the t times, correcting for multiple testing. The spatial variables used are the
dbMEM constructed for Models 3a, 4 and 5. This variant is not implemented in
the stimodels() function. Another approach, called Model 6b in the paper
(and used in the R function), involves a simultaneous test for spatial structure in
all t times, using a staggered matrix of spatial dbMEM variables. The same
approaches can be applied to test for temporal structure in the presence of
space-time interaction, by constructing a staggered matrix of temporal dbMEM
for each site. Readers are referred to the Appendix C of the Legendre et al. (2010)
paper for more details. Beware: in the stimodels() function presented below,
the models called 6a and 6b are both of the staggered type, 6a for testing space
and 6b for testing time.
Models 3, 4 and 5 allow the testing of the interaction, but the drawback is that
there is some lack-of-fit in the terms coded with dbMEM variables and in the
interaction (which contains dbMEM in the three models). Since this lack-of-fit is
part of the residual error, some power is lost in the tests. Also, the permutation tests
of the interaction and main factors can handle the lack-of-fit of the various terms
differently depending on the model. In all, Legendre et al. recommend Model 5 to
test the interaction, mainly because its permutation test has a correct type I error and
this Model provides the highest power to detect an interaction. The authors’ recommendations are as follows: “if one has replicates, Model 1 is the correct choice. If
not, then space-time interaction can safely (in terms of Type I error rate) be tested
using Model 5. If interaction turns out to be nonsignificant, one can test for the main
effects using Model 2. [. . .] In contrast, if the interaction is significant, one should
perform separate analyses for the different sampling campaigns and/or separate
time series analyses for the different spatial units”. Another argument that strongly
speaks in favour of Model 5 for testing interaction is that this model leaves the
smallest number of d.f. in the residuals, which increases the power of the test. These
analyses of the main factors in the presence of an interaction can then be carried out
under Models 6a or 6b of function stimodels() or directly with function
quicksti() (see below).
7.6.2 Testing the Space-Time Interaction with the sti
Functions
As a Supplement to their paper, Legendre et al. (2010) distributed a package called
STI that has never been submitted to CRAN. Recently, the functions of this
package, now called stimodels() and quicksti(), have been integrated in
the package adespatial. Beware: there is an STI (uppercase) package available
on the CRAN web site, that has nothing to do with the one addressed here and carries
out quite different analyses.
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7 Spatial Analysis of Ecological Data
dbMEM constructed for Models 3a, 4 and 5. This variant is not implemented in
the stimodels() function. Another approach, called Model 6b in the paper
(and used in the R function), involves a simultaneous test for spatial structure in
all t times, using a staggered matrix of spatial dbMEM variables. The same
approaches can be applied to test for temporal structure in the presence of
space-time interaction, by constructing a staggered matrix of temporal dbMEM
for each site. Readers are referred to the Appendix C of the Legendre et al. (2010)
paper for more details. Beware: in the stimodels() function presented below,
the models called 6a and 6b are both of the staggered type, 6a for testing space
and 6b for testing time.
Models 3, 4 and 5 allow the testing of the interaction, but the drawback is that
there is some lack-of-fit in the terms coded with dbMEM variables and in the
interaction (which contains dbMEM in the three models). Since this lack-of-fit is
part of the residual error, some power is lost in the tests. Also, the permutation tests
of the interaction and main factors can handle the lack-of-fit of the various terms
differently depending on the model. In all, Legendre et al. recommend Model 5 to
test the interaction, mainly because its permutation test has a correct type I error and
this Model provides the highest power to detect an interaction. The authors’ recommendations are as follows: “if one has replicates, Model 1 is the correct choice. If
not, then space-time interaction can safely (in terms of Type I error rate) be tested
using Model 5. If interaction turns out to be nonsignificant, one can test for the main
effects using Model 2. [. . .] In contrast, if the interaction is significant, one should
perform separate analyses for the different sampling campaigns and/or separate
time series analyses for the different spatial units”. Another argument that strongly
speaks in favour of Model 5 for testing interaction is that this model leaves the
smallest number of d.f. in the residuals, which increases the power of the test. These
analyses of the main factors in the presence of an interaction can then be carried out
under Models 6a or 6b of function stimodels() or directly with function
quicksti() (see below).
7.6.2 Testing the Space-Time Interaction with the sti
Functions
As a Supplement to their paper, Legendre et al. (2010) distributed a package called
STI that has never been submitted to CRAN. Recently, the functions of this
package, now called stimodels() and quicksti(), have been integrated in
the package adespatial. Beware: there is an STI (uppercase) package available
on the CRAN web site, that has nothing to do with the one addressed here and carries
out quite different analyses.
364
7 Spatial Analysis of Ecological Data