8
A. MACFADYEN
abundance figures are essential should now be apparent. If they are
(trophic studies) preliminary samples must be taken to determine the
ratio of variance to mean and thus to establish whether distribution is
patchy ( =aggregated, =contagious, =underdispersed) and, if so, what
must be done to measure and allow for patchiness. When patchiness has
been demonstrated it is usual to attempt to fit the field data to a theoretical distribution containing terms representing the mean numbers per
unit area and the mean size of the patches. The first of these can be used
in trophic studies but the asymmetrical distribution invalidates statistical tests which are based on the assumption of a normal distribution
and to ignore the patchiness is to discard valid biological information.
Distributions which have been fitted to soil sample data include the
Poisson, which describes the frequency of random (i.e. non-patchy)
events. According to this distribution, which rarely applies in natural
soils variance and mean are equal. The negative binomial (see Bliss and
Fisher, 1953; Anscombe, 1950) is a distribution related to the Poisson
but incorporating the two hypotheses that the population is logarithmically distributed within patches and that these patches occur at random. I n addition to m, the mean, the negative binomial uses an extra
parameter k which is given by s2=m +km2 (when s2 is the variance).
As k approaches infinity the distribution becomes identical with the
Poisson whilst as k approaches zero the distribution becomes more
clumped.
In practice, as Healy (in press) has shown, k can be determined from
preliminary sample data by plotting standard deviation (ordinate)
against mean (abscissa) for increasing sample size. The point on the
horizontal axis cut by the regression line (drawn by eye) gives an approximate value for k which can then be used with the definitive
samples to describe spatial distribution.
Methods for testing the closeness of fit of data to distributions of this
kind are discussed by Anscombe (1950)) Waters (1955) and Quenouille
(1950) while Hartenstein (1961) describes a practical study on aggregated soil arthropod populations.
When data obtained from populations which do not fit a normal distribution are to be subjected to statistical tests such as those used to
determine significantly different population levels, the raw data cannot
be used because such tests are based on the assumption of normal clistribution. I n this case the data must be “transformed” by functions
which will vary with the type of distribution. Data which fit a Poisson
distribution should be converted to their square roots (Snedecor, 1946).
When the data fit symmetrically into the groups 0-1, 1-2, 2-4, 4-8,
8-16, etc. logarithms should be taken (Quenouille, 1950) and data which
fit the negative binomial should be transformed by log (z + k).
A. MACFADYEN
abundance figures are essential should now be apparent. If they are
(trophic studies) preliminary samples must be taken to determine the
ratio of variance to mean and thus to establish whether distribution is
patchy ( =aggregated, =contagious, =underdispersed) and, if so, what
must be done to measure and allow for patchiness. When patchiness has
been demonstrated it is usual to attempt to fit the field data to a theoretical distribution containing terms representing the mean numbers per
unit area and the mean size of the patches. The first of these can be used
in trophic studies but the asymmetrical distribution invalidates statistical tests which are based on the assumption of a normal distribution
and to ignore the patchiness is to discard valid biological information.
Distributions which have been fitted to soil sample data include the
Poisson, which describes the frequency of random (i.e. non-patchy)
events. According to this distribution, which rarely applies in natural
soils variance and mean are equal. The negative binomial (see Bliss and
Fisher, 1953; Anscombe, 1950) is a distribution related to the Poisson
but incorporating the two hypotheses that the population is logarithmically distributed within patches and that these patches occur at random. I n addition to m, the mean, the negative binomial uses an extra
parameter k which is given by s2=m +km2 (when s2 is the variance).
As k approaches infinity the distribution becomes identical with the
Poisson whilst as k approaches zero the distribution becomes more
clumped.
In practice, as Healy (in press) has shown, k can be determined from
preliminary sample data by plotting standard deviation (ordinate)
against mean (abscissa) for increasing sample size. The point on the
horizontal axis cut by the regression line (drawn by eye) gives an approximate value for k which can then be used with the definitive
samples to describe spatial distribution.
Methods for testing the closeness of fit of data to distributions of this
kind are discussed by Anscombe (1950)) Waters (1955) and Quenouille
(1950) while Hartenstein (1961) describes a practical study on aggregated soil arthropod populations.
When data obtained from populations which do not fit a normal distribution are to be subjected to statistical tests such as those used to
determine significantly different population levels, the raw data cannot
be used because such tests are based on the assumption of normal clistribution. I n this case the data must be “transformed” by functions
which will vary with the type of distribution. Data which fit a Poisson
distribution should be converted to their square roots (Snedecor, 1946).
When the data fit symmetrically into the groups 0-1, 1-2, 2-4, 4-8,
8-16, etc. logarithms should be taken (Quenouille, 1950) and data which
fit the negative binomial should be transformed by log (z + k).