4.3 Statistical Method,
55
In everyday language and in natural sciences as well, the term correlation is
used synonymously with relation. In statistics its meaning is much more restricted.
The correlation coefficients of Bravais-Pearson and of Spearman (see e.g. Sachs
1992) solely measure the linear relationship between two variables and their ranks,
respectively. Relationships following different functional forms (e.g. quadratic) or
more general relations are not covered by this term. There are more measures of
correlation (Kendall 1990), each of which measures a special kind of relation.
Measurements made in a temporal sequence often show autocorrelation, i.e.
every two or more succeeding values show high correlation (in the sense of the
Bravais-Pearson correlation coefficient). A series of measurements in temporal
order is called time series or longitudinal data in statistics.
In statistics, the terms variability, spread, noise are used synonymously with the
term variance. In ecology they are also used to describe differences in repeated
measurements no matter from which source (Lozan 1992, p 40) or as changes of
structure and parameter values in both spatial and temporal respect.
It is a task for mathematics to provide adequate similarity measures for the assessment of system structures. First of all, however, it is an ecological question,
what is meant by similarity in a specific context. For a comparison of biological
communities, for example, only the presence or absence of species might be of
interest without regard to abundance (qualitative sampling). The ecological requirements form the background for the choice of an adequate similarity measure
to be found by mathematical reasoning. Pfeifer et al. (1998) show the properties of
some common similarity measures by way of example. The sampling properties
are only known for few similarity measures (Dixon 1993). Therefore, it is mostly
not possible to extend similarity-based results from samples to the sampling universe.
4.3
Statistical Methods for the Description and Measurement
of Stability Properties
In this paragraph statistical methods are described, which are relevant for the ecological question of stability properties. They are arranged according to the terms
pattern, process, relation, and distinction between states of a system. In addition to
the presentation of methods, examples are given in the form of case studies. The
aim is to demonstrate the statistical method starting with the ecological question,
explaining the technique as succinctly as possible, and ending up with an indicated
interpretation. It is hoped that the exemplary character of the case studies is strong
enough to make them interesting also for reader outside ELA W A T.
4.3.1
Description of Patterns and Processes
As a first step it is necessary to define the scale on which a pattern or process shall
be investigated, because a pattern or process is not completely defined without its
scale. It depends on the question to be studied which phenomenon may be called a
55
In everyday language and in natural sciences as well, the term correlation is
used synonymously with relation. In statistics its meaning is much more restricted.
The correlation coefficients of Bravais-Pearson and of Spearman (see e.g. Sachs
1992) solely measure the linear relationship between two variables and their ranks,
respectively. Relationships following different functional forms (e.g. quadratic) or
more general relations are not covered by this term. There are more measures of
correlation (Kendall 1990), each of which measures a special kind of relation.
Measurements made in a temporal sequence often show autocorrelation, i.e.
every two or more succeeding values show high correlation (in the sense of the
Bravais-Pearson correlation coefficient). A series of measurements in temporal
order is called time series or longitudinal data in statistics.
In statistics, the terms variability, spread, noise are used synonymously with the
term variance. In ecology they are also used to describe differences in repeated
measurements no matter from which source (Lozan 1992, p 40) or as changes of
structure and parameter values in both spatial and temporal respect.
It is a task for mathematics to provide adequate similarity measures for the assessment of system structures. First of all, however, it is an ecological question,
what is meant by similarity in a specific context. For a comparison of biological
communities, for example, only the presence or absence of species might be of
interest without regard to abundance (qualitative sampling). The ecological requirements form the background for the choice of an adequate similarity measure
to be found by mathematical reasoning. Pfeifer et al. (1998) show the properties of
some common similarity measures by way of example. The sampling properties
are only known for few similarity measures (Dixon 1993). Therefore, it is mostly
not possible to extend similarity-based results from samples to the sampling universe.
4.3
Statistical Methods for the Description and Measurement
of Stability Properties
In this paragraph statistical methods are described, which are relevant for the ecological question of stability properties. They are arranged according to the terms
pattern, process, relation, and distinction between states of a system. In addition to
the presentation of methods, examples are given in the form of case studies. The
aim is to demonstrate the statistical method starting with the ecological question,
explaining the technique as succinctly as possible, and ending up with an indicated
interpretation. It is hoped that the exemplary character of the case studies is strong
enough to make them interesting also for reader outside ELA W A T.
4.3.1
Description of Patterns and Processes
As a first step it is necessary to define the scale on which a pattern or process shall
be investigated, because a pattern or process is not completely defined without its
scale. It depends on the question to be studied which phenomenon may be called a