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Keywords:

  • Bayesian hierarchical spatial models;
  • CAR model;
  • Matérn;
  • MCMC;
  • spatial autocorrelation;
  • species distribution model

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Integrated nested Laplace approximation
  5. Case study: Woodland caribou in the boreal forest of eastern Canada
  6. Results
  7. Discussion
  8. Acknowledgements
  9. References
  10. Supporting Information

1. Spatial analysis of ecological data is central to many interesting questions in ecology. Bayesian implementation of spatially explicit models has received increasing attention from ecologists as Monte Carlo Markov Chain (MCMC) methods have become freely accessible. MCMC simulations offer a flexible framework for modelling extensive ecological data, but they also come with a wide range of problems regarding convergence, processing time and implementation.

2. We introduce to ecologists an alternative procedure for fitting Bayesian hierarchical spatial models (BHSM) with quite general spatial covariance structures. This procedure uses integrated nested Laplace approximations (INLA) as an alternative to MCMC.

3. We show, using a case study of species distribution model with binary areal data, that implementing BHSM with INLA does not require advanced programming skills, yields accurate results compared with MCMC and is rapid (e.g. a few seconds with small to moderate data sets). BHSMs efficiently removed spatial autocorrelation in the residuals and fairly evaluated uncertainty in parameter estimates and predictions.

4. The rapidity of INLA significantly decreased the processing time and allowed both sensitivity analyses on priors and cross-validation tests to be performed within a reasonable amount of time, which ultimately increased model transparency.


Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Integrated nested Laplace approximation
  5. Case study: Woodland caribou in the boreal forest of eastern Canada
  6. Results
  7. Discussion
  8. Acknowledgements
  9. References
  10. Supporting Information

Spatial analysis of ecological data is central to most contemporary issues in applied ecology. Species distribution models provide key information regarding species–environment relationships, and particularly about how environmental stressors limit species distributions (Elith & Leathwick 2009). In the global context of increasing human impacts on ecosystems, predictions derived from such models can inform management plans for endangered species (Cabeza et al. 2004), and assessments of distributions under future land-use (Bomhard et al. 2005) and climate (Thuiller 2003) scenarios. Unfortunately, acquiring reliable inferences and predictions from statistical analysis of distributional data is not straightforward. One problem that must be addressed is spatial autocorrelation.

Analyses of species distributions are sensitive to spatial dependency in model residuals, or in other words, residual spatial autocorrelation (RSA; Latimer et al. 2006). RSA occurs when model residuals at nearby locations are not independent (Legendre 1993). RSA may arise from missing covariates that have or induce spatial structure, from incorrect specification of the functional relationship between a covariate and the response, or from neglecting to account for a spatially contagious process such as nonrandom dispersal of individuals. RSA is associated with biased Type-I error estimates owing to inflation of the effective sample size, which invalidates standard hypothesis tests (Legendre 1993). RSA may also reduce model performance (Latimer et al. 2006) and lead to underestimation of prediction errors (Gelfand et al. 2006). Hence, RSA should be avoided by correct model specification. In practice, though, important environmental covariates are often neglected because they have not been identified or are unmeasured.

Several statistical methods have emerged for modelling spatial data while accounting for RSA (e.g. see Dormann et al. 2007 for frequentist methods and Banerjee, Carlin & Gelfand 2003 for Bayesian methods). With freely available Monte Carlo Markov Chain (MCMC) methods, Bayesian implementation of spatially explicit models has received increasing attention from ecologists. The conditional autoregressive (CAR) model is one example of such models that is now routinely used in ecology for modelling spatial association in data sampled within areal units (Latimer et al. 2006), along roads (Thogmartin, Sauer & Knutson 2004) or transects (Aing et al. 2011) or at points (Haas et al. 2011). In the context of spatial regression, Beale et al. (2010) showed that Gaussian Bayesian CAR models fitted using MCMC yielded precise and unbiased parameter estimates with low Type-I and Type-II errors. While they are always possible to implement in principle, MCMC algorithms applied to complex hierarchical spatial models have a wide range of problems related to convergence and computation time: the fitting procedure is not guaranteed to converge or may converge very slowly. Moreover, implementation of the algorithms can prove problematic in itself, especially for users who are not expert in programming.

