Caribou movement as a correlated random walk

Movement is a primary mechanism coupling animals to their environment, yet there exists little empirical analysis to test our theoretical knowledge of this basic process. We used correlated random walk (CRW) models and satellite telemetry to investigate long-distance movements of caribou, the most vagile, non-volant terrestrial vertebrate in the world. Individual paths of migratory and sedentary female caribou were quantified using measures of mean move length and angle, and net squared displacements at each successive move were compared to predictions from the models. Movements were modelled at two temporal scales. For paths recorded through one annual cycle, the CRW model overpredicted net displacement of caribou through time. For paths recorded over shorter intervals delineated by seasonal behavioural changes of caribou, there was excellent correspondence between model predictions and observations for most periods for both migratory and sedentary caribou. On the smallest temporal scale, a CRW model significantly overpredicted displacements of migratory caribou during 3 months following calving; this was also the case for sedentary caribou in late summer, and in late winter. In all cases of overprediction there was significant positive autocorrelation in turn direction, indicating that movements were more tortuous than expected. In one case of underprediction, significant negative autocorrelation of sequential turn direction was evident, indicating that migratory caribou moved in straightened paths during spring migration to calving grounds. Results are discussed in light of known migration patterns and possible limiting factors for caribou, and indicate the applicability of CRW models to animal movement at vast spatial and temporal scales, thus assisting in future development of more sophisticated models of population spread and redistribution for vertebrates. Received: 14 July 1999 / Accepted: 15 November 1999     

The consequences of spatial structure for the design and analysis of ecological field surveys

In ecological field surveys, observations are gathered at different spatial locations. The purpose may be to relate biological response variables (e.g., species abundances) to explanatory environmental variables (e.g., soil characteristics). In the absence of prior knowledge, ecologists have been taught to rely on systematic or random sampling designs. If there is prior knowledge about the spatial patterning of the explanatory variables, obtained from either previous surveys or a pilot study, can we use this information to optimize the sampling design in order to maximize our ability to detect the relationships between the response and explanatory variables?
The specific questions addressed in this paper are: a) What is the effect (type I error) of spatial autocorrelation on the statistical tests commonly used by ecologists to analyse field survey data? b) Can we eliminate, or at least minimize, the effect of spatial autocorrelation by the design of the survey? Are there designs that provide greater power for surveys, at least under certain circumstances? c) Can we eliminate or control for the effect of spatial autocorrelation during the analysis? To answer the last question, we compared regular regression analysis to a modified t‐test developed by Dutilleul for correlation coefficients in the presence of spatial autocorrelation.
Replicated surfaces (typically, 1000 of them) were simulated using different spatial parameters, and these surfaces were subjected to different sampling designs and methods of statistical analysis. The simulated surfaces may represent, for example, vegetation response to underlying environmental variation. This allowed us 1) to measure the frequency of type I error (the failure to reject the null hypothesis when in fact there is no effect of the environment on the response variable) and 2) to estimate the power of the different combinations of sampling designs and methods of statistical analysis (power is measured by the rate of rejection of the null hypothesis when an effect of the environment on the response variable has been created).
Our results indicate that: 1) Spatial autocorrelation in both the response and environmental variables affects the classical tests of significance of correlation or regression coefficients. Spatial autocorrelation in only one of the two variables does not affect the test of significance. 2) A broad‐scale spatial structure present in data has the same effect on the tests as spatial autocorrelation. When such a structure is present in one of the variables and autocorrelation is found in the other, or in both, the tests of significance have inflated rates of type I error. 3) Dutilleul's modified t‐test for the correlation coefficient, corrected for spatial autocorrelation, effectively corrects for spatial autocorrelation in the data. It also effectively corrects for the presence of deterministic structures, with or without spatial autocorrelation.
The presence of a broad‐scale deterministic structure may, in some cases, reduce the power of the modified t‐test.     

