carroll et al. 1 1 carlos carroll 6 hierarchical bayesian ... · 6 hierarchical bayesian spatial...
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Carroll et al. 1
Carlos Carroll 1
PO Box 104 2
Orleans, CA 95556-0104 3
530-627-3512 5
HIERARCHICAL BAYESIAN SPATIAL MODELS FOR MULTI-SPECIES 6
CONSERVATION PLANNING AND MONITORING 7
RH : HIERARCHICAL SPATIAL HABITAT MODELS 8
6254 words, abstract through literature cited 9
CARLOS CARROLL1, DEVIN S. JOHNSON2, JEFFREY R. DUNK3,4, AND WILLIAM J. 10
ZIELINSKI4. 11
1 Klamath Center for Conservation Research, Orleans, CA 95556 USA 12
2 National Marine Mammal Laboratory, Alaska Fisheries Science Center, NOAA, 7600 Sand 13
Point Way N.E., F/AKC3, Seattle, WA 98115, USA 14
3 Department of Environmental and Natural Resource Sciences, Humboldt State University, 15
Arcata, California 95521, USA 16
4 Redwood Sciences Laboratory, Pacific Southwest Research Station, USDA Forest Service, 17
1700 Bayview Drive, Arcata, California 95521, USA 18
19
Keywords: focal species, hierarchical Bayesian model, Martes pennanti, Northwest Forest Plan, 20
spatial autoregressive model, spatial dependence, species distribution model 21
22
Carroll et al. 2
ABSTRACT 1
Biologists and managers who develop and apply habitat models are often familiar with 2
the challenges posed by their data's spatial structure but unsure whether use of complex spatial 3
models will increase the utility of model results in planning. We compared the relative 4
performance of non-spatial and hierarchical Bayesian spatial models across three vertebrate and 5
invertebrate taxa of conservation concern (Church’s sideband snail, Monadenia churchi, the red 6
tree vole, Arborimus longicaudus, and the Pacific fisher, Martes pennanti pacifica) which 7
provide examples of a range of distribution extents and dispersal abilities. We used presence-8
absence data sets derived from regional monitoring programs to develop models based on both 9
landscape-scale and plot or site-scale environmental covariates. We fit spatial models using 10
Markov Chain Monte Carlo algorithms and a conditional autoregressive or intrinsic conditional 11
autoregressive model framework. Bayesian spatial models improved model fit between 35 and 12
55% over their non-spatial analogue models and also outperformed analogous models developed 13
using maximum entropy (Maxent) methods. Although best spatial and non-spatial models 14
included similar environmental variables, spatial models provided estimates of residual spatial 15
effects that facilitated development of hypotheses concerning the ecological processes 16
structuring distribution patterns. Spatial models using presence-absence data improved fit most 17
for localized endemics with ranges constrained by poorly-known biogeographic factors and for 18
widely-distributed species where strong effects of unmeasured environmental variables or 19
population processes were suspected. By treating spatial effects as a variable of interest rather 20
than a nuisance, hierarchical Bayesian spatial models, especially when based on a common 21
broad-scale spatial lattice (here the national Forest Inventory and Analysis grid of 24 km2 22
hexagons), can increase the relevance of habitat models to multi-species conservation planning. 23
Carroll et al. 3
INTRODUCTION 1
Conservation planning and wildlife management increasingly draw on models of habitat 2
relationships derived from regional surveys of species occurrence in order to further 3
understanding of a species' ecology and the factors limiting its distribution. Information from 4
such models can also facilitate protection and enhancement of habitat, predict distribution in 5
unsurveyed areas, and help evaluate suitability of currently unoccupied areas for reintroduction. 6
However, commonly-used statistical techniques such as logistic regression may be poorly suited 7
for developing broad-scale distribution models due to their inadequate treatment of 8
autocorrelation and other spatial aspects of the data (Dormann et al. 2007). 9
Spatial autocorrelation is a pervasive characteristic of species distribution data sets, due 10
to spatial autocorrelation of environmental factors and biological processes that result in 11
aggregated distributions of individuals (Clark 2007). Spatial structure linked to environmental 12
factors has been termed exogenous, induced, or extrinsic spatial effects in contrast to 13
