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Die Population des Grauspechts (Picus canus) ist seit den 1990er-Jahren in der Schweiz um zwei Drittel zurückgegangen, weshalb er seit 2021 auf der Roten Liste als stark gefährdet geführt wird. Die Ursachen für den markanten Populationsrückgang sind bislang jedoch unklar. Um mehr über seine Habitatansprüche zu erfahren, wurden im Schweizer Jura in den Jahren 2020 bis 2022 insgesamt 55 Grauspecht-Individuen besendert. In diesem methodischen Beitrag stellen wir vor, wie wir mithilfe von Telemetriedaten und hochaufgelösten Fernerkundungsdaten ein Modell zur kleinräumigen Habitatpräferenz während der Brutzeit erstellen werden. Die in diesem Projekt gewonnenen Erkenntnisse sollen dabei helfen, Massnahmen für die Erhaltung und die Förderung dieser heimlichen Spechtart zu erarbeiten.

International audience; In ecology, the true causal structure for a given problem is often not known, and several plausible models and thus model predictions exist. It has been claimed that using weighted averages of these models can reduce prediction error, as well as better reflect model selection uncertainty. These claims, however, are often demonstrated by isolated examples. Analysts must better understand under which conditions model averaging can improve predictions and their uncertainty estimates. Moreover, a large range of different model averaging methods exists, raising the question of how they differ in their behaviour and performance. Here, we review the mathematical foundations of model averaging along with the diversity of approaches available. We explain that the error in model‐averaged predictions depends on each model's predictive bias and variance, as well as the covariance in predictions between models, and uncertainty about model weights. We show that model averaging is particularly useful if the predictive error of contributing model predictions is dominated by variance, and if the covariance between models is low. For noisy data, which predominate in ecology, these conditions will often be met. Many different methods to derive averaging weights exist, from Bayesian over information‐theoretical to cross‐validation optimized and resampling approaches. A general recommendation is difficult, because the performance of methods is often context dependent. Importantly, estimating weights creates some additional uncertainty. As a result, estimated model weights may not always outperform arbitrary fixed weights, such as equal weights for all models. When averaging a set of models with many inadequate models, however, estimating model weights will typically be superior to equal weights. We also investigate the quality of the confidence intervals calculated for model‐averaged predictions, showing that they differ greatly in behaviour and seldom manage to achieve nominal coverage. Our overall recommendations stress the importance of non‐parametric methods such as cross‐validation for a reliable uncertainty quantification of model‐averaged predictions.

Bird migration and wind energy production exploit a similar airspace. There is a well-founded claim that conservation should aim at facilitating both activities. Negative effects can be mitigated either by avoiding a spatial concurrence or by accounting for the temporal course of migration and stopping wind turbines during peak flight activities. In this study we promote a new methodological approach to reduce potential conflicts in the planning as well as during the operation phase of a wind energy project. The basis is a new spatially explicit model for broad front migration. It allows to quantify the collision risk with respect to topography. We simulated migration of non-soaring birds across Switzerland. Model parameters were tuned to achieve results in accordance with current expert knowledge based on many years of visual observations and radar measurements. The resulting maps were used to define areas with different collision risks. For medium and high risk areas, we propose a permanent monitoring system, which is able to shut down the local turbines during peak migration. We evaluated the impact of such a shutdown regime in five specific sites with quantitative radar data for at least one migration season. The model presented here is a simple preliminary, but robust, approach. The main weakness of the model is the use of large-scale rather than local wind conditions. Within the Alps, local wind fields can differ considerably from the general pattern, and accordingly also the distribution of flight directions. We hope to provide a basis for similar models in other geographic areas. In addition, we call for the use of large scale monitoring data, as hopefully will soon be available from weather radar networks, to validate any kind of spatially explicit migration models.