Integrated nested Laplace approximation (INLA) is a recent alternative to MCMC for fitting a large class of Bayesian models such as latent Gaussian models (Rue, Martino & Chopin 2009). Latent Gaussian models can account for hierarchical structure and non-Gaussian errors, as well as spatial and temporal autocorrelation (see INLA section). In fitting these models, INLA substitutes accurate, deterministic approximations to posterior marginal distributions in place of long MCMC simulations, thereby gaining in speed. The quality of such approximations is extremely high, as shown by comparisons with long MCMC runs (Rue, Martino & Chopin 2009). INLA has two main advantages over MCMC techniques. The first and most outstanding is computational: results can be obtained much faster than with a well-built MCMC-sampler. The INLA algorithm is naturally parallelized, thus making it possible to exploit the new computing trend of having multicore processors. The second advantage is that INLA permits a great deal of automation and, in practice, can be almost used as a black box to analyse latent Gaussian models. This second point is especially important in applied communities where programming expertise is limited. An R-INLA library with a user-friendly R interface (R Development Core Team 2011) is freely available at http://www.r-inla.org.

Although INLA is accurate, fast and freely available, it is still little known to ecologists. The first aim of this study is therefore to introduce this recent procedure for fitting spatially explicit hierarchical models to ecological data. To do so, we first compare results obtained with a Bayesian CAR model fitted with MCMC and INLA using spatial data on the distribution of woodland caribou in eastern Canada. Second, we present and compare the performance of an alternative hierarchical spatial model using a flexible approach with a Matérn correlation function (Minasny & McBratney 2005). Using the INLA-R library, we show how these models can be easily fitted, even by nonexpert programmers. We conclude with discussion of specific issues regarding spatial autocorrelation, the interpretation of latent spatial patterns, and the parameterization of spatial random effects in hierarchical models.

Integrated nested Laplace approximation

  1. Top of page
  2. Summary
  3. Introduction
  4. Integrated nested Laplace approximation
  5. Case study: Woodland caribou in the boreal forest of eastern Canada
  6. Results
  7. Discussion
  8. Acknowledgements
  9. References
  10. Supporting Information

In the following description, we assumed some familiarity with Bayesian analyses and the use of likelihood functions. For additional description of these notions, see Hilborn & Mangel (1997) or Bolker (2008).

Based upon Bayes’ theorem, Bayesian analyses combine prior probability distributions with likelihood to target posterior probability distributions of parameters. In this study, we consider a particular class of Bayesian hierarchical models called latent Gaussian models. In this class of models, a latent Gaussian field x is partially observed through data y. Depending on the structure and the type of model considered, the latent field x may include, simultaneously or not, parameters associated with linear predictors (e.g. linear regression models), spline functions (e.g. additive models) and nonspatial or/and spatial random effects (e.g. hierarchical spatial models). Moreover, the probability distribution of y can follow any distribution belonging to the exponential family such as Gaussian, Poisson, Binomial, Beta and many others. The prior density of the latent field π(x|θ) and the likelihood of the data π(y|xθ) are governed by a vector of hyperparameters θ with prior density π(θ). In practice, we are often interested to make inference about the posterior marginal distribution of parameters such as:

  • image

Getting marginal distribution of parameters, however, is almost always analytically intractable because the likelihood often is not Gaussian and the latent field is of high dimension. Stochastic sampling with MCMC methods (e.g. see Banerjee, Carlin & Gelfand 2003; Gelman & Hill 2007) can be used to solve this issue. In theory, MCMC algorithms give exact results. They also are flexible for a wide range of models. One practical limitation of MCMC for fitting spatial models, however, is its low computational speed. Processing time can be sufficiently long so as to preclude the analysis of even small spatial data sets. Moreover, computational demands can make impractical the use of sensitivity analyses on priors and cross-validation tests. This can lead to poor interpretation of the results.

