RACHEL M. FEWSTER [1,4]
STEPHEN T. BUCKLAND [1]
GAVIN M. SIRIWARDENA [2,3]
STEPHEN R. BAILLIE [2]
JEREMY D. WILSON [3,5]
Abstract. Knowledge of the direction, magnitude, and timing of changes in bird population abundance is essential to enable species of priority conservation concern to be identified, and reasons for the population changes to be understood. We give a brief review of previous techniques for the analysis of large-scale survey data and present a new approach based on generalized additive models (GAMs). GAMs are used to model trend as a smooth, nonlinear function of time, and they provide a framework for testing the statistical significance of changes in abundance. In addition, the second derivatives of the modeled trend curve may be used to identify key years in which the direction of the population trajectory was seen to change significantly. The inclusion of covariates into models for population abundance is also discussed and illustrated, and tests for the significance of covariate terms are given. We apply the methods to data from the Common Birds Census of the British Trust for Ornithology for 13 species of f armland birds. Seven of the species are shown to have experienced statistically significant declines since the mid-1960s. Two species exhibited a significant increase. The population trajectories of all but three species turned downward in the 1970s, although in most cases the 1980s brought either some recovery or a decrease in the rate of decline. The majority of populations have remained relatively stable in the 1990s. The results are comparable with those from other analysis techniques, although the new approach is shown to have advantages in generality and precision. We suggest extensions of the methods and make recommendations for the design of future surveys.
Key words: bird census schemes; British birds; change points; Common Birds Census; farmland birds; genralized additive models; log-linear Poisson regression; nonlinear trend; population trajectory; spatiotemporal models; trend analysis.
INTRODUCTION
Accurate analysis of long-term monitoring data is essential for the effective management and conservation of wildlife populations. An important component of such analysis is the determination of trend in population abundance over time. However, the nature of ecological data is such that obtaining reliable estimates of annual abundance is far from straightforward. The methods that are usually employed for time series data are generally inappropriate, because the time span of the surveys is too short (Bowerman and O'Connell 1987). Difficulties are also caused by missing data, which tend to be characteristic of large-scale census schemes (ter Braak et al. 1994, Thomas 1996). Barker and Snuer (1992) outline some other problems typically associated with wildlife survey data.
In the past, indices of abundance were obtained from census data using the chain method (e.g., Marchant et al. 1990). The ratio of abundance in one year to abundance in the next was calculated as the ratio of summed counts in the two years, with the summation taken over only those sites that were surveyed in both years. This method is now generally regarded as inadequate, because of inefficient use of the data and a tendency to generate spurious trends through random drift (Mountford 1982, 1985, Peach and Baillie 1994, ter Braak et al. 1994). A number of alternative procedures have since been proposed, particularly for bird census schemes (ter Braak et al. 1994, Thomas 1996). Nonetheless, there is still some scope for improvement (Thomas 1996).
In North America, most analyses are conducted using route regression techniques. For each of many surveyed routes, a linear regression of log-count against time is performed to yield a log-linear, route-specific trend. Overall trends, for example over a region or state, are obtained by combining the route-specific results using various weighting schemes (Geissler and Noon 1981, Geissler and Snuer 1990, Sauer and Geissler 1990). In Europe, attention has focused on sites-by-years models, in which the expected count in a given site and year is modeled as a function of quantities known as the site effect and the year effect. The site effect allows for variation in abundance between plots, whereas the year effect allows for fluctuations in abundance over time. These models have been fitted using Poisson regression techniques (Pannekoek and van Strien 1996) and by the Mountford method (Mountford 1982, 1985). In contrast to the log-linear form prescribed by route regression, the estimates of annual abundance derived from sites-by-years models are not constrained to follow any pre-specified curve or shape.
The distinct approaches largely reflect differences in the objectives of the analyses. Models that treat log-counts as a linear function of time are designed to estimate some average rate of population change over a time period, although the fit is likely to be poor if the true pattern of change was markedly nonlinear. Conversely, unconstrained annual abundance estimates reveal every fluctuation in population numbers, but can be hard to interpret in terms of long-term change. Consequently, neither approach is well-suited to the investigation of long-term, but nonlinear, trends in population numbers.
Previous attempts to distinguish genuine patterns of nonlinear change from the "noise" of annual fluctuation have centered on the application of smoothing algorithms to annual abundance estimates. For example, Siriwardena et al. (1998a) used a compound running-median algorithm for the smoothing of Mountford indices derived from British bird census data. However, smoothing the output from one model amounts to the application of a second model, and methods that incorporate nonlinear trend estimation directly into the fitting of the original model would be preferable.
In this paper, we present new methods for the analysis of trends in wildlife census data, using generalized additive models (GAMs). GAMs are not simply smoothing devices, but provide a complete modeling framework. Smoothing procedures are built into the model-fitting process, so that inference based on the resulting smooth curve of abundance indices is made fully within the context of the original model. Instead of requiring a linear form for the log trend, GAMs allow any shape ranging from a straight line, through a range of nonparametric curves of increasing complexity, to unconstrained annual estimates. These latter are equivalent to the estimates obtained for sites-by-years models from Poisson regression. The range of smooth curves available from the GAM allows analyses of counts to be tailored a priori according to the biological question of interest: simple linear changes, long-term nonlinear trends, or annual fluctuation.
Smoothing approaches have also been developed in North America to overcome the restrictions of linearity in the traditional route regression method. For example, Taub (1990) and James et al. (1990, 1996) have applied the smoothing algorithm LOESS to counts at the site level and higher. This method has some similarities with the GAM approach, and these will be discussed.
We begin the present paper by formulating the GAM and demonstrating how the conventional sites-by-years model may be regarded as a special case. Significance tests are described for drawing inference from the smooth series of GAM abundance estimates. Next, we present a method to identify key years in which there was seen to be a significant change in the direction of the population trajectory. This is accomplished by estimating the curve of second derivatives of the abundance index curve, and identifying years in which the second derivative was significantly different from zero. Estimating the timing of population changes in this way helps to suggest causes of the change. We conclude with a discussion of covariate models, which enable factors such as geographical location or climate to be taken into account.
The methods are illustrated using data from the British Trust for Ornithology's Common Birds Census (CBC), although they are applicable to a wide range of census schemes. The CBC is one of the longest running of all wildlife monitoring schemes, with annual data available since 1962. Plots of land, or sites, are selected by volunteer observers for repeated visits throughout the breeding season. Territories of all bird species are mapped as accurately as possible according to standard criteria, and the final site count for a given species is the estimated number of territory-holding males in the site. The survey design is described in detail by Marchant et al. (1990).
