70
4 Stalistical Models and Tcchniljucs
well developed in theory and practice, but its assumptions are often not met in
ecology. If the "target variable" has only few possible values (e.g. season, habitat),
then Cluster Analysis can be applied on the basis of the "inf1uential variables" to
see whether objects similar with respect to the "influential variables" are also
similar with respect to the "target variable". However, the most frequently applied
form of Cluster Analysis, Hierarchical Cluster Analysis, is only applicable to
measurements with an underlying hierarchical structure (in the mathematical sense
of hierarchical). This assumption is often not considered when the technique is
chosen.
If the objective is to investigate the relation between two (or more) groups of
variables (e.g. biological and chemical) the following techniques should be taken
into account.
The graphical result of Canonical Correspondence Analysis is an arrangement
of species (biological variables) and stations as points in a co-ordinate system.
Influential variables, e.g. sedimentological variables, are represented as arrows.
They point into the direction of those species to which they are highly correlated.
Their lengths give information about their importance for the biological variables.
The basis is a Multiple Regression Analysis, in which the co-ordinates of the stations are the "target variables". Canonical Correspondence Analysis rests on the
assumption that the stations represent a gradient, along which abundances of species develop unimodally. Furthermore, all the assumptions of Multivariate Regression Analysis have to be met (e.g. about correlation among the influential variables). Canonical Correspondence Analysis has been successfully applied in
benthic ecology (e.g. Kroncke et al. 1996). Some further multivariate standard
techniques are for example Canonical Correlation Analysis (Van der Meer 1991),
Principal Component Analysis and Redundancy Analysis, which received different
responses in ecological applications (Gittins 1985; Green 1993; James & McCulloch 1990; Minchin 1987). A survey on properties and model assumptions is given
e.g. by Jongman et al. (1995, p. 154).
4.3.3
Tests for a Statistical Distinction Between States of a System
A statistical test is a decision procedure stating in which way a test statistic is to be
calculated from the sampled data, and for which values of the test statistic the null
hypothesis is to be rejected. For parametric tests the statistical distribution of the
test statistic is known, because a parametric model is assumed for the data. If no
parametric model is or can be assumed, non-parametric (distribution-free) techniques must be applied. Lienert (1969) defines them as techniques "which are not
bound to the normal distribution and may be applied even if the distribution of the
sampling universe departs from normality or is unknown."
In ecology, multivariate techniques are required, because the state of a system
cannot usually be measured by one single variable. The classical parametric multivariate test is Multivariate Analysis of Variance, which implicitly uses Euclidean
metric and assumes, besides the normal distribution, that both sampling universes
have the same covariance structure, which will usually not be the case with ecological data. Therefore, non-parametric multivariate test procedures are required.
4 Stalistical Models and Tcchniljucs
well developed in theory and practice, but its assumptions are often not met in
ecology. If the "target variable" has only few possible values (e.g. season, habitat),
then Cluster Analysis can be applied on the basis of the "inf1uential variables" to
see whether objects similar with respect to the "influential variables" are also
similar with respect to the "target variable". However, the most frequently applied
form of Cluster Analysis, Hierarchical Cluster Analysis, is only applicable to
measurements with an underlying hierarchical structure (in the mathematical sense
of hierarchical). This assumption is often not considered when the technique is
chosen.
If the objective is to investigate the relation between two (or more) groups of
variables (e.g. biological and chemical) the following techniques should be taken
into account.
The graphical result of Canonical Correspondence Analysis is an arrangement
of species (biological variables) and stations as points in a co-ordinate system.
Influential variables, e.g. sedimentological variables, are represented as arrows.
They point into the direction of those species to which they are highly correlated.
Their lengths give information about their importance for the biological variables.
The basis is a Multiple Regression Analysis, in which the co-ordinates of the stations are the "target variables". Canonical Correspondence Analysis rests on the
assumption that the stations represent a gradient, along which abundances of species develop unimodally. Furthermore, all the assumptions of Multivariate Regression Analysis have to be met (e.g. about correlation among the influential variables). Canonical Correspondence Analysis has been successfully applied in
benthic ecology (e.g. Kroncke et al. 1996). Some further multivariate standard
techniques are for example Canonical Correlation Analysis (Van der Meer 1991),
Principal Component Analysis and Redundancy Analysis, which received different
responses in ecological applications (Gittins 1985; Green 1993; James & McCulloch 1990; Minchin 1987). A survey on properties and model assumptions is given
e.g. by Jongman et al. (1995, p. 154).
4.3.3
Tests for a Statistical Distinction Between States of a System
A statistical test is a decision procedure stating in which way a test statistic is to be
calculated from the sampled data, and for which values of the test statistic the null
hypothesis is to be rejected. For parametric tests the statistical distribution of the
test statistic is known, because a parametric model is assumed for the data. If no
parametric model is or can be assumed, non-parametric (distribution-free) techniques must be applied. Lienert (1969) defines them as techniques "which are not
bound to the normal distribution and may be applied even if the distribution of the
sampling universe departs from normality or is unknown."
In ecology, multivariate techniques are required, because the state of a system
cannot usually be measured by one single variable. The classical parametric multivariate test is Multivariate Analysis of Variance, which implicitly uses Euclidean
metric and assumes, besides the normal distribution, that both sampling universes
have the same covariance structure, which will usually not be the case with ecological data. Therefore, non-parametric multivariate test procedures are required.