endogenous, inherent, or intrinsic spatial effects arising from population processes such as 14
dispersal and territoriality (Lichstein et al. 2002, Wintle and Bardos 2006). Spatially-15
autocorrelated data violate the assumption of independence in standard statistical tests, 16
potentially leading to inclusion of variables with spuriously-significant parameters (Dormann et 17
al. 2007). This reduces the interpretability of model structure and parameters and limits 18
prediction accuracy when models are extrapolated to new regions or novel conditions such as 19
future climates. Rather than pursue spatial independence in survey data, e.g., through spatially 20
dispersed measurements, it may be more effective to use alternative statistical methods that can 21
partition the spatial component from the hypothesized environmental effects, thus increasing the 22
generality of the habitat relationships described in the model and providing insights concerning 23
Carroll et al. 4
the biological processes creating spatial dependence and the scales at which they operate (Bolker 1
et al. 2009). 2
Wildlife distribution models are commonly based on a form of generalized linear model 3
(GLM)(Bolker et al. 2009). For example, logistic regression models a binary response (e.g., 4
presence-absence data for the species analyzed here) by means of a generalized linear model 5
using the logit link. Diverse approaches have been used to extend this approach to account for 6
spatial autocorrelation. A first group of methods incorporates spatial effects as additional 7
covariates within the model (Beale et al. 2010). For example, trend surface variables may be 8
derived from geographic coordinates to model broad-scale trends not explained by environment 9
(Haining 2003). “Simple autoregressive” methods estimate spatial effect through an additional 10
covariate based on smoothed observed occurrence values at neighboring sites within a spatial 11
neighborhood (Davis et al. 2007; Beale et al. 2010). Because such models first estimate the 12
spatial effect and then the environmental effects, spatial structure in the response data tends to be 13
attributed preferentially to the spatial autocovariate, thus potentially under-estimating the 14
strength of environmental variables (Dormann 2007a). In a recent review, Beale et al. (2010) 15
found that methods that model space as an additional covariate generally underperform those that 16
model space in the error or random effect term. 17
Hierarchical Bayesian autoregressive models take this latter approach by simultaneously 18
estimate environmental variables and a spatial error term, allowing the data themselves speak to 19
the best placement of “spatial effects”. The simultaneous estimation of environmental and spatial 20
effects also allows better estimation in areas with missing response data than is possible with 21
simple autoregressive methods (Gelman et al. 2004; Latimer et al. 2006). Hierarchical spatial 22
autoregressive models, although usually intractable with standard statistical software, can be fit 23
Carroll et al. 5
with Markov Chain Monte Carlo (MCMC) techniques (Spiegelhalter et al. 2003). While non-1
Bayesian methods exist for the methods considered here, these approaches are not yet widely 2
tested (Bolker et al. 2009). 3
Several previous reviews compare Bayesian spatial and non-spatial models using 4
artificial data sets (Wintle and Bardos 2006, Dormann et al. 2007, Beale et al. 2010) or plant 5
survey data (Latimer et al. 2006, 2009). In this study, we 6
1) further develop and apply such techniques to a set of multi-species wildlife survey data typical 7
of regional planning efforts, 8
2) evaluate whether the use of spatial models alters model selection and prediction, 9
3) explore the contrasts in model structure associated with use of data from regional monitoring 10
programs with systematic or non-systematic sampling design, and with use of landscape-scale 11
covariates from geographic information systems (GIS) data and site-scale covariates from 12
sparsely distributed plots, and 13
4) develop and contrast examples from vertebrate and invertebrate taxa with a range of dispersal 14
ability and territory size in order to assess how the utility of spatial models varies across scales. 15
METHODS 16
SURVEY DATA 17
We created and compared models for an invertebrate and two vertebrate species of 18
