7.4.2.5 Combining dbMEM Analysis and Variation Partitioning
A clever and global approach to assess the environmental variation related to all
scales of spatial variation is to perform a variation partitioning analysis with an
environmental data set and up to three subsets of spatial variables. Function
varpart(), studied in Sect. 6.3.2.8, can only handle numerical variables (not
factors), however, so that we will have to recode environmental variables 3–5 into
dummy binary variables.
Variation partitioning aims at quantifying the various unique and combined
fractions of variation explained by several sources. In this context, a linear trend
can be considered to represent a source of variation like any other. The trend is likely
to manifest itself on the response as well as the environmental explanatory variables.
Therefore, in this application we advocate not to detrend the response data prior to
variation partitioning, but rather to test for a linear trend and incorporate it explicitly
in the partitioning procedure if it is significant. Note that we consider the linear trend,
represented by a pair of X-Y coordinates, as a whole. If it is significant we incorporate
it into the variation partitioning without forward-selecting the coordinates. This
ensures that any linear trend, large or small, is included.
We have seen in Sect. 6.3.2.8 that, prior to variation partitioning, forward
selection must be done independently in each block of explanatory variables.
Therefore, it would seem that in the present case forward selection should be
computed separately on the environmental variables and the dbMEM variables, all
this with the undetrended response variables. However, a technical point will lead to
recommend to slightly derogate from this procedure. We will indeed forward-select
the environmental variables using the undetrended response variables. The selection
of the dbMEM variables, however, will be run on the response variables that have
been detrended by the X-Y coordinates (if the trend is significant). We are proceeding
in this way because dbMEM, among other properties, are also able to model linear
trends, in addition to other structures (Borcard et al. 2002). Consequently, if a linear
trend is present in the data, it is bound to be modelled by a subset of the dbMEM as
well as by the X-Y coordinates. Contrary to the environment vs dbMEM case, this
does not tell us anything more about the response data: it is only a consequence of
using two different types of spatial variables in the variation partitioning
4 .
In this example, we will arbitrarily split the significant dbMEM variables into a
broad and a fine scale fraction. The partitioning results are presented in Fig. 7.8.
4 One may consider forward-selecting the dbMEM on the undetrended response data, using the X-Y
coordinates as covariables, by means of function ordiR2step() of vegan. However, this
technique implies that the dbMEM variables are residualized on the X-Y coordinates. As a
consequence, the residualized dbMEM variables would no longer be orthogonal, and they would
not be those that will finally be used in the variation partitioning. Therefore, we do not recommend
this procedure.
7.4 Eigenvector-Based Spatial Variables and Spatial Modelling
329
A clever and global approach to assess the environmental variation related to all
scales of spatial variation is to perform a variation partitioning analysis with an
environmental data set and up to three subsets of spatial variables. Function
varpart(), studied in Sect. 6.3.2.8, can only handle numerical variables (not
factors), however, so that we will have to recode environmental variables 3–5 into
dummy binary variables.
Variation partitioning aims at quantifying the various unique and combined
fractions of variation explained by several sources. In this context, a linear trend
can be considered to represent a source of variation like any other. The trend is likely
to manifest itself on the response as well as the environmental explanatory variables.
Therefore, in this application we advocate not to detrend the response data prior to
variation partitioning, but rather to test for a linear trend and incorporate it explicitly
in the partitioning procedure if it is significant. Note that we consider the linear trend,
represented by a pair of X-Y coordinates, as a whole. If it is significant we incorporate
it into the variation partitioning without forward-selecting the coordinates. This
ensures that any linear trend, large or small, is included.
We have seen in Sect. 6.3.2.8 that, prior to variation partitioning, forward
selection must be done independently in each block of explanatory variables.
Therefore, it would seem that in the present case forward selection should be
computed separately on the environmental variables and the dbMEM variables, all
this with the undetrended response variables. However, a technical point will lead to
recommend to slightly derogate from this procedure. We will indeed forward-select
the environmental variables using the undetrended response variables. The selection
of the dbMEM variables, however, will be run on the response variables that have
been detrended by the X-Y coordinates (if the trend is significant). We are proceeding
in this way because dbMEM, among other properties, are also able to model linear
trends, in addition to other structures (Borcard et al. 2002). Consequently, if a linear
trend is present in the data, it is bound to be modelled by a subset of the dbMEM as
well as by the X-Y coordinates. Contrary to the environment vs dbMEM case, this
does not tell us anything more about the response data: it is only a consequence of
using two different types of spatial variables in the variation partitioning
4 .
In this example, we will arbitrarily split the significant dbMEM variables into a
broad and a fine scale fraction. The partitioning results are presented in Fig. 7.8.
4 One may consider forward-selecting the dbMEM on the undetrended response data, using the X-Y
coordinates as covariables, by means of function ordiR2step() of vegan. However, this
technique implies that the dbMEM variables are residualized on the X-Y coordinates. As a
consequence, the residualized dbMEM variables would no longer be orthogonal, and they would
not be those that will finally be used in the variation partitioning. Therefore, we do not recommend
this procedure.
7.4 Eigenvector-Based Spatial Variables and Spatial Modelling
329