river, and therefore their apparent effect on the fish community may have been
spurious and has been “removed” by the analysis, which controlled for the effect of
the physiographic variables. The scaling 1 triplot shows that the sites are not as
cleanly ordered by their succession along the river. This indicates that the chemical
variables that are important for the fishes do not necessarily follow that order and that
the fish community responds significantly to these chemical constraints irrespective
of their locations along the river.
6.3.2.6 Selection of Explanatory Variables
It happens sometimes that one wishes to reduce the number of explanatory variables.
The reasons vary: search for parsimony, rich data set but poor a priori hypotheses,
very small n, or a method producing a large set of explanatory variables which must
be reduced afterwards (as in eigenvector-based spatial analysis, see Chap. 7). In the
Doubs data, there could be two reasons (albeit not compelling) to reduce the number
of explanatory variables: search for parsimony, and possible strong linear dependencies (correlations) among the explanatory variables in the RDA model, which
could render the regression coefficients of the explanatory variables in the model
unstable.
Linear dependencies can be explored by computing the X variables’ variance
inflation factors (VIF), which measure to what extent each variable in a data set X is
collinear with the others. High VIFs are found when two variables are highly
intercorrelated or when one variable is a strong linear combination of several others.
VIF values above 20 indicate strong collinearity. Ideally, VIFs above 10 should be at
least examined, and avoided if possible. High VIFs may indicate variables that are
functionally related to one another. In that case, one can remove a variable from such
a group on the basis of ecological reasoning. For example, if two variables measure
the same basic environmental property but in different manners, e.g. total N vs NO 3
À
, one of them can be removed without harm.
Contrary to what one sometimes reads, variables X with high VIFs should
generally not be manually removed before the application of a procedure of selection
of variables. Indeed, two highly correlated variables that are both strong predictors of
one or several of the response variables Y may both contribute significantly, in
complementary manners, to the linear model of these response variables. A variable
selection procedure is the appropriate way of determining if that is the case.
For a matrix X containing quantitative variables only, VIF indices can be easily
computed with the following R code:
vif <- diag(solve(cor(X)))
6.3 Redundancy Analysis (RDA)
225
spurious and has been “removed” by the analysis, which controlled for the effect of
the physiographic variables. The scaling 1 triplot shows that the sites are not as
cleanly ordered by their succession along the river. This indicates that the chemical
variables that are important for the fishes do not necessarily follow that order and that
the fish community responds significantly to these chemical constraints irrespective
of their locations along the river.
6.3.2.6 Selection of Explanatory Variables
It happens sometimes that one wishes to reduce the number of explanatory variables.
The reasons vary: search for parsimony, rich data set but poor a priori hypotheses,
very small n, or a method producing a large set of explanatory variables which must
be reduced afterwards (as in eigenvector-based spatial analysis, see Chap. 7). In the
Doubs data, there could be two reasons (albeit not compelling) to reduce the number
of explanatory variables: search for parsimony, and possible strong linear dependencies (correlations) among the explanatory variables in the RDA model, which
could render the regression coefficients of the explanatory variables in the model
unstable.
Linear dependencies can be explored by computing the X variables’ variance
inflation factors (VIF), which measure to what extent each variable in a data set X is
collinear with the others. High VIFs are found when two variables are highly
intercorrelated or when one variable is a strong linear combination of several others.
VIF values above 20 indicate strong collinearity. Ideally, VIFs above 10 should be at
least examined, and avoided if possible. High VIFs may indicate variables that are
functionally related to one another. In that case, one can remove a variable from such
a group on the basis of ecological reasoning. For example, if two variables measure
the same basic environmental property but in different manners, e.g. total N vs NO 3
À
, one of them can be removed without harm.
Contrary to what one sometimes reads, variables X with high VIFs should
generally not be manually removed before the application of a procedure of selection
of variables. Indeed, two highly correlated variables that are both strong predictors of
one or several of the response variables Y may both contribute significantly, in
complementary manners, to the linear model of these response variables. A variable
selection procedure is the appropriate way of determining if that is the case.
For a matrix X containing quantitative variables only, VIF indices can be easily
computed with the following R code:
vif <- diag(solve(cor(X)))
6.3 Redundancy Analysis (RDA)
225