As an alternative to MCMC, INLA substitutes stochastic sampling with deterministic approximations based on a clever use of the Laplace approximation and on numerical integration. INLA can be used when the latent field x is a Markov field, which means that the latent field is endowed with a conditional independent structure, for a detailed description of Gaussian Markov field see Rue & Held (2005). In addition, INLA is most useful when the main interest lies in posterior marginal distributions inline image and inline image. This is the case when, for example, assessing the effect of covariates is the goal. We give here a short description of the INLA procedure and refer the interested reader to Rue, Martino & Chopin (2009). The INLA procedure consists of three successive steps. First, an approximation of the marginal posteriors of hyperparameters inline image is computed using Laplace approximation. The main use of this approximation is to select good estimation points to integrate out the uncertainty with respect to θ when approximating the posterior marginals of each parameter xi. A key feature here is to avoid representing inline image parametrically, which allows more flexibility together with reducing computational demands. The second step consists of approximating the posterior marginal for the xi conditioned on selected values of θ, inline image, using again Laplace approximations. Finally, an approximation to inline image is computed via numerical integration as:

  • image(eqn 1)

Where the points θj are selected in the first step of the INLA algorithm and Δj are associated weights. When the number of hyperparameters is small (e.g. <6), as is the case in most ecological applications, the computation of these approximations is very fast and accurate, as shown by long-run comparisons with MCMC (see Rue, Martino & Chopin 2009). These computational advantages and the fact that there is a user-friendly software make INLA an appealing alternative for latent Gaussian models, including generalized linear (or additive) mixed models, time-series models, geoadditive models and state-space models, all of which are currently applied more or less commonly to the analysis of ecological data.

Case study: Woodland caribou in the boreal forest of eastern Canada

  1. Top of page
  2. Summary
  3. Introduction
  4. Integrated nested Laplace approximation
  5. Case study: Woodland caribou in the boreal forest of eastern Canada
  6. Results
  7. Discussion
  8. Acknowledgements
  9. References
  10. Supporting Information

We illustrate the use of INLA in the context of hierarchical spatial modelling with spatially autocorrelated distributional data. First, we compared the performance of INLA with MCMC using an intrinsic spatial CAR model. For the MCMC method, we used a Gibbs sampler that was implemented in WinBUGS (available online at http://www.mrc-bsu.cam.ac.uk/bugs/winbugs/contents.shtml). Second, we show an alternative hierarchical spatial method, which uses a Matérn correlation function to account for spatial associations among units. Matérn spatial correlation models are not supported in WinBugs, so we only present these result using INLA. Overall, the purpose of these comparisons is to show that powerful alternatives to MCMC exist for fitting a wide range of spatially explicit hierarchical models in ecology and evolution. All models using INLA were fitted in R v2.14.1 using the R-INLA library. Data and R-codes are available in Appendix 1.

