**An
Introduction to Ordination**

**Connie Clark**

** Ordination **is a collective term for multivariate
techniques which adapt a multi-dimensional swarm of data points in such a way that when it
is projected onto a two dimensional space any intrinsic pattern the data may possess
becomes apparent upon visual inspection (Pielou, 1984). Basically, ordination serves to
summarize community data (such as species abundance data) by producing a low-dimensional
ordination space in which similar species and samples are plotted close together, and
dissimilar species and samples are placed far apart.

Generally, ordination techniques are used to describe relationships between species composition patterns and the underlying environmental gradients which influence these patterns (asking, what factors structure the community?). For example, if you wanted to examine the distribution patterns of tree species in the Sierra Nevada Mt. Range, ordination could be used to determine which species are commonly found associated with one another, and how the species composition of the community changes with increase in elevation. Recently, use of ordination techniques have expanded to include analysis of dietary overlap (Schluter and Grant, 1982), and to explore patterns of within species morphological differences with geographic distance between populations (Alisauskas, 1998).

**Data**

Commonly, data interpreted using ordination are collected in a species
by sample data matrix, similar to the matrix presented below. Sample data may include
measures of density, biomass, frequency, importance values, presence/absence, or any
number of abundance measures.

E7000 | E6580 | E6000 | E5400 | E5000 | E4000 | E2850 | E1800 | |

ABMA | 88.6 |
144.4 |
21.7 |
52.2 |
0 |
0 |
0 |
0 |

ABCO | 211.4 |
149.3 |
243.2 |
190.5 |
102.4 |
12.4 |
18.7 |
0 |

ACMA | 0 |
0 |
0 |
0 |
0 |
13.1 |
5.5 |
0 |

ARME | 0 |
0 |
0 |
0 |
0 |
0 |
0 |
34.9 |

CADE | 0 |
0 |
0 |
3 |
65 |
33.8 |
36.4 |
28.7 |

CONU | 0 |
0 |
0 |
0 |
0 |
0 |
0 |
11 |

LIDE | 0 |
0 |
0 |
0 |
0 |
2.4 |
0 |
136.6 |

PICO | 0 |
0 |
11.2 |
2.2 |
0 |
0 |
14.7 |
0 |

PILA | 0 |
0 |
0 |
4 |
0 |
0 |
0 |
12.9 |

PIPO | 0 |
0 |
0.8 |
3.5 |
10 |
85.1 |
23.5 |
64.4 |

PIJE | 0 |
6.4 |
9.8 |
28.1 |
16.7 |
0 |
0 |
0 |

PSME | 0 |
0 |
0 |
17.1 |
105.5 |
48.8 |
125 |
5.5 |

QUCH | 0 |
0 |
0 |
0 |
0 |
52.2 |
19.6 |
0 |

QUWI | 0 |
0 |
0 |
0 |
0 |
10 |
7.7 |
0 |

QUKE | 0 |
0 |
0 |
0 |
0 |
47.5 |
46.4 |
0 |

The above is a relatively simple data set. However, it is easy to imagine that a true
data set may encounter dozens of species over hundreds of samples. Complex sample by
species matrices represent dozens to hundreds of dimensions which are impossible to
visualize or interpret. Even graphed, species response curves of large community data sets
can be nearly impossible to interpret. (As they resemble a mess of overlapping peaks and
depressions as shown here.)

Ordination can help us find structure in these complicated data sets. By using various mathematical calculations (which will not be discussed here), ordination techniques will identify similarity between species and samples. Results are then projected onto two dimensions in such a way that species and samples most similar to one another will be close together, and species and samples most dissimilar from one another will appear farther apart (as shown below).

**Ordination techniques:**

There are several different ordination techniques, all of which differ slightly, in the
mathematical approach used to calculate species and sample similarity/dissimiarity. Rather
than reinventing the wheel by discussing each of these techniques in depth, I will offer
only a brief description of the most commonly used methods here. Further details can be
found in the following **suggested references**:

Gauch, H. G., Jr. 1982. Multivariate Analysis in Community Structure. Cambridge University Press, Cambridge

Causton, D. R. 1988. An introduction to vegetation analysis. Unwin Hyman, London.

Kent, M., and P. Coker. 1992. Vegetation description and analysis: a practical approach. Belhaven Press, London.

Pielou, E. C. 1984. The Interpretation of Ecological Data: A Primer on Classification and Ordination. Wiley, New York

Okland, R. H. 1990. Vegetation ecology: theory, methods and applications with reference
to Fennoscandia.

Sommerfeltia Supplement 1:1-233.

Jongman, R. H. G., C. J. F. ter Braak, and O. F. R. van Tongeren, editors. 1987. Data Analysis in Community and Landscape Ecology. Pudoc, Wageningen, The Netherlands.

Analysis of Ecological Communities. Chapman and Hall, London.