Incorporating spatial autocorrelation in rarefaction methods: Implications for ecologists and conservation biologists

Recently, methods for constructing Spatially Explicit Rarefaction (SER) curves have been introduced in the scientific literature to describe the relation between the recorded species richness and sampling effort and taking into account for the spatial autocorrelation in the data. Despite these methodological advances, the use of SERs has not become routine and ecologists continue to use rarefaction methods that are not spatially explicit. Using two study cases from Italian vegetation surveys, we demonstrate that classic rarefaction methods that do not account for spatial structure can produce inaccurate results. Furthermore, our goal in this paper is to demonstrate how SERs can overcome the problem of spatial autocorrelation in the analysis of plant or animal communities. Our analyses demonstrate that using a spatially-explicit method for constructing rarefaction curves can substantially alter estimates of relative species richness. For both analyzed data sets, we found that the rank ordering of standardized species richness estimates was reversed between the two methods. We strongly advise the use of Spatially Explicit Rarefaction methods when analyzing biodiversity: the inclusion of spatial autocorrelation into rarefaction analyses can substantially alter conclusions and change the way we might prioritize or manage nature reserves.     

Hypothesis testing in animal social networks

Behavioural ecologists are increasingly using social network analysis to describe the social organisation of animal populations and to test hypotheses. However, the statistical analysis of network data presents a number of challenges. In particular the non-independent nature of the data violates the assumptions of many common statistical approaches. In our opinion there is currently confusion and uncertainty amongst behavioural ecologists concerning the potential pitfalls when hypotheses testing using social network data. Here we review what we consider to be key considerations associated with the analysis of animal social networks and provide a practical guide to the use of null models based on randomisation to control for structure and non-independence in the data.     

Towards an integrated computational tool for spatial analysis in macroecology and biogeography

Because most macroecological and biodiversity data are spatially autocorrelated, special tools for describing spatial structures and dealing with hypothesis testing are usually required. Unfortunately, most of these methods have not been available in a single statistical package. Consequently, using these tools is still a challenge for most ecologists and biogeographers. In this paper, we present sam (Spatial Analysis in Macroecology), a new, easy-to-use, freeware package for spatial analysis in macroecology and biogeography. Through an intuitive, fully graphical interface, this package allows the user to describe spatial patterns in variables and provides an explicit spatial framework for standard techniques of regression and correlation. Moran's I autocorrelation coefficient can be calculated based on a range of matrices describing spatial relationships, for original variables as well as for residuals of regression models, which can also include filtering components (obtained by standard trend surface analysis or by principal coordinates of neighbour matrices). sam also offers tools for correcting the number of degrees of freedom when calculating the significance of correlation coefficients. Explicit spatial modelling using several forms of autoregression and generalized least-squares models are also available. We believe this new tool will provide researchers with the basic statistical tools to resolve autocorrelation problems and, simultaneously, to explore spatial components in macroecological and biogeographical data. Although the program was designed primarily for the applications in macroecology and biogeography, most of sam 's statistical tools will be useful for all kinds of surface pattern spatial analysis. The program is freely available at http://www.ecoevol.ufg.br/sam (permanent URL at http://purl.oclc.org/sam/ ).     

Elephants in space and time

Autocorrelation in animal movements can be both a serious nuisance to analysis and a source of valuable information about the scale and patterns of animal behavior, depending on the question and the techniques employed. In this paper we present an approach to analyzing the patterns of autocorrelation in animal movements that provides a detailed picture of seasonal variability in the scale and patterns of movement. We used a combination of moving window Mantel correlograms, surface correlation and crosscorrelation analysis to investigate the scales and patterns of autocorrelation in the movements of three herds of elephants in northern Botswana. Patterns of autocorrelation of elephant movements were long‐range, temporally complicated, seasonally variable, and closely linked with the onset of rainfall events. Specifically, for the three elephant herds monitored there was often significant autocorrelation among locations up to lags of 30 days or more. During many seasonal periods there was no indication of decreasing autocorrelation with increasing time between locations. Over the course of the year, herds showed highly variable and complex patterns of autocorrelation, ranging from random use of temporary home ranges, periodic use of focal areas, and directional migration. Even though the patterns of autocorrelation were variable in time and quite complex, there were highly significant correlations among the autocorrelation patterns of the different herds, indicating that they exhibited similar patterns of movement through the year. These major patterns of autocorrelation seem to be related to patterns of rainfall. The strength of correlation in movement patterns of the different herds decreased markedly at the cessation of major rain events. Also, there was a strong crosscorrelation between strength of autocorrelation of movement and rainfall, peaking at time lags of between three and four weeks. Overall, these approaches provide a powerful way to explore the scales and patterns of autocorrelation of animal movements, and to explicitly link those patterns to temporally variable environmental attributes, such as rainfall or vegetation phenology.     

