single staggered matrix. The MEM variables were arranged in blocks corresponding
to each valley. Within each block, all pools belonging to other valleys received the
value 0, in a way similar to the one presented in Appendix C of Legendre et al.
(2010) in the context of space-time analysis. Declerck et al. (2011) provided a
function called create.MEM.model() to construct the staggered spatial matrix
from a set of Cartesian coordinates and information about the number of groups and
number of sites per group. An updated version of that function, called create.
dbMEM.model(), is available in package adespatial.
7.4.4 MEM with Positive or Negative Spatial Correlation:
Which Ones should Be Used?
In the course of the examples above, dbMEM and MEM eigenfunctions have been
produced, some with positive and some with negative spatial correlation. The
question therefore arises: should one use all the (significant) eigenfunctions as
explanatory variables in the following regression or canonical analyses, or only
those that model positive spatial correlation?
There is no single answer to this question. Ecologically speaking, one is generally
more interested in features that are positively correlated at various ranges, simply
because they are the signature of contagious processes that are frequent in nature. On
the other hand, our experience shows that with real data the significant and negatively correlated variables are either related to very local, almost “accidental” data
structures, or they belong to the pure spatial fraction of variation in partitioning, i.e.,
they are caused by biotic interactions. If these are of interest, then all eigenfunctions
should be considered in the analyses.
The dbMEM procedure generates a maximum of n À 1 eigenfunctions (n ¼ number of sites), with roughly the first n/2 modelling positive spatial correlation on
regular sampling designs, so a forward selection procedure including all variables
cannot be conducted with the Blanchet et al. (2008a) double stopping criterion,
which involves the computation of the R
2
adj of the global analysis. Indeed, the n À 1
spatial variables saturate the regression model if they are all considered together.
This is why Blanchet et al. (2008a) proposed to run separate selections on the MEM
with positive and negative eigenvalues and then apply the Sidák (1967) correction to
the probability values: P S ¼ 1 À (1 À P)
k where P is the P-value to be corrected and
k is the number of tests (here k ¼ 2). Of course, this is useful only in the specific
cases where negative spatial correlation is of interest.
7.4.5 Asymmetric Eigenvector Maps (AEM): When
Directionality Matters
7.4.5.1 Introduction
The dbMEM and MEM analyses presented above are designed for situations where
the physical processes generating the response structures (e.g. in communities) do
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7 Spatial Analysis of Ecological Data
to each valley. Within each block, all pools belonging to other valleys received the
value 0, in a way similar to the one presented in Appendix C of Legendre et al.
(2010) in the context of space-time analysis. Declerck et al. (2011) provided a
function called create.MEM.model() to construct the staggered spatial matrix
from a set of Cartesian coordinates and information about the number of groups and
number of sites per group. An updated version of that function, called create.
dbMEM.model(), is available in package adespatial.
7.4.4 MEM with Positive or Negative Spatial Correlation:
Which Ones should Be Used?
In the course of the examples above, dbMEM and MEM eigenfunctions have been
produced, some with positive and some with negative spatial correlation. The
question therefore arises: should one use all the (significant) eigenfunctions as
explanatory variables in the following regression or canonical analyses, or only
those that model positive spatial correlation?
There is no single answer to this question. Ecologically speaking, one is generally
more interested in features that are positively correlated at various ranges, simply
because they are the signature of contagious processes that are frequent in nature. On
the other hand, our experience shows that with real data the significant and negatively correlated variables are either related to very local, almost “accidental” data
structures, or they belong to the pure spatial fraction of variation in partitioning, i.e.,
they are caused by biotic interactions. If these are of interest, then all eigenfunctions
should be considered in the analyses.
The dbMEM procedure generates a maximum of n À 1 eigenfunctions (n ¼ number of sites), with roughly the first n/2 modelling positive spatial correlation on
regular sampling designs, so a forward selection procedure including all variables
cannot be conducted with the Blanchet et al. (2008a) double stopping criterion,
which involves the computation of the R
2
adj of the global analysis. Indeed, the n À 1
spatial variables saturate the regression model if they are all considered together.
This is why Blanchet et al. (2008a) proposed to run separate selections on the MEM
with positive and negative eigenvalues and then apply the Sidák (1967) correction to
the probability values: P S ¼ 1 À (1 À P)
k where P is the P-value to be corrected and
k is the number of tests (here k ¼ 2). Of course, this is useful only in the specific
cases where negative spatial correlation is of interest.
7.4.5 Asymmetric Eigenvector Maps (AEM): When
Directionality Matters
7.4.5.1 Introduction
The dbMEM and MEM analyses presented above are designed for situations where
the physical processes generating the response structures (e.g. in communities) do
348
7 Spatial Analysis of Ecological Data