Ring re-encounter data, in particular ring recoveries, have made a large contribution to our understanding of bird movements. However, almost every study based on ring re-encounter data has struggled with the bias caused by unequal observer distribution. Re-encounter probabilities are strongly heterogeneous in space and over time. If this heterogeneity can be measured or at least controlled for, the enormous number of ring re-encounter data collected can be used effectively to answer many questions. Here, we review four different approaches to account for heterogeneity in observer distribution in spatial analyses of ring re-encounter data. The first approach is to measure re-encounter probability directly. We suggest that variation in ring re-encounter probability could be estimated by combining data whose re-encounter probabilities are close to one (radio or satellite telemetry) with data whose re-encounter probabilities are low (ring re-encounter data). The second approach is to measure the spatial variation in re-encounter probabilities using environmental covariates. It should be possible to identify powerful predictors for ring re-encounter probabilities. A third approach consists of the comparison of the actual observations with all possible observations using randomization techniques. We encourage combining such randomisations with ring re-encounter models that we discuss as a fourth approach. Ring re-encounter models are based on the comparison of groups with equal re-encounter probabilities. Together these four approaches could improve our understanding of bird movements considerably. We discuss their advantages and limitations and give directions for future research.

The likelihood is a relative measure of the probability of obtaining a specific data set given a model. If the likelihood of model 1 is twice as high as the likelihood of model 2, the probability of obtaining precisely this data set is twice as large for model 1 than for model 2. The likelihood function is the likelihood for a fixed data set and varying parameter values, and is used to obtain maximum likelihood (ML) estimates of model parameters. The predictive density is the probability distribution of the data given the model and is sometimes called the Bayesian likelihood.

The results of the data analysis are the basis for your conclusions. They contain estimates of the effects that have been studied including a measurement of uncertainty. Often, derived parameters are more meaningful than model parameters. In addition, it is helpful to present the information needed to judge the biological (or other) relevance of the results. Such information may include a plot or a measure of the total variance or the residual variance in the data so that the effects can be compared to natural variation. Figures can be an informative and easily understandable tool to communicate results. The description of a linear model in the methods section consists of the error distribution, the link function, the linear predictor, the random structure, the prior distributions, the fitting method, and how conclusions are drawn from the joint posterior distribution of the model parameters.

Generalized linear mixed models (GLMM) are the combination of generalized linear models and linear mixed models, which means that the error distribution can be different from Gaussian and that a random factor is included in the model. We discuss the binomial model used to analyze proportions or binary outcome variables, and the Poisson model used to analyze counts or rates data. How we deal with residual analyses, overdispersion, zero-inflation, or modeling rates or densities using an offset are all analogous to previous chapters. We work step by step through a rather complex Poisson mixed model including an offset and variable transformations as well as a derived parameter.

This chapter starts with some general comments about the software used in this book, focusing on R, the statistical software package used in most chapters. R is complemented by OpenBUGS and Stan, which both allow fitting more complex models using Bayesian methods in a very flexible way ( Chapter 12 ). Then we provide some basic information regarding important statistical terms and link them to the use of R. The subsections touch on various topics: data sets, variables, observations, distributions and summary statistics, R objects, graphics, and writing R functions.

Mixed effects models are used to analyze nonindependent, that is, grouped or hierarchical data. These models are called mixed effects models because they contain fixed and random effects. Fixed effects are factors with a finite (“fixed”) number of levels (e.g., sex), whereas random effects have a theoretically infinite number of levels of which we have measured a random sample. For fixed effects we are interested in the specific differences between levels, whereas for random effects we are interested in the between-level (between-group) variance rather than in differences between specific levels. The chapter introduces the concept of partial pooling and covers fitting and interpreting models with random intercepts, random slopes, nested, and crossed random effects.

How to select a model or to base conclusions on more than one model depends on the purpose and the design of the study. In this chapter, we discuss the difference between confirmatory and predictive analyses and explain the tradeoff between bias and precision in statistical models. We present commonly used methods to rank models according to their predictive performance, such as cross-validation and the widely applicable information criterion (WAIC). We also introduce the least absolute shrinkage and selection operator (LASSO) and ridge regression as flexible tools to shrink model coefficients toward zero. Finally, we present how to do model averaging based on posterior model probabilities. Throughout the chapter, we emphasize the importance of prior knowledge and scientific reasoning when building models or combining information from different models.

Spatial autocorrelation is a general property of most ecological data sets. When data are spatially autocorrelated, the value of a random variable characterizing a site can be partially predicted by the values at neighboring sites. Spatial autocorrelation can thus be described as one of the mechanisms leading to pseudoreplication. Uncertainty will be underestimated when this lack of independence is not properly accounted for. We discuss some methods of modeling such data using genersalized linear mixed models (GLMM), and we also present solutions for moderately large data sets.