conservation concern (Carroll et al. 1999; Dunk et al. 2004; Dunk and Hawley 2009). Area-19
limited vertebrate habitat specialists, such as the fisher (Martes pennanti pacifica) analyzed here, 20
are a common focus of regional conservation plans and monitoring programs that collect 21
extensive data on their occurrence (Carroll et al. 1999; Davis et al. 2007). Less commonly, 22
monitoring effort is focused on a wider range of localized or poorly-known taxa (Molina et al. 23
Carroll et al. 6
2006). We consider two such species here for which survey data was collected under the 1
Northwest Forest Plan's Survey and Manage program (Molina et al. 2006): 1) an endemic 2
mollusk species (Church's sideband snail (Monadenia churchi)), and 2) the red tree vole 3
(Arborimus longicaudus), a mammal with broader geographic range but similarly limited 4
territory size and dispersal capability (Dunk et al. 2004; Dunk and Hawley 2009). The three 5
species span several orders of magnitude of maximum dispersal distance (< 100 m for M. 6
churchi to >30 km for M. p. pacifica) and home range size (Dunk et al. 2004; Carroll et al. 7
1999). 8
Survey data for the three species were obtained from systematic presence-absence 9
surveys whose sampling locations were based on the national Forest Inventory and Analysis 10
(FIA; http://fia.fs.fed.us/) grid or upon similar systematic designs such as the Continuous 11
Vegetation Survey (CVS) implemented by federal land management agencies (Dunk et al. 2004; 12
Davis et al. 2007; Dunk and Hawley 2009). For M. p. pacifica, additional data from non-13
systematic surveys were incorporated to broaden geographic representation of sampled sites to 14
private lands. The FIA program regularly collects standardized vegetation data at plots of 15
approximately 1 ha in size, locating one plot within each cell of a grid of hexagons that has been 16
delineated across the United States (see Figure 2 for an example of plot distribution). The current 17
(post-2001) FIA sampling grid uses hexagons of approximately 24 km2 in area (see description at 18
http://fia.fs.fed.us/). For species modeled in relation to plot-level covariates, wildlife surveys 19
were performed at FIA/CVS plots to allow access to vegetation data collected previously, both at 20
sampled locations and at other FIA/CVS sites to which predictions are then made. For species 21
modeled in relation to landscape-scale covariates, the FIA hexagonal grid provides a lattice that 22
facilitates parameterization of spatial neighborhood effects, reduces problems arising from 23
Carroll et al. 7
uneven sampling intensity or survey effort, and reduces the number of sample units to a level 1
(here < 10,000) that makes regional-scale spatial modeling computationally feasible. The size of 2
FIA hexagons is biologically relevant to territory size for our focal species (10-30 km2) and the 3
landscape scale with highest predictive power in previous analyses (Carroll et al. 1999). 4
Although surveys conformed to standard protocols designed to reduce the probability of false 5
absence records, we did not have access to detection histories from repeated surveys, either 6
because sites were visited only once or data was aggregated into a single presence-absence 7
record per site. Therefore, our analysis does not consider how detectability, as distinct from 8
occurrence probability, varies across space and between species (Webster et al. 2008). Further 9
description of survey data and protocols is given in Supplementary Material S1. 10
MODELING METHODOLOGY 11
In this analysis, we apply two forms of spatial autoregressive models, depending on the 12
spatial scale of the environmental covariates most relevant to the particular species. We refer to 13
the two scales of habitat selection analyzed here as landscape- and site-scale models. Distribution 14
models for wide-ranging species (here M. p. pacifica) typically show highest predictive power at 15
scales approximating that of the territory or above (Carroll et al. 1999). Model predictions at this 16
scale are also relevant to regional conservation planning efforts because they predict occupancy 17
seamlessly across the entire landscape. We used such landscape-scale models to analyze the 18
relationship between our wide-ranging focal species and broad-scale environmental variables 19
developed in a GIS. 20