Data

Our case study consists of species distribution data obtained in 1999 from an extensive aerial winter survey of forest-dwelling caribou (Rangifer tarandus caribou) in a region of 42 071 km2 located in the boreal forest of eastern Québec, Canada (Courtois et al. 2003). This region and data are a subset of those studied by Fortin et al. (2008). The observational units for this study are the presences or absences of intensive caribou snow track networks (ICTN). An ICTN indicates areas that were used briefly and intensively by a small group of caribou for foraging or shelter. A total of 232 ICTNs were detected and mapped by an exhaustive winter aerial survey consisting of a dense grid of fixed-wing transects backed up by helicopter spot-checks (see Courtois et al. 2003 for details). The mean size of an ICTN is 0·53 km2 (SE, ±0·06 km2). ICTN locations were taken to be their centroids. The probability of detection given presence is high (>90%), owing to strong contrasts between ICTN and adjacent undisturbed snow surfaces (Courtois et al. 2003). Given the high detectability of ICTN and the intensity of sampling effort, false-positives and false-negatives can be ignored. We initially selected 13 covariates that could potentially influence the probability of ICTN presence (Table 1). These covariates have been obtained from several data sources, including digital forest inventories, interpolated climate data (McKenney et al. 2006) and a 0·75 arc-second resolution Digital Elevation Model (available online at http://www.geobase.ca/geobase/fr/data/cded/description.html). The spatial resolution of covariates was ∼500 m2 for elevation, 8 ha for forest inventory attributes and ∼100 km2 for climate data. All attributes were calculated as total areas or averages over a common 100 km2 grid defined by the climate data (see Table 1). A linear model of coregionalization revealed that a grain size of 100 km2 was appropriate to delineate homogeneous spatial units within the study region. The chosen spatial grid and grain size also corresponds to that of our interpolated climate data, the covariate of lowest spatial resolution. Hereafter, each elementary unit of this grid is referred to as a cell (N = 465). Caribou presence/absence data within cells, however, was calculated for a 1-km2 resolution subgrid. This resolution corresponds to roughly twice the mean area of an ICTN (see Fig. 1).

Table 1.   Environmental variables used as predictors in our study. Minimum, mean and maximum values of each explanatory variable, at the scale of 100 km2 grid cells, are presented for the entire region
Environmental variablesMinimumMeanMaximum
  1. Variables in bold type were retained in the final candidate model for each statistical method.

  2. * or WoodLichen in Fig. 3.

  3. † or OpenLichen in Fig. 3.

Wetlands (km2)0·12·823·1
Water bodies (km2)1·914·087·8
Lichen woodland (km2)*03·627·8
Bare land dominated with lichen (km2)†03·532·2
Coniferous Forests (km2)054·191·5
Deciduous Forests (km2)03·544·7
Wildfire (km2)09·882·2
Logging (km2)019·983·1
Other land cover (km2)03·143·4
Road density (km/km2)00·55·1
Winter mean temperature (°C)−18·91−16·5−12·67
Winter total precipitation (mm)476073
Elevation (m)239500879
image

Figure 1.  Study region located in the boreal forest of eastern Canada. Bold black lines in the middle-panel delineate boundaries of the aerial survey. Colours define the proportion of intensive winter track networks of woodland caribou (ICTN) in each of 465 cells of 100-km2. The right-side panel highlights two spatial unit structures. The 100-km2 grid represents spatial units at which environmental variables are evaluated, and the 1-km2 grid represents spatial units where the presence (black squares, with grey ICTN centroids inside) or absence (white squares) of ICTN was detected. See text for further description.

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Statistical models

Let ni be the number of 1-km2 subcells within the ith cell, and Yi the number of these subcells where the presence of ICTN is observed. We assume Yi to be a binomial random variable and have modelled the probability pi of ICTN presence within cells using a logit link function and the following generic model:

  • image(eqn 2)

Here, Xi is the vector of covariates for cell i, and β is the vector of parameters to be estimated. The two hierarchical spatial models considered in this study only differ in their intrinsic way of defining the spatial random effect f(s). A spatial random effect is used to model dependence among neighbouring cells that is not explained by the covariates.

All models assumed a vague Gaussian prior for the regression parameters inline image (mean = 0, precision = 0·001), where precision = 1/variance. For the binomial intrinsic CAR model fitted with INLA, the prior for the spatial random effect is defined conditionally as:

  • image

where ss’ indicates that the two cells s and s’ are neighbours and ns is the number of neighbours of cell s. We defined two cells to be neighbours if they directly share a single boundary point. The unknown precision hyperparameter inline image controls the smoothness of the spatial random effect and the prior is defined on a logarithmic scale such as it follows a logGamma (shape = 24·47, scale = 0·001) distribution (see the following section on prior choices). Further details of this model can be found in chapter 3 in Rue & Held (2005).

