orthogonal to the contrasts representing the main factors. Now, the proposition of the
space-time interaction analysis is to replace either the spatial or the temporal
contrasts, or the interaction terms, or all of these, by a subset of the dbMEM
variables than can be computed to represent the spatial or temporal design. The
subset is chosen to capture all variation with positive spatial or temporal correlation.
Since the subset contains less than (s – 1) or (t – 1) variables (space or time are said to
be under-fitted), or since the interaction subset contains less than (s – 1) Â (t – 1)
variables, one is saving some degrees of freedom that can be used to estimate the
residual error, and thus to test the interaction term.
On this basis, Legendre et al. (2010) presented the following ANOVA models
with their partitioning of variance and degrees of freedom:
• Model 1: standard, two-way crossed ANOVA design for data with replication.
(s – 1), (t – 1) and (s – 1) Â (t – 1) Helmert contrasts form the design matrices
representing space, time and interaction. st(r – 1) degrees of freedom remain for
the error term.
• Model 2: standard two-way crossed ANOVA design without replication. (s – 1)
and (t – 1) Helmert contrasts form the design matrices representing space and
time. The remaining (s – 1) Â (t – 1) d.f. are used to estimate the error variance.
Interaction may be present but cannot be tested for.
• Model 3a: two-way crossed ANOVA design without replication, but with space
under-fitted: u dbMEM spatial variables replace the (s – 1) spatial Helmert
contrasts; u < (s – 1); an interaction design matrix can be constructed, containing
u  (t – 1) terms. (s – u – 1)  t d.f. remain to estimate the error variance. A test of
the interaction is thus possible.
• Model 3b: two-way crossed ANOVA design without replication, but with time
under-fitted: v dbMEM spatial variables replace the (t – 1) temporal Helmert
contrasts; v < (t – 1); an interaction design matrix can be constructed, containing
v  (s – 1) terms. (t – v – 1)  s d.f. remain to estimate the error variance. A test of
the interaction is thus possible.
• Model 4: two-way crossed ANOVA design without replication, but with space
and time under-fitted: u dbMEM spatial variables replace the (s – 1) spatial
Helmert contrasts and v dbMEM variables replace the (t – 1) temporal contrasts;
u < (s – 1) and v < (t – 1); an interaction design matrix can be constructed,
containing u  v terms. [s  t – (u + v + u  v) – 1] d.f. remain to estimate the
error variance. A test of the interaction is thus possible.
• Model 5: two-way crossed ANOVA design without replication, but with interaction under-fitted: the main factors are represented by their (s – 1) and (t – 1)
Helmert contrasts, but the interaction variables are created using the u and v
dbMEM variables used in Model 4. The d.f. of the interaction term are u  v and
[((s – 1)  (t – 1) – u  v] d.f. remain for he residual error.
• Model 6 is a special case where one wants to test the significance of a spatial
effect, or of a temporal effect, in the presence of space-time interaction. One
approach, called Model 6a in the Legendre et al. (2010) paper (but not in the R
function; see below), tests for the presence of spatial structure in turn for each of
7.6 Space-Time Interaction Test in Multivariate ANOVA, Without Replicates
363
space-time interaction analysis is to replace either the spatial or the temporal
contrasts, or the interaction terms, or all of these, by a subset of the dbMEM
variables than can be computed to represent the spatial or temporal design. The
subset is chosen to capture all variation with positive spatial or temporal correlation.
Since the subset contains less than (s – 1) or (t – 1) variables (space or time are said to
be under-fitted), or since the interaction subset contains less than (s – 1) Â (t – 1)
variables, one is saving some degrees of freedom that can be used to estimate the
residual error, and thus to test the interaction term.
On this basis, Legendre et al. (2010) presented the following ANOVA models
with their partitioning of variance and degrees of freedom:
• Model 1: standard, two-way crossed ANOVA design for data with replication.
(s – 1), (t – 1) and (s – 1) Â (t – 1) Helmert contrasts form the design matrices
representing space, time and interaction. st(r – 1) degrees of freedom remain for
the error term.
• Model 2: standard two-way crossed ANOVA design without replication. (s – 1)
and (t – 1) Helmert contrasts form the design matrices representing space and
time. The remaining (s – 1) Â (t – 1) d.f. are used to estimate the error variance.
Interaction may be present but cannot be tested for.
• Model 3a: two-way crossed ANOVA design without replication, but with space
under-fitted: u dbMEM spatial variables replace the (s – 1) spatial Helmert
contrasts; u < (s – 1); an interaction design matrix can be constructed, containing
u  (t – 1) terms. (s – u – 1)  t d.f. remain to estimate the error variance. A test of
the interaction is thus possible.
• Model 3b: two-way crossed ANOVA design without replication, but with time
under-fitted: v dbMEM spatial variables replace the (t – 1) temporal Helmert
contrasts; v < (t – 1); an interaction design matrix can be constructed, containing
v  (s – 1) terms. (t – v – 1)  s d.f. remain to estimate the error variance. A test of
the interaction is thus possible.
• Model 4: two-way crossed ANOVA design without replication, but with space
and time under-fitted: u dbMEM spatial variables replace the (s – 1) spatial
Helmert contrasts and v dbMEM variables replace the (t – 1) temporal contrasts;
u < (s – 1) and v < (t – 1); an interaction design matrix can be constructed,
containing u  v terms. [s  t – (u + v + u  v) – 1] d.f. remain to estimate the
error variance. A test of the interaction is thus possible.
• Model 5: two-way crossed ANOVA design without replication, but with interaction under-fitted: the main factors are represented by their (s – 1) and (t – 1)
Helmert contrasts, but the interaction variables are created using the u and v
dbMEM variables used in Model 4. The d.f. of the interaction term are u  v and
[((s – 1)  (t – 1) – u  v] d.f. remain for he residual error.
• Model 6 is a special case where one wants to test the significance of a spatial
effect, or of a temporal effect, in the presence of space-time interaction. One
approach, called Model 6a in the Legendre et al. (2010) paper (but not in the R
function; see below), tests for the presence of spatial structure in turn for each of
7.6 Space-Time Interaction Test in Multivariate ANOVA, Without Replicates
363