We adapted the modeling framework of Latimer et al. (2006; model 2), which overlaid a 21
regular grid of cells (here the grid of FIA hexagons) on the study area, with each cell considered 22
a sample unit which might contain a number of survey locations. The model is defined as 23
Carroll et al. 8
follows. Let yi be the measured distribution variable (abundance or presence/absence) at site i. 1
Then yi is distributed as a binomial variable (presence/absence) with probability pi = exp{ηi}/(1 2
+ exp{ηi}), where ηi = β0 + β1x1i + …+ βpxpi + ρi. The variables x1i,…, xpi are site-specific 3
environmental covariates and the ρi are random effects that are jointly distributed as a Gaussian 4
CAR (Conditional AutoRegressive) spatial model. The CAR model is defined by the conditional 5
normal distributions ρi|ρ-i ~ N(μi, σi2), where ρ-i (spatial random effect) is the vector of all ρ 6
except the one for the ith site, 7
22
1, ,i
i j ij Ni in nφμ ρ σ
τ∈= =∑ 8
ni is the number of cells in the neighborhood, Ni of the ith cell, and τ represents the precision 9
(inverse variance) parameter. 10
The weight specification 1/ni in μi and σi2 is termed row standardized weighting (RSW; 11
see Supplementary Material S3 for code used to produce weight matrix). The φ parameter 12
dictates the amount of spatial correlation between spatial effects and lies in the interval [0, 1) for 13
positive association. Typically values often reside very near 1. This observation motivated 14
development of the intrinsic CAR (ICAR) model, a form of the CAR model in which φ = 1. 15
Although the ICAR conditional distributions do not specify a proper joint distribution (i.e., 16
integrates to 1) for the ρ (rho) vector, the parameter and prediction posterior distributions are 17
proper (Banerjee et al. 2004). We compared ICAR models with first-order (a cell and its six 18
immediate neighbors), second-order, and third-order neighborhoods. 19
Because models constructed using only landscape-scale variables generally show poor fit 20
and predictive power for localized low-vagility species, models for such species typically are 21
constructed using "site-scale" environmental variables (e.g., measured over a 1 ha plot as here). 22
Carroll et al. 9
We followed the framework of previous non-Bayesian habitat modeling for A. longicaudus and 1
M. churchi by building models from site-scale environmental variables augmented with relevant 2
broad-scale (e.g., climate) covariates (Dunk et al. 2004; Dunk and Hawley 2009). We analyzed 3
this data using CAR models. Although in a strict sense, both CAR and ICAR models are most 4
appropriately applied to seamless lattices of cells, we applied the CAR framework here to model 5
site-scale responses that resemble the point data typically analyzed with geostatistical models. 6
CAR models can closely approximate continuous geostatistical processes and are much more 7
computationally efficient than geostatistical models (Rue and Held 2005; Webster et al. 2008; 8
Ibánez et al. 2009), although recent development of ‘spatial predictive process’ models may 9
increase the computational feasibility of non-lattice-based approaches for large data sets (Finley 10
et al. 2007; Latimer et al. 2009). 11
When environmental variables are measured at the site scale, and sampled sites are sparse 12
or unevenly distributed, distance between a site and its nearest neighbors may vary widely. This 13
suggests that site-level data, in contrast to surveys summarized at the cell-level, would benefit 14
from the greater flexibility of the CAR model where weights may vary with distance (Cressie 15
1993). In this analysis, however, we used CAR models with row standardized neighborhood 16
weights (RSW), which approximate the weight structure of ICAR in assigning equal weights to 17
all neighbors. We chose row standardized weighting because, in exploratory analyses of our data, 18
row standardized weighting offered better model fit when compared to more complex distance 19
weighting schemes. Contrasting weighting schemes can be implemented and compared using 20
code provided in Supplementary Material S3. The neighborhoods Ni for the point level data were 21
specified by defining a radius threshold of 6, 8, 14, 20, and 30 km centered around the ith site. All 22
of the other sites contained within this radius were considered neighbors of the ith site. The 23
