Calculation of log collated indices
##### Background

A number of techniques have been suggested to calculate indices of abundance from wildlife monitoring data (ter Braak et al, 1994). The main methods used to analyse butterfly monitoring data are 'chaining' methods and log-linear models. Up to 2001, the method adopted by the UK Butterfly Monitoring Scheme was based on a modified chaining method as described by Moss & Pollard (1993), but from 2002 log-linear models have been used as performed by the statistical software package TRIM (Pannekoek & van Strien 2001) and the method used by the Dutch Butterfly Monitoring Scheme.The log-linear models used are described in detail below, as is the method used to derive standard errors for the collated index using bootstrap techniques.

##### Loglinear Poisson regression and TRIM indices

An alternative method for estimating annual indices is to use a loglinear Poisson regression model (ter Braak et al. 1993). In this approach the expected count at a particular site in a given year is assumed to be a product of a site and a year effect, i.e. Expected count = site effect * year effect. If the expected count for site i in year j is denoted by m ij then the model can be written as :

log ( m ij ) = a i + b j

where a i ­and b j denote the effects on a log scale for the ith site and the jth year

­(i = 1, . . ., a ; j = 1, . . ., b ). The index for year j relative to year 1 is defined as b j - b 1 . Alternatively, an index could be defined relative to the series mean index,

i.e. b j - ( b 1 + b 2 + . . . + b b )/b. Note that the choice of baseline does not affect the pattern of year–to-year fluctuations. However, using the series mean provides a more stable reference for assessing individual years, and also helps to compare indices using different methods (see below).

Observed counts are subject to random variation reflecting natural fluctuations in abundance and sampling error. In the log linear Poisson regression model the variance of the observed count is proportional to the mean, i.e. var [count] = c m ij , where c is the dispersion parameter.

The loglinear Poisson regression model can be fitted using the software TRIM (Pannekoek & van Strien 2001). The program estimates the dispersion parameter, and can allow for serial correlation between counts at the same site in different years. Standard errors of the indices are based on the assumption of variance proportional to mean, and a pattern of serial correlation which declines exponentially with time between counts.

For the BMS data, however, these assumptions appear unrealistic, so standard errors have been obtained by applying the bootstrap method (see below), to indices derived using a loglinear Poisson regression model without serial correlation. The approach relaxes the assumption of variance proportional to mean, and allows for the effect of serial correlation. Bootstrap standard errors are not currently available in TRIM, so they have been calculated here using the statistical package Genstat 5 (Genstat Committee 2000).

##### Bootstrap standard errors for BMS indices

Standard errors for the BMS indices have been obtained by the bootstrap method (Efron, B. & Tibshirani, R.J. 1998, Manly, 1997). Bootstrapping involves drawing repeated random samples, with replacement, from the original sample. The bootstrap samples are then used to calculate properties of estimates, e.g. bias and standard error.

This resampling method is computer-intensive, but requires no theoretical calculations and can be easily implemented for any estimate. For the BMS data, bootstrapping proceeds by drawing random samples of n sites, with replacement, from the original set of n sites. For each bootstrap sample, the BMS index is calculated. The standard error of the index is then estimated from the values in large number of bootstrap samples - in this case 500.

Note that a typical bootstrap sample will generally include some sites more than once, with other sites omitted altogether, but in the calculation of the indices the sites are treated as distinct. The method has been successfully applied in the analysis of long-term trends in birds (e.g. Fewster et al. 2000), and is suited to the BMS data.

##### References

Efron, B. & Tibshirani, R.J. (1998). An Introduction to the Bootstrap. Chapman & Hall/CRC, London.

Fewster, R.M., Buckland, S.T., Siriwardena, G.M., Baillie, S.R. & Wilson, J.D. (2000). Analysis of Population Trends for Farmland Birds Using Generalized Additive Models. Ecology, 81, 1970-1984.

Genstat Committee (2000). Genstat Release 4.2 Reference Manual Part 1: Summary. Numerical Algorithms Group, Oxford.

Manly, B. F. J. (1997). Randomization, Bootstrap and Monte Carlo Methods in Biology. Chapman & Hall, London.

Moss, D. & Pollard, E. (1993). Calculation of collated indices of abundance of butterflies, based on monitored sites. Ecological Entomology, 18, 77-83.

Pannekoek, J. & van Strien, A. (2001). TRIM 3 Manual (Trends & Indices for Monitoring data). Statistics Netherlands.

ter Braak, C.J.F., van Strien, A.J., Meijer, R. & Verstrael, T.J. (1994). Analysis of monitoring data with many missing values: which method? In: W. Hagemeijer & T. Verstrael, Eds. Bird Numbers 1992. Distribution, Monitoring and Ecological Aspects. Proceedings of the 12 th International Conference of the International Bird Census Committee and European Ornithological Atlas Committee. SOVON, Beek-Ubbergen, The Netherlands.