A Method for Detecting Positive Growth Autocorrelation without Marking Individuals

In most ecological studies, within-group variation is a nuisance that obscures patterns of interest and reduces statistical power. However, patterns of within-group variability often contain information about ecological processes. In particular, such patterns can be used to detect positive growth autocorrelation (consistent variation in growth rates among individuals in a cohort across time), even in samples of unmarked individuals. Previous methods for detecting autocorrelated growth required data from marked individuals. We propose a method that requires only estimates of within-cohort variance through time, using maximum likelihood methods to obtain point estimates and confidence intervals of the correlation parameter. We test our method on simulated data sets and determine the loss in statistical power due to the inability to identify individuals. We show how to accommodate nonlinear growth trajectories and test the effects of size-dependent mortality on our method''s accuracy. The method can detect significant growth autocorrelation at moderate levels of autocorrelation with moderate-sized cohorts (for example, statistical power of 80% to detect growth autocorrelation ρ ² = 0.5 in a cohort of 100 individuals measured on 16 occasions). We present a case study of growth in the red-eyed tree frog. Better quantification of the processes driving size variation will help ecologists improve predictions of population dynamics. This work will help researchers to detect growth autocorrelation in cases where marking is logistically infeasible or causes unacceptable decreases in the fitness of marked individuals.     

Sampling Animal Movement Paths Causes Turn Autocorrelation

Animal movement models allow ecologists to study processes that operate over a wide range of scales. In order to study them, continuous movements of animals are translated into discrete data points, and then modelled as discrete models. This discretization can bias the representation of the movement path. This paper shows that discretizing correlated random movement paths creates a biased path by creating correlations between successive turning angles. The discretization also biases statistical tests for correlated random walks (CRW) and causes an overestimate in distances travelled; a correction is given for these biases. This effect suggests that there is a natural scale to CRWs, but that distance-discretized CRWs are in a sense, scale invariant. Perhaps a new null model for continuous movement paths is needed. Authors need to be aware of the biases caused by discretizing correlated random walks, and deal with them appropriately.     

Conditionally autoregressive models improve occupancy analyses of autocorrelated data: An example with environmental DNA

Site occupancy‐detection models (SODMs) are statistical models widely used for biodiversity surveys where imperfect detection of species occurs. For instance, SODMs are increasingly used to analyse environmental DNA (eDNA) data, taking into account the occurrence of both false‐positive and false‐negative errors. However, species occurrence data are often characterized by spatial and temporal autocorrelation, which might challenge the use of standard SODMs. Here we reviewed the literature of eDNA biodiversity surveys and found that most of studies do not take into account spatial or temporal autocorrelation. We then demonstrated how the analysis of data with spatial or temporal autocorrelation can be improved by using a conditionally autoregressive SODM, and show its application to environmental DNA data. We tested the autoregressive model on both simulated and real data sets, including chronosequences with different degrees of autocorrelation, and a spatial data set on a virtual landscape. Analyses of simulated data showed that autoregressive SODMs perform better than traditional SODMs in the estimation of key parameters such as true‐/false‐positive rates and show a better discrimination capacity (e.g., higher true skill statistics). The usefulness of autoregressive SODMs was particularly high in data sets with strong autocorrelation. When applied to real eDNA data sets (eDNA from lake sediment cores and freshwater), autoregressive SODM provided more precise estimation of true‐/false‐positive rates, resulting in more reasonable inference of occupancy states. Our results suggest that analyses of occurrence data, such as many applications of eDNA, can be largely improved by applying conditionally autoregressive specifications to SODMs.     

Incorporating the temporal autocorrelation of demographic rates into structured population models

Population dynamics are typically temporally autocorrelated: population sizes are positively or negatively correlated with past population sizes. Previous studies have found that positive temporal autocorrelation increases the risk of extinction due to ‘inertia’ that prolongs downward fluctuations in population size. However, temporal autocorrelation has not yet been analyzed at the level of life cycle transitions. We developed an R package, colorednoise, which creates stochastic matrix population projections with distinct temporal autocorrelation values for each matrix element. We used it to analyze long-term demographic data on 25 populations from the COMADRE and COMPADRE databases and simulate their stochastic dynamics. We found a broad range of temporal autocorrelation across species, populations and life cycle stages. The number of stage-classes in the matrix strongly affected the temporal autocorrelation of the growth rate. In the plant populations, reproduction transitions had more negative temporal autocorrelation than survival transitions, and matrices dominated by positive temporal autocorrelation had higher extinction risk, while in animal populations transition type was not associated with noise color. Our results indicate that temporal autocorrelation varies across life cycle transitions, even among populations of the same species. We present the colorednoise package for researchers to analyze the temporal autocorrelation of structured demographic rates.     