**Web Links**

**Note: **this web site comes __highly recommended__ as it provides detailed yet
simple explanations of most currently used ordination techniques (see the Indirect Gradient Analysis section of above
mentioned web page). In the General Reference
section of the web site, Palmer offers a fantastic glossary for terms used in ordination,
and clarifies some common confusion in the terminology used to date. In addition, he
provides links to other ordination sites and offers addresses for software links. In the Statistics and Background section of the site, read through Centroids
and Inertia, Similarity, Distance and Difference, and Explorations in Coenspace for the
conceptual background necessary in understanding ordination techniques. The Direct Gradient Analysis section will be of
interest if you have specific environmental data collected in addition to abundance and
species data. You may find this to be a stronger approach to the analysis of your data
set. Ecological Data, Transformations and
Standardization is for more advanced
users who likely have an understanding of ordination and seek more advanced information
regarding data manipulation.

__
Principal
Components Analysis (PCA)__

PCA** **was one of the earliest ordination techniques applied to ecological data. PCA
uses a rigid rotation to derive orthogonal axes, which maximize the variance in the data
set. Both species and sample ordinations result from a single analysis. Computationally,
PCA is basically an eigenanalysis. The sum of the eigenvalues will equal the sum of the
variance of all variables in the data set. PCA is relatively objective and provides a
reasonable but crude indication of relationships.

__
Reciprocal Averaging (RA)
- Correspondence Analysis__

RA is an ordination technique related conceptually to weighted averages. However,
computationally, RA is related to eigenvector ordinations. RA places sampling units and
species on the same gradients, and maximizes variation between species and sample scores
using a correlation coefficient. It serves as a relatively objective analysis of community
data. Results are generally superior to the results from PCA. However, RA axis ends are
compressed relative to the middle, and the second axis is often a distortion of the first
axis, resulting in an arched effect.

__Detrended Correspondence Analysis (DCA__)

DCA is an eigenvector ordination technique based on Reciprocal Averaging, correcting for
the arch effect produced from RA. Hill and Gauch (1980) report DCA results are superior to
those of RA. Other ecologists criticize the detrending process of DCA. DCA is widely used
for the analysis of community data along gradients. It has also been found effective for
niche ordination of birds by foraging heights (Sabo 1980). DCA ordinates samples and
species simultaneously. It is not appropriate for the analysis of a matrix of similarity
values *between* community data (Gauch, 1982).

__Nonmetric Multidimensional Scaling (NMS)__

NMS actually refers to an entire related family of ordination techniques. These techniques
use rank order information to identify similarity in a data set. NMS is a truly
nonparametric ordination method which seeks to best reduce space portrayal of
relationships. The verdict is still out on this type of ordination. Gauch (1982) claims
NMS is not worth the extra computational effort, and that it gives effective results only
for easy data sets with low diversity. Others hold NMS is extremely effective (Kenkel and
Orloci, 1986, Bradfield and Kenkel, 1987).

**Appropriate uses of ordination:**

It is important to keep in mind that the purpose of ordination is to assist a researcher
to find pattern in data sets that are otherwise too complicated to interpret. A good
ordination technique will be able to identify the most important dimensions in a data set,
and ignore the "noise", in order to show these patterns. However, ordination
techniques should not be used in hypothesis driven analysis. They are meant as **exploratory**
tools. Thus, *post-hoc* analysis is acceptable, and many different techniques can be
tried on the same data set. No null hypothesis can be rejected, nor are p- values
generated to test statistical significance. When p-values are offered, they can only be
used as a rough guide or indicator of underlying processes that MAY BE explaining
community patterns.

**Bibliography**

Alisauskas, R. T. 1998. Winter range expansion and relationships between landscape and
morphometrics of midcontinent Lesser Snow Geese. Auk 115(4 ):851-862.

Brandfield, G. E., and N. C. Kenkel. 1987. Nonlinear ordination using flexible shortest path adjustment of ecological distances. Ecology 68(3):750-753.

Gauch, H. G., Jr. 1982. Multivariate Analysis in Community Structure. Cambridge University Press, Cambridge.

Hill, M. O. and Gauch, H. G. 1980. Deterended correspondence analysis, an improved ordination technique. Vegetatio 42:47-58.

Kenkel, N. C., and L. Orloci. 1986. Applying metric and nonmetric multidimensional scaling to ecological studies: some new results. Ecology 67:919-928.

Pielou, E. C. 1984. The Interpretation of Ecological Data: A Primer on Classification and Ordination. Wiley, New York

Sabo, S. R. 1980. Niche and habitat relations in subalpine bird communities of the White Mountains of New Hampshire. Ecological Monographw:50:241-259.

Schluter. D., and P. R. Grant. 1982. The distribution of *Geospiza difficilis* on
Galapagos islands: test of three hypotheses. Evolution 36:1213-1226

*This page was last updated on
06/02/05*