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2.9
Statistical methods used for alerts
The
Alert
System page contains a general overview of how the alert
system works. More detailed information is given below about the
statistical methods used to estimate population indices, population
changes and their confidence intervals.
2.9.1 General
structure of data and models
2.9.2 Fitting smoothed models
2.9.3 CBC/BBS trends
2.9.4 WBS/WBBS trends
2.9.5 Constant Effort Sites Scheme
2.9.6 Heronries Census
2.9.1
General structure of data
The data for
all of the schemes reported here consist of annual counts made over
a period of years at a series of sites. They can thus be summarised
as a data matrix of sites x years, within which a proportion of
the cells contain missing values because not all of the sites are
covered every year. Such data can be represented as a simple model:
log
(count) = site effect + year effect
Each site has
a single siteeffect parameter. These site parameters are not usually
of biological interest but they are important because abundance
is likely to differ between sites. The main parameters of interest
are the year effects. These can be modelled either with as many
parameters as years (an annual model), or with a smaller number
of parameters, representing a smoothed curve.
A simple annual
model would be fitted as a generalised linear model with Poisson
errors and a log link function. This is the main model provided
by the program TRIM (Pannekoek
& van Strien 1996), which is widely used for population
monitoring.
2.9.2
Fitting smoothed models
Our preferred
method for generating a smoothed population trend is to fit a smoothed
curve to the data directly using a generalised additive model (GAM)
(Hastie & Tibshirani
1990, Fewster
et al. 2000). Thus the model from the previous section
becomes:
log
(count) = site effect + smooth (year)
where smooth
(year) represents some smoothing function of year. It was not straightforward
to fit GAMs to the CBC/BBS or Heronries Census data and we have
therefore fitted smoothed curves with a similar degree of smoothing
to the annual indices (details below).
The nonparametric
smoothed curve fitted in our models is based on a smoothing spline.
The degree of smoothing is specified by the number of degrees of
freedom (df). A simple linear trend has df = 1, whereas the full
annual model has df = t1, where t is the number of years in the
time series. Here we set df to be approximately 0.3 times the number
of years in the time series (Fewster
et al. 2000). The degrees of freedom used for the
main data sets presented in this report are summarised below.
Note that the
numbers of years shown here are different from those available for
calculating change measures, because we use the whole time series
available for analysis (i.e. prior to the truncation of end points),
and because we count the number of years in the time series rather
than the number of annual change measures.
2.9.3
CBC/BBS trends
The model fitted
to the combined CBC and BBS data is that historically employed for
the BBS, a generalised linear model with counts assumed to follow
a Poisson distribution and a logarithmic link function. Standard
errors were calculated via a bootstrapping procedure. For presentation
in the figures, both the population trend and its confidence limits
were also subsequently smoothed using a thinplate smoothing spline.
The overall result is a smoothed trend that is mathematically equivalent
to that produced from a generalised additive model.
A similar method as employed for the joint CBC/BBS trend has been
used for the BBS alone. This adopted a generalised
linear model with counts assumed to follow a Poisson distribution
and a logarithmic link function. Standard errors were calculated
via a bootstrapping procedure involving 199 bootstraps. For presentation
in the figures, both the population trend and its confidence limits
were also subsequently smoothed using a thinplate smoothing spline.
2.9.4
WBS/WBBS trends
The model fitted to the combined WBS and WBBS data is identical
to that employed for the joint CBC/BBS trend, a generalised
linear model with counts assumed to follow a Poisson distribution
and a logarithmic link function. Standard errors were calculated
via a bootstrapping procedure involving 199 bootstraps. For presentation
in the figures, both the population trend and its confidence limits
were also subsequently smoothed using a thinplate smoothing spline.
The overall result is a smoothed trend that is mathematically equivalent
to that produced from a generalised additive model, as used in earlier
reports for the WBS data alone.
2.9.5
Constant Effort Sites
GAMs were fitted to the CES data for catches of adults and juveniles
separately with the addition of an offset to correct for missing
visits. Confidence limits were fitted using a bootstrap technique
to avoid restrictive assumptions about the distribution of the data.
Bootstrap samples were drawn from the data by sampling plots with
replacement. We generated 199 bootstrap samples from each data set
and fitted a GAM to each of them. Confidence limits for the smoothed
population indices (85% cl) and change measures (90% cl) were determined
by taking the appropriate percentiles from the distributions of
the bootstrap estimates, in a similar manner to that employed for
the WBS/WBBS trends.
2.9.6
Heronries Census
The Heronries
Census data were analysed using a modified sites x years model based
on ratio estimation which incorporates information about new colonies
(sites) that have been established and other colonies from the sample
that are known to have become extinct. The method was developed
by Thomas (1993) specifically
in relation to the heronries data set. Since then the heronries
database has been substantially upgraded and the method has been
applied to the full data set (Marchant
et al. 2004).
The above method
of analysis cannot be easily applied within a GAM framework. Therefore
we fitted a smooth curve to the annual indices. This was done using
PROC TSPLINE of SAS (SAS
2009). This procedure should give very similar estimates to
a GAM analysis but it does not provide confidence intervals for
the smoothed population trend or the change measures derived from
it. This is not a serious limitations as there are no potential
alerts for Grey Heron, whose
populations have generally been increasing.
Section
3 – Species pages
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