Model assumptions need to be checked before drawing conclusions from any model. Usually, the residuals are independent and identically distributed (iid), when the model assumptions are met. We discuss several graphical methods for assessing independence and distribution of the residuals such as the Tukey–Anscombe or the QQ plot. We further discuss how to detect temporal and spatial autocorrelation as well as heteroscedasticity in the residuals.

We introduce a Bayesian method to assess goodness of fit and also discuss the use of the proportion of variance explained, R 2 , a classical measurement of goodness of fit. Posterior predictive model checking is the comparison of the simulated data from the model with the observed data. The simulated data can differ from the observed data in various aspects and to various degrees. The description of the discrepancies between the simulated and the observed data shows how well the model reflects different aspects of the data. The Bayesian p -value is the probability that a test statistic calculated from simulated data is larger than the test statistic calculated from the observed data. Bayesian p -values based on specific test statistics can be used to formally evaluate model fit, but the most important comparison between the observed and the simulated data is primarily done graphically. We show how R 2 values can be calculated for different levels of a mixed model.

In a generalized linear model (GLM), the distribution of the data can be chosen from a variety of statistical distributions. Therefore, these models are used when the outcome variable is a count or proportions or something else that will produce nonnormally distributed residuals in a normal linear model. We introduce the binomial, the Bernoulli, and the Poisson distributions and explain the link functions used together with these distributions in generalized linear models. We discuss how overdispersion and zero-inflation can be diagnosed and what can be done if such data structures occur. We also explain how offsets can be used to analyze rates or densities.

Bayesian data analysis is the process of fitting a probability model to data and drawing inferences based on posterior distributions of the model parameters or derived quantities. Bayes’ theorem provides the theoretical background and modern computer techniques, such as the R function sim , allow for the application of these processes. The posterior distribution of model parameters expresses our belief about the parameter values after having looked at the data and given that the model is realistic. Frequentists express the uncertainty about parameter estimates by showing where hypothetical new parameter values would have been if the experiment had been repeated many times. We like the Bayesian methods for several reasons: (1) because of the ease of obtaining uncertainty estimates for derived quantities (error propagation), often answering our question more directly than estimates for model parameters; (2) because they make it feasible for those who are not mathematicians to fit more complex, and thus more realistic, models, and (3) because they formally allow including prior knowledge and combining information from different data sources.

The checklist provides a step-by-step guide through a typical data analysis. After data checking, we start with defining a model structure depending on the nature of the data-generating process. Then, the error distribution is chosen. The explanatory variables are prepared using transformations, centering, and scaling. After model fitting, goodness of fit is assessed using residual analyses, posterior predictive model checking, or sensitivity analyses. Some residual plots and test statistics for posterior predictive checking are suggested. We discuss some patterns in the residuals that indicate lack of fit and what can be done in these cases. In the last step, when the model fits the data well, conclusions are drawn.

Markov chain Monte Carlo (MCMC) is a simulation technique that can be used to find the posterior distribution and to sample from it. Thus, it is used to fit a model and to draw samples from the joint posterior distribution of the model parameters. The different MCMC algorithms differ in their performance in relation to speed and convergence depending on the model structure. Gibbs sampling, Metropolis–Hasting algorithm, and Hamiltonian Monte Carlo are briefly presented in a nontechnical way. The software OpenBUGS and Stan are MCMC samplers. Convergence of the chains is assessed graphically using traceplots and diagnostic statistics such as the -value or the Monte Carlo error.

Prior distributions, or “priors” are important parts of the model. Improper, flat, and weakly informative prior distributions are discussed. Informative or weakly informative prior distributions have computational as well as conceptual advantages over flat priors. Conjugate priors ease the calculation of posterior distributions by hand. The influence of the prior on the estimated slope of a regression line is assessed using a prior sensitivity analysis. We also use different prior distributions for the residual variance in a regression and the between-nest variance in a zero-inflated mixed model. As prior for the variance parameters, the uniform, inverse-gamma, half-Cauchy, and folded t -distributions are compared. Finally, the relative information in the data compared to the prior distribution is measured by the overlap between the prior and the posterior distribution.

