4.3 Statistical Methods
69
Technique: One of the series is used as an "anchoring series" (Ofenheimer
1971). The time points are arranged in such a way that the anchoring series (silicate in this case) is monotonic. Then, the other series is rearranged according to the
new ordering (Fig. 4.3.14). The null hypothesis is: the order of the values of the
newly sorted phosphate series is purely random. According to the principle of
randomization the argument proceeds as follows. Under the null hypothesis, a
strictly or nearly monotonous ordering is so improbable, that the null hypothesis is
rejected. To what extent the series may depart from monotonicity without rejection
of the null hypothesis depends upon the significance level and the test procedure.
One test of monotonicity is the Moore-Wallis test of first differences (compare
4.3.1.8).
Result: There are three situations in the newly arranged phosphate series, where
the series does not increase but decreases. This is the value of the Moore-Wallis
test statistic, which corresponds to a p-value of 0.022 (Bortz et al. 1990, Table 40).
The test result indicates a parallel development of silicate and phosphate during
one tide.
4.3.2
Analysis of Relations
In ELA W AT, processes were investigated with the objective of linking them to
patterns.
From the mathematical point of view, the simplest situation is the one of independent replicates with two variables per object. In statistical text books (see e.g.
Kendall 1990) there are many measures of dependence, especially of correlation.
Which measure is suitable depends mostly on the question and also whether a test
is required. The sample properties are known for only some of the measures.
The standard techniques are no longer applicable if the replicates are not independent, but show spatial or temporal dependence, as is the case by definition with
patterns and processes. For those cases specially elaborated models are required
(Smith et al. 1993; Cressie 1993; Mardia & Goodall 1993). A modification of
known tests exists for binary data (Cerioli 1997).
In ELA W A T a special technique was developed for the investigation of the
relation between the abundances of two species (the tubeworm L. conchilega and
the copepod Harpacticus obscurus), where abundances of H. obscurus in
neighbouring samples were not independent from each other (Pfeifer et al. 1996b).
If the replicates are independent, but several variables were measured per
object, multivariate techniques will have to be applied. An introduction to many
techniques of multivariate statistics with examples from terrestrial ecology is given
e.g. by Jongman et a!. (1995) and Manly (1986).
If the objective is to investigate the relation of one variable, which is viewed as
"target variable" (e.g. abundance of one species), and several other variables,
which are viewed as "intluential variables", then multiple regression has to be
considered as an adequate technique. Its solution consists of the estimation of statistical parameters, which quantify a possible dependence of the "target variable"
on the "influential variable". An example of the application of multiple regression
to the analysis of a relation between the distribution of birds in an area and the
abiotic conditions there is given by Scheiffarth et a!. (1996). Regression analysis is
69
Technique: One of the series is used as an "anchoring series" (Ofenheimer
1971). The time points are arranged in such a way that the anchoring series (silicate in this case) is monotonic. Then, the other series is rearranged according to the
new ordering (Fig. 4.3.14). The null hypothesis is: the order of the values of the
newly sorted phosphate series is purely random. According to the principle of
randomization the argument proceeds as follows. Under the null hypothesis, a
strictly or nearly monotonous ordering is so improbable, that the null hypothesis is
rejected. To what extent the series may depart from monotonicity without rejection
of the null hypothesis depends upon the significance level and the test procedure.
One test of monotonicity is the Moore-Wallis test of first differences (compare
4.3.1.8).
Result: There are three situations in the newly arranged phosphate series, where
the series does not increase but decreases. This is the value of the Moore-Wallis
test statistic, which corresponds to a p-value of 0.022 (Bortz et al. 1990, Table 40).
The test result indicates a parallel development of silicate and phosphate during
one tide.
4.3.2
Analysis of Relations
In ELA W AT, processes were investigated with the objective of linking them to
patterns.
From the mathematical point of view, the simplest situation is the one of independent replicates with two variables per object. In statistical text books (see e.g.
Kendall 1990) there are many measures of dependence, especially of correlation.
Which measure is suitable depends mostly on the question and also whether a test
is required. The sample properties are known for only some of the measures.
The standard techniques are no longer applicable if the replicates are not independent, but show spatial or temporal dependence, as is the case by definition with
patterns and processes. For those cases specially elaborated models are required
(Smith et al. 1993; Cressie 1993; Mardia & Goodall 1993). A modification of
known tests exists for binary data (Cerioli 1997).
In ELA W A T a special technique was developed for the investigation of the
relation between the abundances of two species (the tubeworm L. conchilega and
the copepod Harpacticus obscurus), where abundances of H. obscurus in
neighbouring samples were not independent from each other (Pfeifer et al. 1996b).
If the replicates are independent, but several variables were measured per
object, multivariate techniques will have to be applied. An introduction to many
techniques of multivariate statistics with examples from terrestrial ecology is given
e.g. by Jongman et a!. (1995) and Manly (1986).
If the objective is to investigate the relation of one variable, which is viewed as
"target variable" (e.g. abundance of one species), and several other variables,
which are viewed as "intluential variables", then multiple regression has to be
considered as an adequate technique. Its solution consists of the estimation of statistical parameters, which quantify a possible dependence of the "target variable"
on the "influential variable". An example of the application of multiple regression
to the analysis of a relation between the distribution of birds in an area and the
abiotic conditions there is given by Scheiffarth et a!. (1996). Regression analysis is