Carroll et al. 10
sampling design based on FIA plots separated by 5-6 km did not allow us to assess smaller 1
spatial neighborhoods. 2
We used the program WinBUGS (version 1.4.3; Spiegelhalter et al. 2003) to fit ICAR 3
models and their non-spatial generalized linear model analogues (see Supplementary Material S3 4
for BUGS code). We used OpenBUGS (v3.0.3; Thomas et al. 2008) to fit CAR models and their 5
non-spatial generalized linear model analogues. Both WinBUGS and OpenBUGS simulations 6
used three MCMC chains, each with a burn-in period of 10,000 iterations followed by 40,000 7
iterations for estimation. Convergence was evaluated using the Brooks-Gelman-Rubin statistic 8
(Spiegelhalter et al. 2003). We assumed no pre-existing knowledge of model parameters and thus 9
specified the least informative priors that allowed model convergence (Supplementary Material 10
S3). In order to facilitate convergence of model estimates, we standardized variables by 11
subtracting the mean, and then dividing by two SD (Gelman 2008). 12
DESCRIPTIONS OF CANDIDATE MODELS BY SPECIES 13
We sought to evaluate, by comparison with previously-published results, whether the use 14
of spatial models altered model selection and prediction. Therefore, rather than exhaustively 15
explore the set of all possible candidate models, we developed a set of models for each species 16
based on the best model identified in the publications from which the dataset was drawn. We 17
compared this model with alternate candidate models which varied by assigning either linear and 18
quadratic terms to each variable or dropping one or more variables. We thus evaluated whether 19
addition of spatial random effects reduced the structure of the environmental component of the 20
best model. Interaction terms were only evaluated if present in the previously-published best 21
model. This resulted in 13 to 15 candidate models per species, depending on the number of 22
variables in the previously-published model (Supplementary Material S2). We identified best 23
Carroll et al. 11
spatial models by means of a two-step process. First we identified the best scale of analysis 1
(spatial neighborhood size) by comparing mean squared predictive error (MSPE; see below) of 2
the full model for a species across a range of scales. Second, we identified the best of the 3
candidate models at the best scale. 4
For M. churchi, we developed 15 candidate models from the best model of Dunk et al. 5
(2004), which included four variables: mean dbh of conifers (CDBH), percentage canopy cover 6
of hardwoods (HC), coefficient of variation of December and July precipitation (CVPRE), and a 7
moisture stress index relating summer temperature and precipitation mean annual precipitation 8
(SMRT)(Table 1). For A. longicaudus, we developed 14 candidate models from the best model 9
of Dunk and Hawley (2009), which included four variables: percent slope (SLOPE), basal area 10
of trees with dbh between 45 and 90 cm (BAREA), maximum tree dbh (MAXDBH), and the 11
standard deviation of conifer dbh (SDDBH)(Table 1). For M. p. pacifica, the candidate set of 13 12
models was built using seven variables identified in previous habitat modeling studies for the 13
species using generalized linear or generalized additive models (Carroll et al. 1999; Carroll et al. 14
2001; Davis et al. 2007): tree canopy closure (DEN), tree size class (SIZ), annual snowfall 15
(SNOW) annual precipitation (PPT), terrain ruggedness (TRI), summer tasseled-cap greenness 16
(GRN), a metric derived from satellite imagery, and the fisher habitat value rating derived from 17
the California Wildlife Habitat Relationships system (WHR)(Table 1). 18
Spatial trend surface variables (functions of the x and y coordinates) included in 19
previously-published models (Dunk et al. 2004; Dunk and Hawley 2009) were not considered in 20
this analysis as they would duplicate trends modeled within the spatial random effect term. Our 21
non-spatial analogue models, which lacked both trend surfaces and spatial random effects, 22
should thus be seen as 'naive' versions of previously-published non-spatial models for these 23
Carroll et al. 12
species. To better compare the performance of the CAR models with the most commonly-used 1
alternative methods for analyzing species distributional data, we also developed distribution 2