Multilevel modeling analysis of dyadic network data with an application to Ye'kwana food sharing

Behavioral ecologists have recently begun using multilevel modeling for the analysis of social behavior. We present a multilevel modeling formulation of the Social Relations Model that is well suited for the analysis of dyadic network data. This model, which we adapt for count data and small datasets, can be fitted using standard multilevel modeling software packages. We illustrate this model with an analysis of meal sharing among Ye'kwana horticulturalists in Venezuela. In this setting, meal sharing among households is predicted by an association index, which reflects the amount of time that members of the households are interacting. This result replicates recent findings that interhousehold food sharing is especially prevalent among households that interact and cooperate in multiple ways. We discuss opportunities for human behavioral ecologists to expand their focus to the multiple currencies and cooperative behaviors that characterize interpersonal relationships in preindustrial societies. We discuss possible extensions to this statistical modeling approach and applications to research by human behavioral ecologists and primatologists. Am J Phys Anthropol 157:507–512, 2015. © 2015 Wiley Periodicals, Inc.     

Temporal autocorrelation functions for movement rates from global positioning system radiotelemetry data

Autocorrelation has been viewed as a problem in telemetry studies because sequential observations are not independent in time or space, therefore violating assumptions for statistical inference. Yet nearly all ecological and behavioural data are autocorrelated in both space and time. We argue that there is much to learn about the structure of ecological and behavioural data from patterns of autocorrelation. Such patterns include periodicity in movement and patchiness in spatial data, which can be characterized by an autocorrelogram, semivariogram or spectrum. We illustrate the utility of temporal autocorrelation functions (ACFs) for analysing step-length data from GPS telemetry of wolves (Canis lupus), cougars (Puma concolor), grizzly bears (Ursus arctos) and elk (Cervus elaphus) in western Alberta. ACFs often differ by season, reflecting differences in foraging behaviour. In wilderness landscapes, step-length ACFs for predators decay slowly to apparently random patterns, but sometimes display strong daily rhythms in areas of human disturbance. In contrast, step lengths of elk are consistently periodic, reflecting crepuscular activity.     

Quantitative approaches in climate change ecology

Contemporary impacts of anthropogenic climate change on ecosystems are increasingly being recognized. Documenting the extent of these impacts requires quantitative tools for analyses of ecological observations to distinguish climate impacts in noisy data and to understand interactions between climate variability and other drivers of change. To assist the development of reliable statistical approaches, we review the marine climate change literature and provide suggestions for quantitative approaches in climate change ecology. We compiled 267 peer‐reviewed articles that examined relationships between climate change and marine ecological variables. Of the articles with time series data (n = 186), 75% used statistics to test for a dependency of ecological variables on climate variables. We identified several common weaknesses in statistical approaches, including marginalizing other important non‐climate drivers of change, ignoring temporal and spatial autocorrelation, averaging across spatial patterns and not reporting key metrics. We provide a list of issues that need to be addressed to make inferences more defensible, including the consideration of (i) data limitations and the comparability of data sets; (ii) alternative mechanisms for change; (iii) appropriate response variables; (iv) a suitable model for the process under study; (v) temporal autocorrelation; (vi) spatial autocorrelation and patterns; and (vii) the reporting of rates of change. While the focus of our review was marine studies, these suggestions are equally applicable to terrestrial studies. Consideration of these suggestions will help advance global knowledge of climate impacts and understanding of the processes driving ecological change.     