Statistical models serve to communicate information in data, to think about systems, to learn from data, and to make predictions and decisions. Our daily life is governed by models. This book is about linear models and extensions of these. In a linear model, the expected value of the outcome variable itself, or a transformation thereof, is a linear function of the predictor variables (the deterministic part of the model). The scatter of the observations around the expected value is described by a probability distribution (the stochastic part of the model). The statistical models are applied using R, BUGS, and Stan and inference is drawn in a Bayesian framework. The book is written by practitioners for practitioners.

Normal linear models (LM) are used when the outcome variable can be modeled directly with predictor variables, that is, no link function is needed, and when the resulting residuals are approximately normally distributed. These models include widely used techniques: simple linear regression for a continuous predictor, one-way ANOVA for a categorical predictor (which includes comparing two samples as in a t -test), their analogs with more than one predictor (i.e., multiple regression and multiway ANOVA, respectively), and ANCOVA when continuous as well as categorical predictors are used. Thus, it becomes clear that all these techniques are special cases of one and the same technique: the normal linear model. The chapter covers fitting and interpreting such models and the related topics of collinearity, ordered factors, and polynomials. The chapter is fundamental for understanding the subsequent models covered in following chapters.

This chapter introduces five hierarchical ecological models and corresponding BUGS and Stan code. First, a hierarchical multinomial model is used to study the use of different roosting site types by an owl species based on telemetry data in relation to environmental variables. Second, we analyze the breeding success of the black stork using a zero-inflated Poisson model with random effects. Third, the distribution of a toad species is estimated from presence-absence data taking into account that the species may have gone undetected during a visit. To do so, an occupancy model is fitted to the data using Stan. Fourth, an extension of the occupancy model, the territory occupancy model, is used to estimate survival based on observations of unmarked birds. Finally, we introduce the Stan code to fit a Cormack–Jolly–Seber (CJS) model including random effects. The CJS model is widely used in ecology to estimate survival based on mark-recapture data.

International audience; The effects of patch size and isolation on metapopulation dynamics have received wide empirical support and theoretical formalization. By contrast, the effects of patch quality seem largely underinvestigated, partly due to technical difficulties in properly assessing quality. Here we combine habitat-quality modeling with four years of demographic monitoring in a metapopulation of greater white-toothed shrews (Crocidura russula) to investigate the role of patch quality on metapopulation processes. Together, local patch quality and connectivity significantly enhanced local population sizes and occupancy rates (R2 = 14% and 19%, respectively). Accounting for the quality of patches connected to the focal one and acting as potential sources improved slightly the model explanatory power for local population sizes, pointing to significant source-sink dynamics. Local habitat quality, in interaction with connectivity, also increased colonization rate (R2 = 28%), suggesting the ability of immigrants to target high-quality patches. Overall, patterns were best explained when assuming a mean dispersal distance of 800 m, a realistic value for the species under study. Our results thus provide evidence that patch quality, in interaction with connectivity, may affect major demographic processes.

Recent developments in metacommunity theory have raised awareness that processes occurring at regional scales might interfere with local dynamics and affect conditions for the local coexistence of competing species. Four main paradigms are recognized in this context (namely, neutral, patch-dynamics, species-sorting, and mass-effect), which differ according to the role assigned to ecological or life-history differences among competing species, as well as to the relative time scale of regional vs. local dynamics. We investigated the patterns of regional and local coexistence of two species of shrews (Crocidura russula and Sorex coronatus) sharing a similar diet (generalist insectivores) over four generations, in a spatially structured habitat at the altitudinal limit of their distributions. Local populations were small, and regional dynamics were strong, with high rates of extinction and recolonization. Niche analysis revealed significant habitat differentiation on a few important variables, including temperature and availability of winter resting sites. In sites suitable for both species, we found instances of local coexistence with no evidence of competitive exclusion. Patterns of temporal succession did not differ from random, with no suggestion of a colonization-competition trade-off. Altogether, our data provide support for the mass-effect paradigm, where regional coexistence is mediated by specialization on different habitat types, and local coexistence by rescue effects from source sites. The strong regional dynamics and demographic stochasticity, together with high dispersal rates, presumably contributed to mass effects by overriding local differences in specific competitive abilities.