models for the three species using Maxent (Phillips et al. 2006; Phillips & Dudik 2008). Maxent 3
uses a “maximum entropy” approach that compares presence locations to a set of available 4
locations (Phillips & Dudik 2008), and has performed well in comparison with 15 alternate 5
methods on a wide variety of taxa in diverse regions (Elith et al. 2006). Although Maxent is 6
more typically applied to presence-only data, it can also be used to analyze presence-absence 7
data, as here, by basing the set of available habitat on the total set of sampled locations (Phillips 8
et al. 2009). We used the default Maxent setting that allow flexible response forms such as 9
thresholds and hinge effects (Phillips & Dudik 2008). However, as with the GLM and CAR 10
models, only those interaction effects present in previously-published models were evaluated in 11
the Maxent analysis. We also summarized results from previous studies of the three species 12
using generalized additive models (GAM) which contained spatial covariates in the form of trend 13
surfaces (Dunk et al. 2004; Dunk and Hawley 2009) or a “simple autoregressive” term (Davis et 14
al. 2007). 15
COMPARISON OF MODELS 16
We fitted candidate models and ranked competing models by their mean squared 17
predictive error (MSPE)(Gelfand & Ghosh 1998). Although Deviance Information Criterion 18
(DIC), a metric that identifies parsimonious models through use of a penalty term based on the 19
number of model parameters, is the most common model diagnostic used to evaluate fit of 20
Bayesian models, its utility depends the accuracy of its estimation of the effective number of 21
parameters in the model (pD)(Spiegelhalter et al. 2002; Gelman et al. 2004). In those cases where 22
DIC’s normality assumptions are violated, pD may be estimated as negative, thus biasing DIC-23
Carroll et al. 13
based model ranking. In contrast, MSPE lacks such assumptions. MSPE judges a model based on 1
its ability to accurately produce replicated data “close” to the data that was observed (Gelman et 2
al. 2004). Because more complex models tend to do this slightly better, MSPE tends to choose 3
slightly more complex models than does DIC and shows less differentiation between alternate 4
closely-competing models. Unlike DIC, MSPE does not have an explicit penalty as the number 5
of model parameters increases. However, as model complexity becomes too large, the variability 6
in the replicated data becomes large which, in turn, raises MSPE and reduces model rank. Thus 7
MSPE and DIC-based model rankings are often similar (Carroll and Johnson 2008). 8
Diagnostic metrics also have been developed for Bayesian models based on comparing 9
observed responses with simulations of predicted responses (Gelman et al. 2004). We used the 10
posterior predictive p-value (PPPV) to evaluate how likely it is that the observed data and the 11
predicted data were drawn from the same distribution (Gelman et al. 2004). Similarly, we 12
produced a percent correct classification metric (PCC) based on the proportion of simulated 13
responses matching observed responses. We evaluated the contrast between spatial and non-14
spatial models by means of the Spearman rank correlation between predictions from the best 15
spatial model and its non-spatial analogue (corrS) and the percent reduction in mean squared 16
predictive error (MSPE) from non-spatial to spatial models. 17
For spatial, non-spatial, and Maxent models, we evaluated the AUC (Area Under the 18
receiver-operating Curve), a threshold-independent metric used to evaluates a model’s 19
discriminatory ability (Swets 1988). AUC ranges between 0.5 for a model that performs equal to 20
random expectation to 1.0 for a model with perfect classification. To evaluate the potential for 21
over-fitting and the sensitivity of model results to subsetting of the input data, we used a cross-22
validation procedure that randomly withheld 10% of the samples and evaluated the ability of 23
Carroll et al. 14
models developed from the remaining samples to predict the holdout data. We repeated cross-1
validation 10 times for the best model for each species and reported the mean of the resultant 2
AUC values. 3
We compared general patterns in model results across species, including the size of the 4
spatial neighborhood associated with the best model. We evaluated the statistical and ecological 5
implications of contrasts between the species in correlation between predictions of spatial and 6
