Mark J. Brewer

Regression analysis of spatial data

Many of the most interesting questions ecologists ask lead to analyses of spatial data. Yet, perhaps confused by the large number of statistical models and fitting methods available, many ecologists seem to believe this is best left to specialists. Here, we describe the issues that need consideration when analysing spatial data and illustrate these using simulation studies. Our comparative analysis involves using methods including generalized least squares, spatial filters, wavelet revised models, conditional autoregressive models and generalized additive mixed models to estimate regression coefficients from synthetic but realistic data sets, including some which violate standard regression assumptions. We assess the performance of each method using two measures and using statistical error rates for model selection. Methods that performed well included generalized least squares family of models and a Bayesian implementation of the conditional auto-regressive model. Ordinary least squares also performed adequately in the absence of model selection, but had poorly controlled Type I error rates and so did not show the improvements in performance under model selection when using the above methods. Removing large-scale spatial trends in the response led to poor performance. These are empirical results; hence extrapolation of these findings to other situations should be performed cautiously. Nevertheless, our simulation-based approach provides much stronger evidence for comparative analysis than assessments based on single or small numbers of data sets, and should be considered a necessary foundation for statements of this type in future.

Green Space and Stress: Evidence from Cortisol Measures in Deprived Urban Communities

Contact with green space in the environment has been associated with mental health benefits, but the mechanism underpinning this association is not clear. This study extends an earlier exploratory study showing that more green space in deprived urban neighbourhoods in Scotland is linked to lower levels of perceived stress and improved physiological stress as measured by diurnal patterns of cortisol secretion. Salivary cortisol concentrations were measured at 3, 6 and 9 h post awakening over two consecutive weekdays, together with measures of perceived stress. Participants (n = 106) were men and women not in work aged between 35–55 years, resident in socially disadvantaged districts from the same Scottish, UK, urban context as the earlier study. Results from linear regression analyses showed a significant and negative relationship between higher green space levels and stress levels, indicating living in areas with a higher percentage of green space is associated with lower stress, confirming the earlier study findings. This study further extends the findings by showing significant gender differences in stress patterns by levels of green space, with women in lower green space areas showing higher levels of stress. A significant interaction effect between gender and percentage green space on mean cortisol concentrations showed a positive effect of higher green space in relation to cortisol measures in women, but not in men. Higher levels of neighbourhood green space were associated with healthier mean cortisol levels in women whilst also attenuating higher cortisol levels in men. We conclude that higher levels of green space in residential neighbourhoods, for this deprived urban population of middle-aged men and women not in work, are linked with lower perceived stress and a steeper (healthier) diurnal cortisol decline. However, overall patterns and levels of cortisol secretion in men and women were differentially related to neighbourhood green space and warrant further investigation.

Possible human impacts on adaptive radiation: beak size bimodality in Darwin's finches

Adaptive radiation is facilitated by a rugged adaptive landscape, where fitness peaks correspond to trait values that enhance the use of distinct resources. Different species are thought to occupy the different peaks, with hybrids falling into low-fitness valleys between them. We hypothesize that human activities can smooth adaptive landscapes, increase hybrid fitness and hamper evolutionary diversification. We investigated this possibility by analysing beak size data for 1755 Geospiza fortis measured between 1964 and 2005 on the island of Santa Cruz, Galápagos. Some populations of this species can display a resource-based bimodality in beak size, which mirrors the greater beak size differences among species. We first show that an historically bimodal population at one site, Academy Bay, has lost this property in concert with a marked increase in local human population density. We next show that a nearby site with lower human impacts, El Garrapatero, currently manifests strong bimodality. This comparison suggests that bimodality can persist when human densities are low (Academy Bay in the past, El Garrapatero in the present), but not when they are high (Academy Bay in the present). Human activities may negatively impact diversification in ‘young’ adaptive radiations, perhaps by altering adaptive landscapes.

Protected area networks and savannah bird biodiversity in the face of climate change and land degradation.

The extent to which climate change might diminish the efficacy of protected areas is one of the most pressing conservation questions. Many projections suggest that climate-driven species distribution shifts will leave protected areas impoverished and species inadequately protected while other evidence suggests that intact ecosystems within protected areas will be resilient to change. Here, we tackle this problem empirically. We show how recent changes in distribution of 139 Tanzanian savannah bird species are linked to climate change, protected area status and land degradation. We provide the first evidence of climate-driven range shifts for an African bird community. Our results suggest that the continued maintenance of existing protected areas is an appropriate conservation response to the challenge of climate and environmental change.

A Bayesian model for local smoothing in kernel density estimation

A new procedure is proposed for deriving variable bandwidths in univariate kernel density estimation, based upon likelihood cross-validation and an analysis of a Bayesian graphical model. The procedure admits bandwidth selection which is flexible in terms of the amount of smoothing required. In addition, the basic model can be extended to incorporate local smoothing of the density estimate. The method is shown to perform well in both theoretical and practical situations, and we compare our method with those of Abramson (The Annals of Statistics 10: 1217–1223) and Sain and Scott (Journal of the American Statistical Association 91: 1525–1534). In particular, we note that in certain cases, the Sain and Scott method performs poorly even with relatively large sample sizes.We compare various bandwidth selection methods using standard mean integrated square error criteria to assess the quality of the density estimates. We study situations where the underlying density is assumed both known and unknown, and note that in practice, our method performs well when sample sizes are small. In addition, we also apply the methods to real data, and again we believe our methods perform at least as well as existing methods.

