S4
4 Statistical Models and Tcchmquo
As part of mathematics the field of mathematical statistics is based on the axioms of probability, it rests on the laws of chance. Therefore it does not matter
whether biological phenomena only seem to function randomly (possibly because
of many deterministic influences working simultaneously) or whether they really
do function randomly. For example, the mean values from two samples will be
considered significantly different if the probability of observing or measuring two
values with this difference is very low, given that the samples have the same underlying mean. Statistics gives no evidence whether a difference, which is considered significant in the statistical sense, is an ecologically relevant difference. In the
field of applied statistics, data are analysed by means of graphical, mathematical
and statistical techniques. Often, the technique is based on a probability concept. It
is assumed that the measurements or observations are random samples from a real
or imaginary sampling universe. (In statistical textbooks the term population is
used synonymously with sampling universe. In the following the word population
will be reserved for populations in the biological sense.) Results are to be referred
to that sampling universe, which cannot be examined as a whole for reasons of
time and money or in principle. Only for a random sample is it possible to draw
inference from the sample to the sampling universe by means of probability theory.
Otherwise, the principle of randomisation can be used to make probability arguments about the sample (for details see Edgington 1995). The techniques belonging
to descriptive and exploratory statistics are permitted in any case. However, generalization of results to a larger set than the one observed is rarely allowed.
In the following, measurements and observations are termed "variables".
Statistical hypothesis testing means falsification of the null model. A statistically significant conclusion is only permitted if the observed data would have
occurred under a valid null hypothesis (null model) with a very small probability
only. Thus, in contrast to the models mentioned above, the null model serves as a
reference. A formal requirement is the knowledge of the statistical distribution of
the test statistic under the null hypothesis.
A stochastic process is a random function. For statistical modelling it does not
really matter whether this IS a function of time or of space. In ecology, generally,
the word process denotes a phenomenon, which is characterized by a temporal
change (e.g. biochemical transformation processes in a mussel bed). In the following, the word process will be used in this general sense. If a process is meant in
the mathematical sense, it will be specified by using the term stochastic process.
A pattern is regarded as a structure of objects, like e.g. their spatial
arrangement. In Sect. 4.3. I the term is defined more precisely depending on the
context. In ecology the terms pattern and spatial and temporal distribution are often
used as synonyms.
The statistical distribution (probability distribution, distribution function, probability function) of a random variable specifies the probabilities of the possible
values (e.g. for a fair dice the statistical distribution is: 1-\, 2-'/", ... , 6-'/J. The
empirical distribution of an observed variable specifies the cumulative frequencies
of the observed values. In statistics, the term parameter denotes a value characterizing a statistical distribution. The mean is a parameter, for example. In other
fields of science, parameter is used synonymously with variable and distribution
denotes for example the spatial arrangement in an area.
4 Statistical Models and Tcchmquo
As part of mathematics the field of mathematical statistics is based on the axioms of probability, it rests on the laws of chance. Therefore it does not matter
whether biological phenomena only seem to function randomly (possibly because
of many deterministic influences working simultaneously) or whether they really
do function randomly. For example, the mean values from two samples will be
considered significantly different if the probability of observing or measuring two
values with this difference is very low, given that the samples have the same underlying mean. Statistics gives no evidence whether a difference, which is considered significant in the statistical sense, is an ecologically relevant difference. In the
field of applied statistics, data are analysed by means of graphical, mathematical
and statistical techniques. Often, the technique is based on a probability concept. It
is assumed that the measurements or observations are random samples from a real
or imaginary sampling universe. (In statistical textbooks the term population is
used synonymously with sampling universe. In the following the word population
will be reserved for populations in the biological sense.) Results are to be referred
to that sampling universe, which cannot be examined as a whole for reasons of
time and money or in principle. Only for a random sample is it possible to draw
inference from the sample to the sampling universe by means of probability theory.
Otherwise, the principle of randomisation can be used to make probability arguments about the sample (for details see Edgington 1995). The techniques belonging
to descriptive and exploratory statistics are permitted in any case. However, generalization of results to a larger set than the one observed is rarely allowed.
In the following, measurements and observations are termed "variables".
Statistical hypothesis testing means falsification of the null model. A statistically significant conclusion is only permitted if the observed data would have
occurred under a valid null hypothesis (null model) with a very small probability
only. Thus, in contrast to the models mentioned above, the null model serves as a
reference. A formal requirement is the knowledge of the statistical distribution of
the test statistic under the null hypothesis.
A stochastic process is a random function. For statistical modelling it does not
really matter whether this IS a function of time or of space. In ecology, generally,
the word process denotes a phenomenon, which is characterized by a temporal
change (e.g. biochemical transformation processes in a mussel bed). In the following, the word process will be used in this general sense. If a process is meant in
the mathematical sense, it will be specified by using the term stochastic process.
A pattern is regarded as a structure of objects, like e.g. their spatial
arrangement. In Sect. 4.3. I the term is defined more precisely depending on the
context. In ecology the terms pattern and spatial and temporal distribution are often
used as synonyms.
The statistical distribution (probability distribution, distribution function, probability function) of a random variable specifies the probabilities of the possible
values (e.g. for a fair dice the statistical distribution is: 1-\, 2-'/", ... , 6-'/J. The
empirical distribution of an observed variable specifies the cumulative frequencies
of the observed values. In statistics, the term parameter denotes a value characterizing a statistical distribution. The mean is a parameter, for example. In other
fields of science, parameter is used synonymously with variable and distribution
denotes for example the spatial arrangement in an area.