adonis2(). Let us briefly present adonis2(), which is based on McArdle and
Anderson (2001). See also Sect. 6.3.3. In this context, this is how the overall
MANOVA would be computed:
adonis2(spe.hel[1:27, ] ~ ele.fac * pH.fac,
method = "euc",
by = "term"
)
Hint The argument by = term, which is the default option, asks the function to
compute the sums of squares and assess the significance of the terms in the order
of their mention in the code. In balanced designs this is not important: reversing
the order of the two factors will produce the exact same result. In the case of
unbalanced designs, however, changing the order changes the results because of
the non-orthogonality of the factors, which in fact introduces a [b] fraction in the
sense of Sect. 6.3.2.8. This is another good reason to avoid unbalanced designs
whenever possible.
Finally, Laliberté et al. (2009) offer a function called manovRDa.R to compute a
two-way MANOVA by RDA for fixed and random factors. It is available on the
Web page http://www.elaliberte.info/code.
6.3.2.10 Nonlinear Relationships in RDA
Another point is worth mentioning. RDA carries out a multivariate linear regression
analysis followed by a PCA of the fitted values. Consequently, other methods based
upon the multiple regression equation can be used in RDA as well. Consider the fact
that all RDA models presented above only used explanatory variables to the power
1. However, it is frequent that raw species responses actually have unimodal
distributions along an environmental gradient, showing an ecological optimum and
some tolerance to the variation of a given environmental constraint. A strictly linear
model would strongly suffer from lack-of-fit in such a case. Plotting all possible pairs
of response versus explanatory variables to detect such nonlinearities would be too
cumbersome. An interesting shortcut to identify and model unimodal responses is to
provide second-degree explanatory variables along with the first-degree terms (i.e.,
provide the terms for a quadratic model), then run forward selection. This procedure
will retain the relevant terms (called monomials), be they of the first or second
degree. Of course, the interpretation of such results is more complex, so that it should
be applied only when one has serious reasons to suspect unimodal distributions
causing nonlinear relationships. Third-order monomials of explanatory variables
may even be useful in the case of unimodal but highly skewed response variable
distributions (Borcard et al. 1992). Let us compute two examples of polynomial
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6 Canonical Ordination
Anderson (2001). See also Sect. 6.3.3. In this context, this is how the overall
MANOVA would be computed:
adonis2(spe.hel[1:27, ] ~ ele.fac * pH.fac,
method = "euc",
by = "term"
)
Hint The argument by = term, which is the default option, asks the function to
compute the sums of squares and assess the significance of the terms in the order
of their mention in the code. In balanced designs this is not important: reversing
the order of the two factors will produce the exact same result. In the case of
unbalanced designs, however, changing the order changes the results because of
the non-orthogonality of the factors, which in fact introduces a [b] fraction in the
sense of Sect. 6.3.2.8. This is another good reason to avoid unbalanced designs
whenever possible.
Finally, Laliberté et al. (2009) offer a function called manovRDa.R to compute a
two-way MANOVA by RDA for fixed and random factors. It is available on the
Web page http://www.elaliberte.info/code.
6.3.2.10 Nonlinear Relationships in RDA
Another point is worth mentioning. RDA carries out a multivariate linear regression
analysis followed by a PCA of the fitted values. Consequently, other methods based
upon the multiple regression equation can be used in RDA as well. Consider the fact
that all RDA models presented above only used explanatory variables to the power
1. However, it is frequent that raw species responses actually have unimodal
distributions along an environmental gradient, showing an ecological optimum and
some tolerance to the variation of a given environmental constraint. A strictly linear
model would strongly suffer from lack-of-fit in such a case. Plotting all possible pairs
of response versus explanatory variables to detect such nonlinearities would be too
cumbersome. An interesting shortcut to identify and model unimodal responses is to
provide second-degree explanatory variables along with the first-degree terms (i.e.,
provide the terms for a quadratic model), then run forward selection. This procedure
will retain the relevant terms (called monomials), be they of the first or second
degree. Of course, the interpretation of such results is more complex, so that it should
be applied only when one has serious reasons to suspect unimodal distributions
causing nonlinear relationships. Third-order monomials of explanatory variables
may even be useful in the case of unimodal but highly skewed response variable
distributions (Borcard et al. 1992). Let us compute two examples of polynomial
244
6 Canonical Ordination