Reciprocal averaging

Graphics

Developed in the 1930's; first introduced in ecology in 1973 (Hill, M.O., J. Ecol. 61:237-249)

Synonymous w/ correspondence analysis, reciprocal ordering, dual scaling

Hill noted that a problem w/ PCA on variance/covariance matrix is that information in rare species is not used in the analysis. RA was proposed as a solution to this problem; other advantages were noted later.

Consider an environmental gradient:

Assume that species B has the same mode as A (i.e., maximum expression is in same place along gradient), but B is less abundant

i.e., A and B have identical distributions, but if you put them in a V-C matrix,

and then do PCA on the V-C matrix, B (w/ small variance) will be near the origin; A (w/ large variance) will be far from the origin:

i.e., A and B appear to have ecologically different behaviors (distributions), but they don't! They have identical modes

PCA on the correlation matrix is more intuitively palatable than PCA on the variance-covariance matrix in this case

RA will give a result similar to PCA on the correlation matrix for this simple example

An additional problem associated w/ PCA is called the "arch" (or horseshoe) effect

PCA forces data onto 2 (or more) axes, even if only one gradient is influencing data

RA reduces the "arch" problem compared to PCA [e.g., Fig. 4.14, Pielou 1984]

Algorithm of RA involves simple matrix algebra on weighted averages

Weighted averages were used in DGA (ref. soil moisture gradient of Dix and Smeins)

RA scores converge to a unique solution which is independent of initial arbitrary species scores

RA tends to ordinate species together based on similar distributions (regardless of abundances)

i.e., species scores are adjusted for abundance

Species w/ restricted distributions exert considerable influence over ordination results

These results may not be ecologically meaningful (i.e., outlying points may not be reflecting environment, but sampling intensity)

Therefore, species which occur infrequently (arbitrarily, > 5% of quadrats) are often removed from data set before analysis

All species and quadrats in any ordination influence the final ordination

Both PCA and RA are linear models

i.e., they assume species are distributed in a linear manner

Projections of non-linear data onto fewer axes than found in the original data gives a misleading picture of the original data (and probably misinterpretation) [ref. Fig. 4.12 Pielou 1984, p. 189]

Departures from linearity in real data are expressed by [ref. Fig. 3.15 Gauch 1982, p. 106; Fig. 4.14 Pielou 1984, p. 193]:

"arch" effect

involution of secondary axes

scale contraction at the ends

These problems are bad enough w/ simulated data (when we know what the gradient is)

With real data, gradients influencing vegetation probably are unknown--that's why we're doing ordination

Note that RA is more robust than PCA to nonlinear relationships between species performances and environment ("arch" effect, involution, and scale contraction are all less pronounced)

A further "advantage" of RA over PCA is the ability to include environmental data in the ordination, according to Gauch and Stone (1979, Amer. Midl. Nat. 102:332-345; cited in Gauch 1982)

Environmental interpretation follows ordination w/ PCA, but can be included w/ RA

Environmental variables are treated as "species" for each quadrat, and numerical values of environmental variables are analogous to species abundances

With any ordination algorithm, correlation coefficients can be calculated between environmental variables and ordination axes

Detrended correspondence analysis

More graphics

Detrended correspondence analysis (i.e., detrended reciprocal averaging)

developed in software form (DECORANA) in 1979

published in a journal in 1980 (Hill & Gauch Vegetatio 42:47-58)

Goals are to remove "arch" effect and compression of first-axis ends

Method:

To get rid of "arch", divide axis 1 into several segments, and adjust axis 2 to have a mean of 0 in each segment

To calculate a third DCA axis, sample scores are detrended w/ respect to the second axis as well as the first ... and so on for higher axes

Several authors (e.g., Pielou, ter Braak) have suggested that DCA is "overzealous" in its removal of the "arch"

A second method for eliminating the "arch" was suggested in 1987 (ter Braak, C.F. Vegetatio 69:69-77)

"Arch" reflects quadratic relationship between first axis and second axis

In addition to the constraint that axes be orthogonal, merely add a second constraint: that axes be uncorrelated w/ the square (cube, etc.) of previous axes

termed "detrending by polynomials", done w/ ter Braak's (1987) canonical correspondence analysis computer program, CANOCO

The second goal of DCA is to "stretch" axis ends

this causes distances in the ordination space to have consistent meaning in terms of compositional differences of samples, or distributional differences of species

DCA is currently popular w/ ecologists, esp. w/ nonlinear data

The most common criticism of DCA is that detrending is artificial

problems ("arch", compression of axes) are "fixed" whether they are real or not

w/ field data, we don't know relationships between species and quadrats

Interpretation and presentation of ordination results (common display options):

1. Quadrat or species lists

   rank order of ordination scores may present a clear gradient (e.g., moisture, successional status)

2. Arranged matrix

   not very useful if 2 gradients (still appears random)

3. Quadrat and species ordination graphs

   if 2 axes, use a series of graphs (e.g., plot 1*2, 1*3)

4. Graph environmental parameters on quadrat ordination

   instead of plotting points for each quadrat, plot value of some environmental variable (e.g., pH); isolines are sometimes drawn

5. Hybrid ordination

   allows various kinds of data to be plotted on one figure

   however, since each "species" (incl. environmental data) and quadrat affects the ordination, it may be undesirable to include environmental data in ordination

6. Trace diagram

   ordination score for one axis (usu. axis 1) and environmental data, plotted by plot location