Consequently, the method presented here must involve a “trick” to spare some
degrees of freedom from the main factors. How can this be done? In this section,
we are examining the case where s sites have been surveyed t times.
In a two-way crossed and balanced ANOVA design involving two factors with
s and t levels (hence there are s  t combinations of levels) and r replicates per cell,
the number of degrees of freedom of the explainable part of the variance is equal to
(s  t) – 1. The tests of the main factors take up (s – 1) and (t – 1) d.f. respectively.
When there are r replicates within each cell (r > 1), then the interaction term uses
(s – 1) Â (t – 1) d.f. and st(r – 1) d.f. remain for the error term. In other words, it is
the within-cell replicates that provide the d.f. of the residuals. Without replication,
r ¼ 1 and st(r – 1) ¼ 0. Therefore, no test of the interaction is possible since the (s –
1) Â (t – 1) d.f. remaining after the main factors have been taken into account are in
fact attributable to a combination of interaction and error, but without the possibility
to untangle them.
This being clear, how could one conceive a plan with two crossed factors, no
replication but with a possibility of testing the interaction? The answer is:
in limited cases, when at least one of the two factors can be coded with less than
(s – 1) or (t – 1) d.f., and/or if the interaction can be coded with less than (s – 1) Â (t –
1) d.f. The situation addressed here is the one of community surveys made at a given
set of locations and repeated over time, for instance for long-term monitoring
purposes such as those performed by governmental agencies in many countries
around the world. This type of sampling produces an unreplicated repeated-measures
design, which is a type of two-way factorial design. Sites and survey times cannot be
replicated; at best, an adjacent site would be a pseudo-replicate. The same applies to
survey times.
In such data, there is no replication at the level of individual sites: each site is
sampled only once in a given survey. However, interaction is an important indicator
in such situations: if present, it can mean either that the spatial structure of the
community has changed over time, or, conversely, that the temporal evolution of the
community is not the same at all sites. In fact, interaction is so important for such
repeated surveys that it should be “the first statistical indication ecologists should
look for” (Legendre et al. 2010). As in any ANOVA, the presence of an interaction
means that one should run separate analyses of the time factor at each site and
separate analyses of the spatial structure at each point in time.
Let us consider the ANOVA as a multiple regression using a design matrix of
explanatory variables, i.e., a matrix of dummy variables coding the levels of the
factors. We saw in Sect. 6.3.2.9 how to code a factor into a set of orthogonal Helmert
contrasts. In the present case, to code for the spatial and temporal factors with s sites
and t times, we need (s – 1) and (t – 1) contrasts, which correspond to the number of
d.f. of the two factors. If the interaction could be tested, its design matrix would be
created by multiplying each Helmert contrast in the space matrix by each contrast of
the temporal matrix; the result would be a set of (s – 1) Â (t – 1) new contrasts,
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7 Spatial Analysis of Ecological Data
degrees of freedom from the main factors. How can this be done? In this section,
we are examining the case where s sites have been surveyed t times.
In a two-way crossed and balanced ANOVA design involving two factors with
s and t levels (hence there are s  t combinations of levels) and r replicates per cell,
the number of degrees of freedom of the explainable part of the variance is equal to
(s  t) – 1. The tests of the main factors take up (s – 1) and (t – 1) d.f. respectively.
When there are r replicates within each cell (r > 1), then the interaction term uses
(s – 1) Â (t – 1) d.f. and st(r – 1) d.f. remain for the error term. In other words, it is
the within-cell replicates that provide the d.f. of the residuals. Without replication,
r ¼ 1 and st(r – 1) ¼ 0. Therefore, no test of the interaction is possible since the (s –
1) Â (t – 1) d.f. remaining after the main factors have been taken into account are in
fact attributable to a combination of interaction and error, but without the possibility
to untangle them.
This being clear, how could one conceive a plan with two crossed factors, no
replication but with a possibility of testing the interaction? The answer is:
in limited cases, when at least one of the two factors can be coded with less than
(s – 1) or (t – 1) d.f., and/or if the interaction can be coded with less than (s – 1) Â (t –
1) d.f. The situation addressed here is the one of community surveys made at a given
set of locations and repeated over time, for instance for long-term monitoring
purposes such as those performed by governmental agencies in many countries
around the world. This type of sampling produces an unreplicated repeated-measures
design, which is a type of two-way factorial design. Sites and survey times cannot be
replicated; at best, an adjacent site would be a pseudo-replicate. The same applies to
survey times.
In such data, there is no replication at the level of individual sites: each site is
sampled only once in a given survey. However, interaction is an important indicator
in such situations: if present, it can mean either that the spatial structure of the
community has changed over time, or, conversely, that the temporal evolution of the
community is not the same at all sites. In fact, interaction is so important for such
repeated surveys that it should be “the first statistical indication ecologists should
look for” (Legendre et al. 2010). As in any ANOVA, the presence of an interaction
means that one should run separate analyses of the time factor at each site and
separate analyses of the spatial structure at each point in time.
Let us consider the ANOVA as a multiple regression using a design matrix of
explanatory variables, i.e., a matrix of dummy variables coding the levels of the
factors. We saw in Sect. 6.3.2.9 how to code a factor into a set of orthogonal Helmert
contrasts. In the present case, to code for the spatial and temporal factors with s sites
and t times, we need (s – 1) and (t – 1) contrasts, which correspond to the number of
d.f. of the two factors. If the interaction could be tested, its design matrix would be
created by multiplying each Helmert contrast in the space matrix by each contrast of
the temporal matrix; the result would be a set of (s – 1) Â (t – 1) new contrasts,
362
7 Spatial Analysis of Ecological Data