The spatial random effect in the Matérn model is a Markov representation on a regular grid of a continuous Gaussian field with a Matérn covariance function. The Matérn function offers a flexible way of modelling spatial dependence among units as it includes the exponential and squared exponential covariance functions as special cases (see section 2.1.3. in Banerjee, Carlin & Gelfand 2003 or Minasny & McBratney 2005 for details).

The correlation function in the Matérn model is defined as:

  • image

Where d is the Euclidean distance, Kv() is the Bessel function of order v. The range is defined to be inline image and it is the distance at which two cells are practically uncorrelated. The Matérn model has two hyperparameters: the precision of the spatial random effect and the range as defined above. As it is a difficult task to estimate simultaneously both range and precision parameters, we fixed the range to be five times the distance between two neighbouring cells. This value for the range was meant to capture the fine scale spatial association among cells. As for the CAR model, the precision hyperparameter is defined internally on a logarithmic scale and we assigned it a logGamma (shape = 23·36, scale = 0·001) prior. The Matern covariance function implies a dense covariance matrix that greatly increases computational demands and processing time. The INLA software uses a Markov representation of the Matern field that has been introduced by Lindgren, Lindstrøm & Rue (2011). The Markov representation offers several computational advantages and greatly reduces the running time to fit such model. The description of such Markov representation is beyond the scope of this study and the interested reader is referred to Lindgren, Lindstrøm & Rue (2011).

Finally, to ensure parameter identifiability, a sum-to-zero constraint is imposed on f(s) in all cases.

Variable selection and prior choices

Logistic, CAR and Matérn models were fit using the same design matrix Xi of environmental covariates, which were selected from the initial set of 13 covariates by a preliminary screening process. Selection was carried out by fitting a full logistic model without spatial random effect and then calculating variance inflation factors (VIF) as a test for multicollinearity (Zuur et al. 2009). Covariates with VIF > 4 were removed one-by-one using a backwards, stepwise procedure. We then fit every model with the same eight covariates using linear effects on the logit scale (Table 1). For the CAR and Matérn models, the choice of the shape parameter for the precision of the spatially structured effect f(s) determines the smoothness of the spatial effect and, through this, the spatial scale at which it operates. Misspecification of this parameter can lead to poor interpretation of the results (e.g. overfitting), so special care must be taken. To avoid pitfalls, we manually investigated the sensitivity of the results for a wide range of shape parameter values. In particular, we investigated the effect of various shape parameter values on coefficient estimates and credible intervals of explanatory variables, deviance information criterion (DIC), the number of effective parameters and the number of effective replicates. As model comparisons that are based on DIC can underpenalize for model complexity and thus encourage overfitting (Plummer 2008), we retained values of the shape parameters such that the ratio of data/parameters was always >20 (Burnham & Anderson, 2002).

Model outputs

For each fitted model, we calculated the mean predicted value of ICTN presence in each cell (with 95% credible intervals) and estimated RSA using variograms. We compared inferential properties among models using parameter estimates and their credible limits. We assessed model goodness-of-fit by calculating the Pearson’s correlation coefficient (r) between predicted values and observed proportions of ICTN presence at the cell level. Finally, we used ‘leave-one-out’ cross-validation to assess the predictive power of each model, based on the conditional predictive ordinate (CPO) statistic (Held, Schrödle & Rue 2010). Following Held, Schrödle & Rue (2010), the CPO value for the ith cell is defined as CPOi = π(yi,obs|y-(i),obs), where yi,obs is the binomial outcome of the ICTN presence for cell i, and y-(i),obs denotes the data without the ith cell. The number of cells is 465. We used the mean logarithmic-CPOi defined as:

  • image(eqn 3)

The lower the LCPO for a model, the better is its predictive power.

