# Population Trend Analysis

In order to produce a clear picture of the long-term trend for each species, GAMs (General Additive Models) have been used to fit a smoothed line to each dataset, allowing for factors that could influence the means (e.g. bat detector make, temperature) where appropriate (see Barlow et.al. 2015 for more details). These smoothed curves are quite robust against random variation between years, except at the ends of the series where annual fluctuations and extreme outliers can have an unacceptably large impact on the first and last few years. To counteract this problem, it is best not to use the first year of a survey as the baseline year (where the index equals 100) and in this report the year 1999 has been taken as the baseline year wherever possible. Most surveys start from 1997, although there are a few exceptions. The Field Survey starts from 1998, and some Hibernation Surveys and Roost Counts have earlier years of data for some species. Where these data are available and improve trend estimation, they have been included in the GAMs but as they comprise small amounts of data, the start year is still shown as 1997. In all cases, the estimate for the most recent year should be regarded as provisional and a dotted line is used on the graphs to indicate this.

The average annual percentage change is an approximation based on the assumption that the trend during the period considered is constant and linear. It is estimated by calculating the annual percentage change that would take the population from 100 in the base year to the index value in the current year.

The Generalised Additive Models (GAMs) are based on the method described by Fewster et al. (2000). These involve fitting a log-linear generalised linear model (i.e. a regression model with a logarithmic relationship to the explanatory variables and a Poisson error distribution) to the counts on each survey. A site term is fitted in the model to allow for differences in abundance between sites and the time trend is modelled using the GAM framework to fit a smoothed curve. These GAM models are essentially a more sophisticated version of a polynomial curve, and are less likely to display misleading trends at the extremes of the data than a polynomial. The degree of smoothing is controlled by specifying the degrees of freedom for the smoothing process; this may vary between 1 (equivalent to a simple linear trend) and one less than the number of years (a ‘saturated model’ equivalent to fitting individual annual means). For the results presented here the degrees of freedom are generally set to the default value suggested by Fewster et al. (2000), which is 0.3 times the number of years. However curves for different degrees of freedom are always checked to ensure that the model provides an appropriate degree of smoothing to the annual means without being unduly influenced by individual outlying years. The index values are derived from the fitted curve, taking the base year to be 100. Annual means from the saturated model are also shown on the graphs in order to give a visual impression of any deviations from the smoothed curve.

The other feature of these models is that confidence limits based on standard theory will not be valid due to temporal correlations. In addition NBMP data suffer from other complications not present in the data examined by Fewster *et al.* which also invalidate the usual method of calculating confidence limits. Firstly the data are much more variable than would be expected from a Poisson distribution. This phenomenon, known as ‘overdispersion’, is very common in biological data, but is particularly extreme in these datasets. Fewster et al. (2000) suggest a negative binomial distribution might be an alternative but simulations suggest that, whilst it sometimes produces more precise results, this is not always the case. Secondly the repeat counts in each year add a further complexity to the correlation structure of the data. All these problems are avoided by using the bootstrap approach recommended by Fewster in which the model is fitted to a large number of new datasets created by resampling sites with replacement from the original sites. At least 400 bootstrap samples are used for each model to ensure robust 95% confidence limits. The same bootstrapping approach can be used to produce confidence limits to other quantities of interest, including the short- and long-term assessments used in the Defra biodiversity indicators.

Data for Great Britain are weighted to allow for the different sampling rates in England, Scotland and Wales. This is achieved by weighting each site in proportion to the ratio of non-upland area to the number of sites surveyed for the relevant country, thus ensuring that each country contributes equally to the trends based on land area. Weighting is not applied to those species, such as serotine and horseshoe bats, which have a restricted range within the UK.

Overdispersion is a particular problem for the bat detector surveys, where a single bat repeatedly flying past the observer may give rise to a large count of bat passes. This results in wide confidence limits for Poisson or negative binomial GAM models and so we have instead presented results for a binomial model of the proportion of observation points on each survey where the species was observed. Apart from this difference in the response variable, the same GAM approach, with bootstrap confidence limits, is adopted. Simulations suggest that these binomial models have greater power to detect trends with the high levels of overdispersion seen in the bat detector surveys.

In order to test whether the smoothed curves differed between different countries or regions Fewster et al. (2000) suggest a deviance test. However, simulations have suggested that this test can produce too many significant results, and so the results presented here use a randomisation approach to obtain a probability value from the change in deviance.

Analyses were conducted in Genstat and in R (version 3.4)

**References**

Barlow, K.E., Briggs, P.A., Haysom K.A., Hutson A.M., Lechiara, N.L., Racey, P.A., Walsh A.L. & Langton, S.D. (2015) Citizen science reveals trends in bat populations: the National Bat Monitoring Programme in Great Britain. Biological Conservation 182: 14-26

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

Toms, M.P. Siriwardena, G.M. & Greenwood, J.D. (1999) Developing a mammal monitoring programme for the UK. BTO research Report 223. British Trust for Ornithology, Thetford, Norfolk.

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