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BBWC home > Contents > Methodology > Statistical methods used for alerts 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 2.9.1 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 site-effect 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 widely used program TRIM (Pannekoek & van Strien 1996). 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 smooth function of year. It was not straightforward to fit GAMs to the CBC/BBS, CES or Heronries Census data and we have therefore fitted smooth curves with a similar degree of smoothing to the annual indices (details given below). The non-parametric smooth 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 while the full annual model has df = t-1, 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 on the web site 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. The present ‘Breeding Birds in the Wider Countryside’ is the first since the close of the CBC and the first to present joint CBC/BBS indices, in place of those derived solely from the CBC. The model fitted to these combined data is that historically employed for the BBS, a Generalized 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 thin-plate smoothing spline with 11 degrees of freedom. GAMs were fitted to the WBS data using the approach described above (Fewster et al. 2000). 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 The section on confidence limits and statistical testing (2.8.4) gives the reasons for choosing these particular confidence limits. The GAMs were fitted using a modified version of the FORTRAN program GAIM (Hastie & Tibshirani 1990). Annual indices were fitted to catches of adults and juveniles separately using the method described by Peach et al. (1998). This is essentially the annual 'sites x years' model described above but with the addition of an offset to correct for missing visits. Offsets could not easily be incorporated in the GAM software that we have available. Therefore we fitted a smooth curve to the annual indices. This was done using PROC TSPLINE of SAS with 6 degrees of freedom. This procedure should give very similar estimates to a GAM analysis, but it does not provide confidence intervals for the smoothed population trends, nor for the change measures derived from it. Therefore all alert flags relating to the CES are shown in square brackets. The Heronries Census data were analysed using a modified sites x years model which incorporates information about new colonies (sites) that have been established and other colonies from the sample that are known to have gone 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 with 23 degrees of freedom. 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. |
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