Results

  1. Top of page
  2. Summary
  3. Introduction
  4. Integrated nested Laplace approximation
  5. Case study: Woodland caribou in the boreal forest of eastern Canada
  6. Results
  7. Discussion
  8. Acknowledgements
  9. References
  10. Supporting Information

Conditional autoregressive and Matérn models adequately removed RSA present in the nonspatial logistic model (Fig. 2). We detect no difference in parameter estimates for the Bayesian CAR model using MCMC vs. INLA approaches (Fig. 3). Nor did we find any significant difference in parameter estimates between Matérn and CAR models fitted with INLA (Fig. 3). When fitted with MCMC (100 000 iterations, three chains), the CAR model took several hours to converge, whereas INLA took c. 5 s. Estimation of the Matérn model with INLA was also a matter of seconds.

image

Figure 2.  Spatial variograms of (i) the raw data; (ii) residuals of the nonspatial logistic model; (iii) residuals of the hierarchical conditional autoregressive (CAR) model; and (iv) residuals of the spatial model with a Matérn correlation function. Note: A likelihood-ratio (LR) test between the following two models shows that residuals of logistic model are spatially autocorrelated (LR = 10·1, = 0·007): model 1: Semi-variance = β0. model 2: Semi-variance = β0 + β1 distance + β2 distance2.

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image

Figure 3.  Parameter estimates (±95% credible intervals) of every environmental variable used for each statistical method in this study. See Table 1 for a description of each variable.

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As expected, results of hierarchical spatial model outcomes are sensitive to specification of the shape parameter (Fig. 4, Appendix 2). The shape parameter governs the smoothness of the spatial random effect: low values (e.g. uninformative priors) mirror spatial associations at short distances, whereas high values mirror spatial correlation at large distances (Fig. 4). Our result shows that accounting for autocorrelation at short distances comes with the cost of increasing number of estimated parameters, which in turn increases the risk of overfitting (Fig. 4). We found that DIC-based comparisons underpenalized for model complexity and encouraged overfitting (Fig. 4). To avoid this issue and to balance the amount of information contained in the data with the number of estimated parameters, we selected a shape parameter so that the number of effective replicates was greater than, but close to, 20 (Fig. 4). Lower values of effective replicates encouraged overfitting, while higher values only mirrored spatial trend at broad scales (Fig. 4). When values of the shape parameter are ≥24·5, the binomial CAR model no longer differs from a nonspatial logistic model.

image

Figure 4. Left panel: sensitivity analyses for Bayesian conditional autoregressive (CAR) model showing the effect of different values of the shape parameter on (a) deviance information criteria (DIC) and the number of effective parameters (pD), and (b) the number of effective replicates. Letters C, D and E correspond to shape parameters of 24·52, 24·47 and 22·50, respectively. Right panel: posterior mean of the spatial random effect of Bayesian CAR model for each shape parameter defined above.

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The occurrence probability of track networks of woodland caribou increased with the proportion of lichen woodland and water bodies in a cell, but decreased with the portion of deciduous stands and disturbances caused by fire and logging (Fig. 3). The effect of logging was consistently more negative than that of burning. No significant selection pattern was found for bare lands dominated by terrestrial lichens or for wetlands (Fig. 3). The occurrence probability of track networks tended to increase with elevation, but elevation also showed a greater sensitivity to specification of the shape parameter (Appendix 2).

The distribution of spatial random effects among Bayesian models revealed strong spatial patterns at multiple scales (Figs 4 and 5). At coarse scales, favourable winter areas for woodland caribou increased from northeast to west. At intermediate scales, two suitable subregions emerged in the south and northwest parts of the study area. Within these subregions, finer-scale local variation was evident. Although CAR and Matérn models roughly had the same number of effective parameters, the Matérn model allowed describing a latent spatial structure at finer scale, likely because of the flexibility of the Matérn correlation function and because data are on a regular lattice.

image

Figure 5.  Posterior mean of the spatial random effect for hierarchical spatial model with Matérn correlation function.

