted assuming linear or unimodal relationships among variables:

• ReDundancy Analysis (RDA) assumes a linear relationship among variables
• Canonical Correspondence Analysis (CCA) and Detrended Canonical Correspondence Analysis (DCCA) assume a unimodal relationship among variables
• Distance-based Redundancy Analysis (db-RDA; aka Canonical Analysis of Principal Coordinates (CAP)) is analogous to RDA but, as the name suggests, focuses on the distance matrix

References

Wildi, O. 2010. Data analysis in vegetation ecology. Wiley-Blackwell, West Sussex, UK.