An updated perspective on the role of environmental autocorrelation in animal populations

Ecological theory predicts that the presence of temporal autocorrelation in environments can considerably affect population extinction risk. However, empirical estimates of autocorrelation values in animal populations have not decoupled intrinsic growth and density feedback processes from environmental autocorrelation. In this study, we first discuss how the autocorrelation present in environmental covariates can be reduced through nonlinear interactions or by interactions with multiple limiting resources. We then estimated the degree of environmental autocorrelation present in the Global Population Dynamics Database using a robust, model-based approach. Our empirical results indicate that time series of animal populations are affected by low levels of environmental autocorrelation, a result consistent with predictions from our theoretical models. Claims supporting the importance of autocorrelated environments have been largely based on indirect empirical measures and theoretical models seldom anchored in realistic assumptions. It is likely that a more nuanced understanding of the effects of autocorrelated environments is necessary to reconcile our conclusions with previous theory. We anticipate that our findings and other recent results will lead to improvements in understanding how to incorporate fluctuating environments into population risk assessments.     

Local environmental effects and spatial effects in macroecological studies using mapped abundance classes: the case of the rook Corvus frugilegus in Scotland

1. The study of the spatial pattern of species abundance is complicated by statistical problems, such as spatial autocorrelation of the abundance data, which lead to the confusion of environmental effects and dispersal. 2. Atlas-derived data for the rook in Scotland are used as a case study to propose an approach for assessing the likely contribution of dispersal and local environmental effects, based on a Bayesian Conditional Autoregressive (CAR) approach. 3. The availability of moist grasslands is a key factor explaining the spatial pattern of abundance. This is influenced by a combination of climatic and soil-related factors. A direct link to soil properties is for the first time reported for the wide-scale distribution of a bird species. In addition, for this species, dispersal seems to contribute significantly to the spatial pattern and produces a smoother than expected decline in abundance at the north-western edge of its distribution range. Areas where dispersal is most likely to be important are highlighted. 4. The approach described can help ecologists make more efficient use of atlas data for the investigation of the structure of species abundance, and can highlight potential sink areas at the landscape and regional scale. 5. Bayesian spatial models can deal with data autocorrelation in atlas-type data, while clearly communicating uncertainty through the estimation of the full posterior probability distribution of all parameters.     

The persistent spatial patchiness of limpet grazing

The characteristic variability of grazing has potential consequences for intertidal productivity and community structure, particularly as many of the underlying functional relationships are thought to be non linear. As a first approximation, it can be hypothesised that grazing is patchy over short time periods before a more uniform coverage is established over longer time scales. This prediction is supported by relatively short term observations previously made of limpet foraging. We used eight arrays of wax disks on each of four shores to test the hypothesis that grazing is patchy in the short term, but that this pattern is lost as the pattern of grazing averages out over longer time scales. Wax disk arrays were exposed for two weeks at a time for a period of six weeks in 2001 and in 2002 using the same set of disk holes each time. Grazing at the same disk location could therefore be measured over two weeks and over longer periods by averaging successive deployments. We used all three successive deployments to estimate the average grazing at each disk location over a six week period in 2001 and 2002. All six deployments were used to characterise the pattern of grazing at longer time scales. The spatial pattern of grazing in arrays was summarized using semivariogram analyses. For two-week deployments, the average standardized semivariance of grazing for disks separated by 20 cm was less than one. This pattern implies spatial autocorrelation of grazing at this scale. There was no support for the hypothesis that small scale patchiness in grazing would disappear over time. The average strength of spatial autocorrelation increased when data were integrated over longer periods. A preliminary analysis indicated that the degree of autocorrelation within arrays increased with grazing intensity at short time scales. Surface roughness disrupted autocorrelation of grazing over both short and long time scales. The persistent patchiness of grazing is likely to have implications for biofilm productivity, particularly on smoother shores.     

Study design and parameter estimability for spatial and temporal ecological models

The statistical tools available to ecologists are becoming increasingly sophisticated, allowing more complex, mechanistic models to be fit to ecological data. Such models have the potential to provide new insights into the processes underlying ecological patterns, but the inferences made are limited by the information in the data. Statistical nonestimability of model parameters due to insufficient information in the data is a problem too‐often ignored by ecologists employing complex models. Here, we show how a new statistical computing method called data cloning can be used to inform study design by assessing the estimability of parameters under different spatial and temporal scales of sampling. A case study of parasite transmission from farmed to wild salmon highlights that assessing the estimability of ecologically relevant parameters should be a key step when designing studies in which fitting complex mechanistic models is the end goal.     