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Overall, CAR and Matérn models improved predictive power by 10% and the percentage of explained deviance by 14% compared with nonspatial logistic model (Fig. 6). Uncertainty of the estimates, however, is 5–20% larger for the Bayesian than for the nonspatial logistic model (Fig. 3). As expected, models showing higher uncertainty in parameter estimates had higher prediction errors (Fig. 6). In no case, was a parameter estimate significant in a spatial and nonspatial model but of differing sign (Fig. 3).

image

Figure 6.  (a) Mean predicted probability of occurrence of woodland caribou track networks (values have been rescaled to range from 0 to 1); and (b) Standard deviation of the predicted probability of occurrence for the statistical methods tested in this study. Below panel (a), a summary table shows for each statistical method: (1) Deviance information criteria (DIC); (2) the effective number of parameters (pD); (3) the correlation coefficient between observed and predicted values (r); (4) the log-score statistic (LCPO) measuring the predictive power of each model through cross-validation (the lower, the better; see text for further description).

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Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Integrated nested Laplace approximation
  5. Case study: Woodland caribou in the boreal forest of eastern Canada
  6. Results
  7. Discussion
  8. Acknowledgements
  9. References
  10. Supporting Information

Making reliable inferences and robust predictions from the analysis of spatial data is central to many pressing issues in ecology and evolution. Hierarchical spatial models are now routinely used to analyse spatially autocorrelated data as they offer convenient properties such as a flexibility regarding the probability distribution of the response variable, together with their ability to evaluate simultaneously the contributions of fixed predictors and spatial random effects to the likelihood. This last property ensures that variability is properly attached to predictions (Gelfand et al. 2006) and, if needed, allows predictions to be made at unsampled locations, thereby accommodating gaps in sampling and irregular sampling intensity. In this study, we evaluated a relatively new statistical methodology for estimating hierarchical spatial models, which uses the INLA (Rue, Martino & Chopin 2009) as an alternative to MCMC simulations. With this novel numerical inference approach, MCMC sampling becomes redundant as the posterior marginal distributions are accurately approximated in a fully automated fashion (Held, Schrödle & Rue 2010).

Several remarks regarding INLA can be drawn from our study. First, the results of CAR model that were obtained with INLA were so accurate that no tangible difference in model outcomes could be detected with MCMC. This confirms previous comparative results obtained for a wide range of Gaussian latent models (Rue, Martino & Chopin 2009). Second, the processing time for fitting hierarchical spatial models was very rapid, which ultimately allowed us to perform cross-validation tests and sensitivity analyses on priors within a reasonable amount of time. Such a gain in processing time with INLA opens new perspectives in modelling spatial data under the Bayesian framework, as it alleviates one of the most important bottlenecks associated with MCMC. Third, the use of INLA with the R interface greatly facilitates implementation of Bayesian hierarchical spatial models. A strong expertise in programming is thus no longer an obstacle to fitting these models. Fourth, an increasing variety of hierarchical spatial models are now available for end-users (e.g. spatial GLMM, geographic weighted regression, thin-plate splines, among others), which ultimately broadens the range of tools available to ecologists. These models, together with the models presented in this study, allow a wider range of questions to be addressed in ecology and evolution. As with any statistical procedure, INLA also has limitations: precision of approximations becomes less accurate as the number of hyperparameters to be estimated increases above 6; occupancy models containing an observational process to cope with imperfect detection (MacKenzie et al. 2002) are not yet implemented in INLA and will require further developments. Site-occupancy models developed for spatially correlated observations in an active area of research (Aing et al. 2011); although INLA greatly reduces the processing time relative to MCMC on small to moderate-sized data sets, the spatial modelling of large data sets (N > 106) remains challenging.

