When interpreting such a complex variation partitioning diagram, keep in mind
that the R
2 adjustment is done for the fractions that are directly fitted by a linear RDA
model, without resorting to partial RDA or multiple regression (here the first 15 rows
of the table of results); the individual fractions [a] to [p] are then computed by
subtraction. Small negative R
2
adj values frequently appear in this process. Small
negative R
2
adj values generally correspond to explanatory components that explain
less of the response variables’ variation than would be expected by chance for
random normal deviates; so, for all practical purposes, they can be interpreted as
zeros and neglected during interpretation, although they must be taken into account
when computing sums of subsets, or sums of all fractions (the latter is equal to 1)
5 .
The whole set of environmental and spatial variables explains 52.5% of the
variation of the undetrended mite data (see the R
2
adj for “All” fractions). The
environmental variables alone (matrix X1 in the partitioning results) explain
40.8% of the variation, of which a mere 7.3% is not spatially structured (fraction
[a]). This fraction represents species-environment relationships associated with local
environmental conditions.
The fractions involving environmental and spatial variables (in this example,
essentially fractions [e], [g] and [h]) represent spatially structured environmental
variation. Fraction [e] (27.4% variation explained) is common to the environmental
data and the linear (X-Y ) gradient. This is a typical case of induced spatial
dependence, where the spatial structure of environmental factors produces a similar
spatial structure in the response data. Fraction [g] (8.0% variation explained) and
fraction [h] (3.4% variation explained) are common to the environmental and,
respectively, the broad-scale and fine-scale dbMEM variables. These fractions
might also be interpreted as signatures of induced spatial dependence, but with
spatial patterns more complex than linear gradients. However, when some variation
is explained jointly by environmental and spatial variables, one should be careful
when inferring causal species-environment relationships: the correlations may be
due to a direct influence of the environmental variables on the species (direct induced
spatial dependence), or to some unmeasured underlying process that is spatially
structured and is influencing both the mite community and the environmental variables (e.g., spatial variation induced by a historical causal factor).
The variation partitioning results also show that the four sources of variation have
unequal but significant unique contributions: the environment alone ([a], 7.3%) and
the broad-scaled ([c], 7.0%) variation are the largest, while the trend alone ([b],
3.7%) and the fine-scaled variation ([d], 1.6%) play smaller roles.
The variation explained by spatial variables independently of the environment is
represented by fractions [b], [c], [d], [f], [i], [j] and [m]. Together, these fractions
explain 11.6% variation. Most likely, some of this variation, especially at broad and
5 A negative common fraction of variation [b] may also occur if two variables or groups of variables
X and W, together, explain Y better than the sum of their individual effects (Legendre and Legendre
2012, p. 573). This happens, for instance, when the correlation between X and W is negative, but
both are correlated positively with Y.
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7 Spatial Analysis of Ecological Data
that the R
2 adjustment is done for the fractions that are directly fitted by a linear RDA
model, without resorting to partial RDA or multiple regression (here the first 15 rows
of the table of results); the individual fractions [a] to [p] are then computed by
subtraction. Small negative R
2
adj values frequently appear in this process. Small
negative R
2
adj values generally correspond to explanatory components that explain
less of the response variables’ variation than would be expected by chance for
random normal deviates; so, for all practical purposes, they can be interpreted as
zeros and neglected during interpretation, although they must be taken into account
when computing sums of subsets, or sums of all fractions (the latter is equal to 1)
5 .
The whole set of environmental and spatial variables explains 52.5% of the
variation of the undetrended mite data (see the R
2
adj for “All” fractions). The
environmental variables alone (matrix X1 in the partitioning results) explain
40.8% of the variation, of which a mere 7.3% is not spatially structured (fraction
[a]). This fraction represents species-environment relationships associated with local
environmental conditions.
The fractions involving environmental and spatial variables (in this example,
essentially fractions [e], [g] and [h]) represent spatially structured environmental
variation. Fraction [e] (27.4% variation explained) is common to the environmental
data and the linear (X-Y ) gradient. This is a typical case of induced spatial
dependence, where the spatial structure of environmental factors produces a similar
spatial structure in the response data. Fraction [g] (8.0% variation explained) and
fraction [h] (3.4% variation explained) are common to the environmental and,
respectively, the broad-scale and fine-scale dbMEM variables. These fractions
might also be interpreted as signatures of induced spatial dependence, but with
spatial patterns more complex than linear gradients. However, when some variation
is explained jointly by environmental and spatial variables, one should be careful
when inferring causal species-environment relationships: the correlations may be
due to a direct influence of the environmental variables on the species (direct induced
spatial dependence), or to some unmeasured underlying process that is spatially
structured and is influencing both the mite community and the environmental variables (e.g., spatial variation induced by a historical causal factor).
The variation partitioning results also show that the four sources of variation have
unequal but significant unique contributions: the environment alone ([a], 7.3%) and
the broad-scaled ([c], 7.0%) variation are the largest, while the trend alone ([b],
3.7%) and the fine-scaled variation ([d], 1.6%) play smaller roles.
The variation explained by spatial variables independently of the environment is
represented by fractions [b], [c], [d], [f], [i], [j] and [m]. Together, these fractions
explain 11.6% variation. Most likely, some of this variation, especially at broad and
5 A negative common fraction of variation [b] may also occur if two variables or groups of variables
X and W, together, explain Y better than the sum of their individual effects (Legendre and Legendre
2012, p. 573). This happens, for instance, when the correlation between X and W is negative, but
both are correlated positively with Y.
332
7 Spatial Analysis of Ecological Data