non-spatial models, and the increase in quality of fit of the model (proportional reduction in 7
MSPE) between spatial and non-spatial models. We summarized for each species the structure of 8
the best spatial and non-spatial models. We described the spatial pattern of the random effect 9
term. For species modeled at the cell-level, we described the spatial pattern of predicted 10
probability based on environmental effects alone (without the rho term). Evaluating predicted 11
abundance without the rho term allowed us to assess the effects of predicted species-habitat 12
relationships without the influence of unmeasured environmental variables. 13
RESULTS 14
GENERAL PATTERNS ACROSS SPECIES 15
Across all species, the smallest spatial neighborhoods generally produced the best-fitting 16
models (Figure 1). Neighborhoods with an eight km radius, the smallest evaluated at the cell 17
level, performed best for M. p. pacifica. Neighborhoods with a six km radius, the smallest 18
evaluated in the site-scale models, produced the best models for M. churchi, whereas an eight km 19
neighborhood radius produced the best models for A. longicaudus (Figure 1). 20
Spatial models outperformed non-spatial models for all species (Table 2). The 21
improvement in fit conferred by the spatial model, as measured by the percent reduction in 22
MSPE from non-spatial to spatial models, was 35.3, 54.7, and 54.0 for A. longicaudus, M. p. 23
Carroll et al. 15
pacifica, and M. churchi, respectively (Figure 1, Table 2). The correlation between spatial and 1
non-spatial model predictions generally paralleled the degree of contrast in fit between spatial 2
and non-spatial models (Table 2). Percent correct classification improved from 70.4, 84.8, and 3
73.6% (non-spatial models) to 81.3, 91.8, and 88.3% (spatial models) for A. longicaudus, M. p. 4
pacifica, and M. churchi, respectively. 5
For all species, for both the full data set and cross-validation runs, AUC values for the 6
spatial models were greater than AUC shown by non-spatial or Maxent models (Table 2). 7
Although PPPV for both spatial and non-spatial models for all species supported the conclusion 8
that predicted and observed values were drawn for the same distribution, i.e., the model was not 9
over-fit (Table 2), contrast between AUC and CV AUC was greater for spatial than for non-10
spatial models. Thus the improvement in AUC associated with use of spatial models remained, 11
but was less pronounced under the cross-validation runs than for the full data set. Maxent models 12
showed AUC intermediate between non-spatial and spatial models for all species under both full 13
and cross-validation runs. 14
SUMMARY OF MODELS 15
For M. churchi, a model with linear terms for 3 of 4 variables (CDBH, CVPRE, and 16
SMRT) and quadratic terms for the remaining variable (HC) was the best spatial model, while a 17
model with quadratic terms for three variables (CDBH, HC, SMRT) and a linear term for 18
CVPRE was the best non-spatial model (Supplementary Material S2). For A. longicaudus, a 19
model with quadratic terms for all of four variables (SLOPE, BAREA, SDDBH, MAXDBH) 20
showed lowest MSPE among the candidate non-spatial models. The best spatial model was 21
similar except for a linear term on one variable (MAXDBH). For M. p. pacifica, the best spatial 22
model contained linear terms for the variables DEN, GRN, and TRI, and a quadratic term for 23
Carroll et al. 16
SNOW. The best non-spatial model was similar, but with a linear term for snowfall. Generally, 1
spatial models showed similar ranking to their non-spatial analogues (Supplementary Material 2
S2). 3
The broad-scale pattern of spatial random effects was relatively smooth (low rugosity) for 4
M. churchi and M. p. pacifica (Figure 2a, 3a). However, spatial random effects for the mollusk 5
were concentrated in a single cohesive population and for M. p. pacifica in two disjunct 6
populations. Greater meso-scale variation in spatial random effects was evident for A. 7
longicaudus, (Figure 2b). For M. p. pacifica, the environmental component from the selected 8
model (without the spatial random effects) predicts high habitat suitability for a large area in the 9
northern Sierra Nevada (Figure 3b). 10
DISCUSSION 11
Biologists and managers who develop and apply habitat models are often familiar with 12