In hierarchical spatial models, the shape parameter of the gamma distribution that was used as a prior for the precision of the spatial model controls the smoothness of the spatial random effect. The degree of smoothness can be viewed as the spatial equivalent of the pooling factor ω (Gelman & Hill 2007), which represents the degree to which the estimates are pooled together rather than estimated separately for each group factor. Our case study emphasized that parameterization of the shape parameter requires special care. The choice of an uninformative prior favours a low degree of smoothing, which in turn can yield a model that is overfitted to the given observations (see Fig. 4a). As described by Plummer (2008), this situation typically occurs when the choice of shape parameter is solely based on DIC comparisons, as DIC can underpenalize model complexity in this class of models. In contrast, highly informative priors increase the degree of spatial smoothing to such a point that the full model becomes equivalent to a nonspatial one (see Fig. 4a, b). Up to now, there is no Bayesian equivalent to AICc (see Burnham & Anderson 2002). Hence, we recommend that, based on information theory, the value of the shape parameter be defined so that the ratio between sample size and the number of effective parameters be >20. This stopping rule maintains a balance between information contained in the data and the number of effective parameters (Rue, Martino & Chopin 2009). Sensitivity analyses allow such thresholds to be identified and should be an integral part of any modelling exercise with hierarchical models. Additional work is needed to define Bayesian comparative criteria sensitive to small numbers of equivalent replicates.

Our results on landscape selection patterns largely support previous findings that show woodland caribou avoid regeneration areas following recent disturbance (Vors et al. 2007; Courbin et al. 2009) and deciduous stands (Courbin et al. 2009), which likely mimics a behavioural response to increasing predation risk (Wittmer et al. 2007). For example, deciduous stands and regenerating areas are used by moose (Alces alces L.), and increasing moose abundance often translates into higher risk of predation by wolves (Seip 1992; Wittmer et al. 2007). The positive selection of water bodies presumably reflects a similar strategy as, on one hand, these large ice-covered open areas improve the ability to detect predators and, on the other hand, surrounding forests enhance the ability to escape from predators (Mysterud & Ostbye 1999). We found a strong positive selection for lichen woodland that confirms the importance of this habitat in providing arboreal and terrestrial lichens, two major components of woodland caribou winter diet (Johnson, Parker & Heard 2001). Our results also show that, in addition to removing RSA, the inclusion of a spatial random effect yields strong latent spatial patterns at multiple scales that were not explained by environmental covariates. We argue that these latent spatial patterns offer opportunities for researchers to investigate and make further ecological hypotheses about the underlying processes that generated these patterns. For instance, the scale and distribution of spatial random effects could be used as a surrogate in inferring processes from spatial patterns (McIntire & Fajardo 2009). These effects might also be indicative of important missing covariates and further serve sampling strategies by prioritizing locations where data acquisition seems to be the most urgent.

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Integrated nested Laplace approximation
  5. Case study: Woodland caribou in the boreal forest of eastern Canada
  6. Results
  7. Discussion
  8. Acknowledgements
  9. References
  10. Supporting Information

We are grateful to Daniel Fortin, Réhaume Courtois, Claude Dussault and the Ministère des Ressources naturelles et de la Faune du Québec for providing access to the caribou data, and the Géo/Stat Centre at Laval University for the forest inventory data. We also thank D. McKenney and P. Papadopol of the Canadian Forest Service, Natural Resources Canada, for providing climate data over the study region. Funding for this research was provided by an NSERC (Natural Sciences and Engineering Research Council of Canada) scholarship to J.B. We sincerely thank P. Racine for his contributions in manipulating GIS data, E.J.B. McIntire and G. Daigle for stimulating discussions on spatial analyses, W.F.J. Parsons for English revisions, and the associate editor together with three anonymous reviewers for their constructive comments.

References

  1. Top of page
  2. Summary
  3. Introduction
  4. Integrated nested Laplace approximation
  5. Case study: Woodland caribou in the boreal forest of eastern Canada
  6. Results
  7. Discussion
  8. Acknowledgements
  9. References
  10. Supporting Information
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Supporting Information

  1. Top of page
  2. Summary
  3. Introduction
  4. Integrated nested Laplace approximation
  5. Case study: Woodland caribou in the boreal forest of eastern Canada
  6. Results
  7. Discussion
  8. Acknowledgements
  9. References
  10. Supporting Information

Appendix S1. R-code for running with INLA all hierarchical models presented in the paper.

Appendix S2. Effect of shape parameter on coefficients of regression (? 95% CI).

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