(uncorrected) p-values in increasing order. Divide the smallest by k. If, and only if
the result is smaller than or equal to α, divide the second smallest p-value by k À 1.
Proceed in the same way, each time relaxing the correcting factor (i.e., k À 2,
k À 3...), until a nonsignificant value is encountered.
Other corrections have been proposed in addition to the two presented above.
Several are available in a function called p.adjust() in package stats. This
function can be called whenever one has run several simultaneous tests of significance. The data submitted to that function must be a vector of p-values.
7.2.7 Modelling Spatial Structures
Beyond the methods described above, there are other, more modelling-oriented
approaches to spatial analysis. Finding spatial structures in ecological data indicates
that some process has been at work to generate them; the most important are
environmental forcing (past or present) and biotic processes. Therefore, it is interesting to identify the spatial structures in the data and model them. Spatial structures
can then either be related to explanatory variables representing hypothesized causes,
or help generate new hypotheses as to which processes may have generated them.
Spatial structures can be present at many different scales. Identifying these scales
and modelling the corresponding spatial structures separately is a long-sought goal
for ecologists. A first, rather coarse approach in multivariate analysis is the adaptation of trend-surface analysis to canonical ordination. As suggested by ter Braak
(1987) and demonstrated by Legendre (1990), response data may be explained by a
polynomial function of the (centred) site coordinates. Borcard et al. (1992) have
shown how to integrate this method into variation partitioning to identify, among
other fractions, the pure spatial component of the ecological variation of species
assemblages.
Multivariate trend-surface analysis is limited to the extraction of rather simple
broad-scaled spatial structures, because polynomial terms become rapidly cumbersome, and highly correlated if one uses raw polynomials. In practice, its use is
restricted to third-degree polynomials. A breakthrough came with the development
of eigenvector-based spatial functions, which will be described in Sect. 7.4, after a
short example of trend-surface analysis.
7.3 Multivariate Trend-Surface Analysis
7.3.1 Introduction
Most ecological data have been sampled on geographic surfaces. Therefore, the
crudest way to model the spatial structure of the response data is to regress the
7.3 Multivariate Trend-Surface Analysis
309
the result is smaller than or equal to α, divide the second smallest p-value by k À 1.
Proceed in the same way, each time relaxing the correcting factor (i.e., k À 2,
k À 3...), until a nonsignificant value is encountered.
Other corrections have been proposed in addition to the two presented above.
Several are available in a function called p.adjust() in package stats. This
function can be called whenever one has run several simultaneous tests of significance. The data submitted to that function must be a vector of p-values.
7.2.7 Modelling Spatial Structures
Beyond the methods described above, there are other, more modelling-oriented
approaches to spatial analysis. Finding spatial structures in ecological data indicates
that some process has been at work to generate them; the most important are
environmental forcing (past or present) and biotic processes. Therefore, it is interesting to identify the spatial structures in the data and model them. Spatial structures
can then either be related to explanatory variables representing hypothesized causes,
or help generate new hypotheses as to which processes may have generated them.
Spatial structures can be present at many different scales. Identifying these scales
and modelling the corresponding spatial structures separately is a long-sought goal
for ecologists. A first, rather coarse approach in multivariate analysis is the adaptation of trend-surface analysis to canonical ordination. As suggested by ter Braak
(1987) and demonstrated by Legendre (1990), response data may be explained by a
polynomial function of the (centred) site coordinates. Borcard et al. (1992) have
shown how to integrate this method into variation partitioning to identify, among
other fractions, the pure spatial component of the ecological variation of species
assemblages.
Multivariate trend-surface analysis is limited to the extraction of rather simple
broad-scaled spatial structures, because polynomial terms become rapidly cumbersome, and highly correlated if one uses raw polynomials. In practice, its use is
restricted to third-degree polynomials. A breakthrough came with the development
of eigenvector-based spatial functions, which will be described in Sect. 7.4, after a
short example of trend-surface analysis.
7.3 Multivariate Trend-Surface Analysis
7.3.1 Introduction
Most ecological data have been sampled on geographic surfaces. Therefore, the
crudest way to model the spatial structure of the response data is to regress the
7.3 Multivariate Trend-Surface Analysis
309