Do multiple herbivores maintain chemical diversity of Scots pine monoterpenes

A central issue in our understanding of the evolution of the diversity of plant secondary metabolites (PSMs) is whether or not compounds are functional, conferring an advantage to the plant, or non-functional. We examine the hypothesis that the diversity of monoterpene PSMs within a plant species (Scots pine Pinus sylvestris) may be explained by different compounds acting as defences against high-impact herbivores operating at different life stages. We also hypothesize that pairwise coevolution, with uncorrelated interactions, is more likely to result in greater PSM diversity, than diffuse coevolution. We tested whether up to 13 different monoterpenes in Scots pine were inhibitory to herbivory by slugs (Arion ater), bank voles (Clethrionomys glareolus), red deer (Cervus elaphus) and capercaillie (Tetrao urogallus), each of which attack trees at a different life stage. Plants containing more α-pinene were avoided by both slugs and capercaillie, which may act as reinforcing selective agents for this dominant defensive compound. Herbivory by red deer and capercaillie were, respectively, weakly negatively associated with δ3-carene, and strongly negatively correlated with the minor compound β-ocimene. Three of the four herbivores are probably contributory selective agents on some of the terpenes, and thus maintain some, but by no means all, of the phytochemical diversity in the species. The correlated defensive function of α-pinene against slugs and capercaillie is consistent with diffuse coevolutionary processes.

A new statistical framework for the quantification of covariate associations with species distributions

Summary Identifying processes that shape species geographical ranges is a prerequisite for understanding environmental change. Currently, species distribution modelling methods do not offer credible statistical tests of the relative influence of climate factors and typically ignore other processes (e.g. biotic interactions and dispersal limitation). We use a hierarchical model fitted with Markov Chain Monte Carlo to combine ecologically plausible niche structures using regression splines to describe unimodal but potentially skewed response terms. We apply spatially explicit error terms that account for (and may help identify) missing variables. Using three example distributions of European bird species, we map model results to show sensitivity to change in each covariate. We show that the overall strength of climatic association differs between species and that each species has considerable spatial variation in both the strength of the climatic association and the sensitivity to climate change. Our methods are widely applicable to many species distribution modelling problems and enable accurate assessment of the statistical importance of biotic and abiotic influences on distributions.

Assessing trends in biodiversity over space and time using the example of British breeding birds

Summary Partitioning biodiversity change spatially and temporally is required for effective management, both to determine whether action is required and whether it should be applied at a regional level or targeted more locally. As biodiversity is a multifaceted concept, comparative analyses of different indices, focussing on different components of biodiversity change (evenness vs. abundance), give better information than a single headline index. We model changes in the spatial and temporal distribution of British breeding birds using generalized additive models applied to count data collected between 1994 and 2011. Abundance estimates, accounting for differences in detectability, are then used in community-specific (farmland and woodland) biodiversity indices. Temporal trends in biodiversity, and change points in those trends, are assessed at different spatial scales. The geometric mean of relative abundance, a headline indicator of biodiversity change, is assessed together with a goodness-of-fit evenness measure focussing separately on the rare and common species in the communities. Our analysis reveals predominantly declining trends in biodiversity indices for farmland and woodland bird communities in southern and eastern England, perhaps signalling environmental deterioration in this part of the country. Conversely, our results also show generally more positive trends in the north of Britain, consistent with north–south gradient expectations from the effects of climate change. We also reveal predominantly positive changes in evenness for the common species and negative changes in evenness for the rarer species in the communities, consistent with previously documented homogenization in bird communities. Synthesis and applications. Bird populations are seen as good indicators of ecosystem health, and trends for different communities can be indicative of wider biodiversity changes within their respective habitats. However, temporal trends in biodiversity at the national level may miss opposing trends occurring at different locations within the nation. We develop methods that allow assessment of how temporal trends vary spatially and whether these trends differ for the rare and common species in the respective communities. Our methods may be used to test hypotheses about the processes that generate the trends.

A Hierarchical Model for Compositional Data Analysis

This article introduces a hierarchical model for compositional analysis. Our approach models both source and mixture data simultaneously, and accounts for several different types of variation: these include measurement error on both the mixture and source data; variability in the sample from the source distributions; and variability in the mixing proportions themselves, generally of main interest. The method is an improvement on some existing methods in that estimates of mixing proportions (including their interval estimates) are sure to lie in the range [0, 1]; in addition, it is shown that our model can help in situations where identification of appropriate source data is difficult, especially when we extend our model to include a covariate. We first study the likelihood surface of a base model for a simple example, and then include prior distributions to create a Bayesian model that allows analysis of more complex situations via Markov chain Monte Carlo sampling from the likelihood. Application of the model is illustrated with two examples using real data: one concerning chemical markers in plants, and another on water chemistry.

Discretisation for inference on normal mixture models

The problem of inference in Bayesian Normal mixture models is known to be difficult. In particular, direct Bayesian inference (via quadrature) suffers from a combinatorial explosion in having to consider every possible partition of n observations into k mixture components, resulting in a computation time which is O(kn). This paper explores the use of discretised parameters and shows that for equal-variance mixture models, direct computation time can be reduced to O(Dknk), where relevant continuous parameters are each divided into D regions. As a consequence, direct inference is now possible on genuine data sets for small k, where the quality of approximation is determined by the level of discretisation. For large problems, where the computational complexity is still too great in O(Dknk) time, discretisation can provide a convergence diagnostic for a Markov chain Monte Carlo analysis.