variable. In other words, an interaction is present when the effect of one factor
changes across the levels of the other factor. An interaction is most easily measured
when the two factors are linearly independent (uncorrelated, scalar product of 0) as
in balanced ANOVA, a case where the [b] fraction is equal to 0. This completes the
demonstration that a [b] fraction is not an interaction.
6.3.2.9 RDA as a Tool for Multivariate ANOVA
In its classical, parametric form, multivariate analysis of variance (MANOVA) has
stringent conditions of application and restrictions (e.g. multivariate normality of
each group of data, homogeneity of the variance-covariance matrices, number of
response variables smaller than the number of objects minus the number of degrees
of freedom of the MANOVA model). It is practically never well-adapted to ecological data, despite its obvious interest for the analysis of the results of ecological
experiments.
Fortunately, for MANOVA, RDA offers an elegant alternative to parametric
analysis, while adding the versatility of permutation tests and the possibility of
representation of the results in triplots. The trick is to use factor variables and their
interactions as explanatory variables in RDA. In the example below, the factors are
coded as orthogonal Helmert contrasts to allow the testing of the factors and
interaction in a way that provides the correct F values. The interaction is represented
by the products of the variables coding for the main factors. The properties of
Helmert contrasts are the following for a balanced design: (1) the sum of each
coding variable is zero; (2) all variables coding for a factor or the interaction are
orthogonal (their scalar products are all zero); (3) the groups of variables coding for
the main factors and their interaction are all orthogonal to one another.
To illustrate this application, let us use a part of the Doubs data to construct a
fictitious balanced two-way ANOVA design. We will use the first 27 sites (site 8 has
already been excluded in Sect. 6.3.2.1), leaving the two last out.
We create a first factor representing elevation. This factor will have 3 levels, one
for each group of sites 1–10, 11–19 and 20–28. Remember that the empty site 8 has
been removed from the data, so the design is balanced with 9 sites per group.
The second factor will mimic the pH variable as closely as possible. In the real
data, pH is fairly independent from elevation (r ¼ À0.05), but no direct codification
allows the creation of a 3-level factor orthogonal to our first factor. Therefore, we
will simply create an artificial, 3-level factor orthogonal to elevation and approximately representing pH. Be aware that we do this for illustration purposes only, and
that such a manipulation would not be tolerable in the analysis of real data.
The end result is a balanced two-way crossed design with two factors of three
levels each. After having tested for the homogeneity of variance-covariance matrices
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6 Canonical Ordination
changes across the levels of the other factor. An interaction is most easily measured
when the two factors are linearly independent (uncorrelated, scalar product of 0) as
in balanced ANOVA, a case where the [b] fraction is equal to 0. This completes the
demonstration that a [b] fraction is not an interaction.
6.3.2.9 RDA as a Tool for Multivariate ANOVA
In its classical, parametric form, multivariate analysis of variance (MANOVA) has
stringent conditions of application and restrictions (e.g. multivariate normality of
each group of data, homogeneity of the variance-covariance matrices, number of
response variables smaller than the number of objects minus the number of degrees
of freedom of the MANOVA model). It is practically never well-adapted to ecological data, despite its obvious interest for the analysis of the results of ecological
experiments.
Fortunately, for MANOVA, RDA offers an elegant alternative to parametric
analysis, while adding the versatility of permutation tests and the possibility of
representation of the results in triplots. The trick is to use factor variables and their
interactions as explanatory variables in RDA. In the example below, the factors are
coded as orthogonal Helmert contrasts to allow the testing of the factors and
interaction in a way that provides the correct F values. The interaction is represented
by the products of the variables coding for the main factors. The properties of
Helmert contrasts are the following for a balanced design: (1) the sum of each
coding variable is zero; (2) all variables coding for a factor or the interaction are
orthogonal (their scalar products are all zero); (3) the groups of variables coding for
the main factors and their interaction are all orthogonal to one another.
To illustrate this application, let us use a part of the Doubs data to construct a
fictitious balanced two-way ANOVA design. We will use the first 27 sites (site 8 has
already been excluded in Sect. 6.3.2.1), leaving the two last out.
We create a first factor representing elevation. This factor will have 3 levels, one
for each group of sites 1–10, 11–19 and 20–28. Remember that the empty site 8 has
been removed from the data, so the design is balanced with 9 sites per group.
The second factor will mimic the pH variable as closely as possible. In the real
data, pH is fairly independent from elevation (r ¼ À0.05), but no direct codification
allows the creation of a 3-level factor orthogonal to our first factor. Therefore, we
will simply create an artificial, 3-level factor orthogonal to elevation and approximately representing pH. Be aware that we do this for illustration purposes only, and
that such a manipulation would not be tolerable in the analysis of real data.
The end result is a balanced two-way crossed design with two factors of three
levels each. After having tested for the homogeneity of variance-covariance matrices
238
6 Canonical Ordination