Inference in MCMC step selection models

Habitat selection models are used in ecology to link the spatial distribution of animals to environmental covariates and identify preferred habitats. The most widely used models of this type, resource selection functions, aim to capture the steady-state distribution of space use of the animal, but they assume independence between the observed locations of an animal. This is unrealistic when location data display temporal autocorrelation. The alternative approach of step selection functions embed habitat selection in a model of animal movement, to account for the autocorrelation. However, inferences from step selection functions depend on the underlying movement model, and they do not readily predict steady-state space use. We suggest an analogy between parameter updates and target distributions in Markov chain Monte Carlo (MCMC) algorithms, and step selection and steady-state distributions in movement ecology, leading to a step selection model with an explicit steady-state distribution. In this framework, we explain how maximum likelihood estimation can be used for simultaneous inference about movement and habitat selection. We describe the local Gibbs sampler, a novel rejection-free MCMC scheme, use it as the basis of a flexible class of animal movement models, and derive its likelihood function for several important special cases. In a simulation study, we verify that maximum likelihood estimation can recover all model parameters. We illustrate the application of the method with data from a zebra.     

Methods to account for spatial autocorrelation in the analysis of species distributional data: a review

Species distributional or trait data based on range map (extent‐of‐occurrence) or atlas survey data often display spatial autocorrelation, i.e. locations close to each other exhibit more similar values than those further apart. If this pattern remains present in the residuals of a statistical model based on such data, one of the key assumptions of standard statistical analyses, that residuals are independent and identically distributed (i.i.d), is violated. The violation of the assumption of i.i.d. residuals may bias parameter estimates and can increase type I error rates (falsely rejecting the null hypothesis of no effect). While this is increasingly recognised by researchers analysing species distribution data, there is, to our knowledge, no comprehensive overview of the many available spatial statistical methods to take spatial autocorrelation into account in tests of statistical significance. Here, we describe six different statistical approaches to infer correlates of species’ distributions, for both presence/absence (binary response) and species abundance data (poisson or normally distributed response), while accounting for spatial autocorrelation in model residuals: autocovariate regression; spatial eigenvector mapping; generalised least squares; (conditional and simultaneous) autoregressive models and generalised estimating equations. A comprehensive comparison of the relative merits of these methods is beyond the scope of this paper. To demonstrate each method's implementation, however, we undertook preliminary tests based on simulated data. These preliminary tests verified that most of the spatial modeling techniques we examined showed good type I error control and precise parameter estimates, at least when confronted with simplistic simulated data containing spatial autocorrelation in the errors. However, we found that for presence/absence data the results and conclusions were very variable between the different methods. This is likely due to the low information content of binary maps. Also, in contrast with previous studies, we found that autocovariate methods consistently underestimated the effects of environmental controls of species distributions. Given their widespread use, in particular for the modelling of species presence/absence data (e.g. climate envelope models), we argue that this warrants further study and caution in their use. To aid other ecologists in making use of the methods described, code to implement them in freely available software is provided in an electronic appendix.     

Incorporating ecological principles into statistical models for the prediction of species’ distribution and abundance

Understanding the determinants of species’ distributions and abundances is a central theme in ecology. The development of statistical models to achieve this has a long history and the notion that the model should closely reflect underlying scientific understanding has encouraged ecologists to adopt complex statistical methods as they arise. In this paper we describe a Bayesian hierarchical model that reflects a conceptual ecological model of multi‐scaled environmental determinants of riverine fish species’ distributions and abundances. We illustrate this with distribution and abundance data of a small‐bodied fish species, the Empire gudgeon Hypseleotris galii, in the Mary and Albert Rivers, Queensland, Australia. Specifically, the model sought to address; 1) the extent that landscape‐scale abiotic variables can explain the species’ distribution compared to local‐scale variables, 2) how local‐scale abiotic variables can explain species’ abundances, and 3) how are these local‐scale relationships mediated by landscape‐scale variables. Overall, the model accounted for around 60% of variation in the distribution and abundance of H. galii. The findings show that the landscape‐scale variables explain much of the distribution of the species; however, there was considerable improvement in estimating the species’ distribution with the addition of local‐scale variables. There were many strong relationships between abundance and local‐scale abiotic variables; however, several of these relationships were mediated by some of the landscape‐scale variables. The extent of spatial autocorrelation in the data was relatively low compared to the distances among sampling reaches. Our findings exemplify that Bayesian statistical modelling provides a robust framework for statistical modelling that reflects our ecological understanding. This allows ecologists to address a range of ecological questions with a single unified probability model rather than a series of disconnected analyses.