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Significant fractions are identified by asterisks (a = 0.05). Eigenvalues of the first canonical axis (canonical A1) are reported as fractions of the total variance in Y. The second canonical axis was never significant; since each analysis only produced two canonical eigenvalues in this example, the portions of variation corresponding to A1 are the differences between columns 2 and 4 from the left. Fraction [b] is not an independently-calculated component of the variation; hence, it cannot be tested for significance nor decomposed into canonical axes (see Meot et al., 1998, for alternative solutions).

The analysis decomposed the total explained variation [a + b + c] into a significant environmental component [a + b] and a significant component [c] which estimates the spatially-structured variation of Y not explained by the environmental variables. The table shows that [a + b], which is the variation of Y explained by the environmental variables, is mostly spatially structured since [b] represents 89% of [a + b] and [a] is not significant.

Figure 13.28 shows maps of the "fitted site scores" of the first canonical axis of fraction [a + b + c] and of its two components, [a + b] and [c]. These maps were obtained by kriging (Subsection 13.2.2) using program OKB2D of the GSLIB library (Deutsch & Journel, 1992); all-directional spherical variogram models were fitted to the empirical variograms prior to kriging (Subsection 13.1.3). While the proportions of variation of [a + b] and [c] add up to that of [a + b + c] (0.450 + 0.334 = 0.784), this is not the case for the proportions of variation represented by the first canonical axes: A^ + b] + A^j ^A^ + b + c]. This is because the partition Figure 13.28 Bacterial variables: map of the "fitted site scores" of the first canonical axes of three fractions of the variation: top [a + b + c], middle [a + b], bottom [c]. Dots represent the 20 sampling sites. North is nearly parallel to the vertical axis of the maps. Compare with Fig. 13.26b which represents fraction [b + c]. Arrows at the base of map [c], labelled "sea water", indicate the positions of connections of the Thau lagoon with the Mediterranean Sea.

Figure 13.28 Bacterial variables: map of the "fitted site scores" of the first canonical axes of three fractions of the variation: top [a + b + c], middle [a + b], bottom [c]. Dots represent the 20 sampling sites. North is nearly parallel to the vertical axis of the maps. Compare with Fig. 13.26b which represents fraction [b + c]. Arrows at the base of map [c], labelled "sea water", indicate the positions of connections of the Thau lagoon with the Mediterranean Sea.

of fraction [a + b + c] into canonical axes is done independently of the partitions of [a + b] or [c]. As a consequence, maps of a given axis of variation (e.g. axis I of the various fractions, mapped in Fig. 13.28) do not exactly add up with this method; they only add up approximately.

Fraction [b + c] (73.4% of the variation in the bacterial data) is the one extracted by the canonical analysis, for the same data, in the numerical example of Section 13.4; Fig. 13.26 presents two maps of this fraction of the variation. In this example, the maps of axis I of fraction [b + c] (Fig. 13.26) are very similar to the map of axis I of [a + b + c] (Fig. 13.28) because [a] is very small.

Before mapping, all signs of axis I of fractions [a + b] and [c] were reversed to make them agree with the signs of the site scores of axis I of [a + b + c]; signs obtained in unconstrained or constrained ordination analyses are arbitrary. With signs reversed, axis I of [a + b], which is the fraction extracted by the environmental variables, is negatively correlated to variables NH4 and phaeopigments.

The map of axis I of fraction [a + b + c] (63% of the variation in the response bacterial variables) and [a + b] (36%) are quite similar, whereas the map of axis I of fraction [c] (33% of the variation) is quite different. The trend surface equation that produced the "fitted site scores" for the 20 sampling stations is written out by program CANOCO under the heading "Regression/canonical coefficients for standardized variables":

-"■""""' ""'"""—■ 2 3 2 2 3

Axis I of [c] = 1.8017X + 2.2817X - 1.0809X Y - 1.3064XY + 1.5563Y

In this equation, the spatial variables are residuals of the standardized terms of the spatial polynomial after controlling for the effect of the three environmental variables. Examination of the map of fraction [c] suggests a hypothesis for the origin of this fraction of variation: that of a marine influence, which had not been included among the explanatory variables in the analysis. Indeed, the negative values on the map form a plume originating at the connections of the Thau lagoon with the sea and extending westwards. To "explain away" fraction [c], i.e. to make it become non-significant, another analysis could be conducted that would include variables quantifying the marine influence on the stations of the lagoon among the environmental variables. Such variables could be derived from a hydrodynamic model of the lagoon.

Applications of this method cover a wide range of ecological problems. Here is a selected list of fields and papers: palaeoecology (Zeeb et al., 1994; see also Ecological application 10.3), vegetation (Heikkinen & Birks, 1996), periphyton (Cattaneo et al., 1993), protozoa (Buttler et al., 1996), zooplankton (Pinel-Alloul, 1995), aquatic macroinvertebrates (Pinel-Alloul et al., 1996), fish (Rodriguez & Magnan, 1995), and birds (Bersier & Meyer, 1994).

Variation partitioning has been applied to more than two explanatory data sets. (1) Pinel-Alloul et al. (1995) tested the hypothesis that biotic and abiotic factors, as well as spatial structuring, explain together the broad-scale spatial heterogeneity of zooplankton assemblages among lakes. The explanatory variables comprised abiotic (physics and chemistry, morphometry) and biotic factors (phytoplankton and fish assemblages); these factors were analysed separately and together, using four approaches described in the paper. (2) Quinghong & Brakenhielm (1995) explained the spatial patterns of epiphytic green algae and lichens using climatic, pollution, and geographic variables. They showed how to isolate the seven components of variation resulting from crossing three sets of explanatory variables. (3) Anderson & Gribble (1998) extended the variation partitioning method to three matrices of explanatory variables representing the environmental, spatial and temporal components, respectively. They also showed how to isolate the seven components of variation. Using this approach, they were able to resolve the confounding of space and time which is often encountered when sampling is conducted over an extensive period, because of the large size of the area to be surveyed.