the challenges posed by their data's spatial structure but unsure whether more complex spatial 13
models will actually increase the utility of model results in planning (Beale et al. 2010). Because 14
we analyzed presence-absence distribution data from three species with a wide range of dispersal 15
ability and home range size, using both landscape and site-scale environmental variables, our 16
conclusions may be generalizable across the spectrum of potential applications of spatial wildlife 17
habitat models, and can assist planners in deciding when to use spatial models, choosing the 18
appropriate model structure, and interpreting model results. 19
Our results suggest that the greater cost of spatial models in computational time and 20
effort is often justified due to improved quality of model fit and greater insights into the spatial 21
processes producing autocorrelation in the data. For all species, for both the full data set and 22
cross-validation runs, model performance as measured by the AUC value was greater for spatial 23
Carroll et al. 17
CAR models than for non-spatial logistic regression or maximum entropy (Maxent) models. 1
However, contrast between model performance under full and cross-validation runs was greater 2
for spatial than for non-spatial models, suggesting that spatial model results may be moderately 3
sensitive to variation in input data. Maxent models showed similar sensitivity, with their 4
advantage over non-spatial models in the full model runs not as evident in the cross-validation 5
runs. Spatial model performance in our results for A. longicaudus and M. churchi was better 6
than that shown by previous spatial GAM models for those species (Table 2). However, our 7
spatial model for M. p. pacifica showed similar but slightly lower performance than did a GAM 8
model from a previous study that incorporated a “simple autoregressive” spatial term based on 9
smoothing of observed responses (Davis et al. 2007). This contrast may also be due to the 10
slightly different distribution dataset and cross-validation procedure used by Davis et al. (2007). 11
Our results agree with those from a recent review using simulated spatially-correlated data, 12
which found that methods with spatial effects in the error term, such as the CAR and ICAR 13
methods used here, generally performed similarly and often considerably better than those with 14
spatial effect as an additional covariate (such as spatial trend surface or “simple autoregressive” 15
methods)(Beale et al. 2010). However, GAM performed best among the latter methods, and both 16
groups of spatial methods outperformed non-spatial approaches (Beale et al. 2010). 17
Lack of explicit treatment of spatial structure may lead to inclusion in the model of 18
spurious environmental variables (Dormann et al. 2007; Dormann 2007b). In our results, the 19
contrast between the fit of analogous spatial and non-spatial models decreased as models become 20
more complex (Supplementary Material S2), suggesting that a subset of the environmental 21
covariates may be acting as surrogates for spatial structure in the non-spatial models. Our results 22
also provided insights concerning the scale at which spatial processes operate to influence 23
Carroll et al. 18
distribution (Figure 1). For the two species where data allowed comparison of 6 and 8 km 1
neighborhoods, the mollusk species with low dispersal ability had best models at a smaller 2
neighborhood size than did the more vagile mammal (A. longicaudus). Although our results for 3
M. churchi and M. p. pacifica allow the possibility that finer-scale sampling designs would 4
support improved models, in regional monitoring programs with limited resources the benefits of 5
increased sampling intensity must be balanced against any resultant decrease in the extent of the 6
region sampled. Our ICAR model results do agree with previous studies in concluding that first-7
order neighborhoods are often sufficient to account for spatial structure in distribution data 8
(Griffith 1996; Kissling and Carl 2008). 9
We hypothesize that the high rugosity of the spatial effect in models for A. longicaudus 10
(Figure 2b) may be due to meso-scale processes such as dispersal limitation rather than broad-11
scale environmental effects. In contrast, we hypothesize that the smoothness of the pattern of 12
spatial effect for M. p. pacifica (Figure 3a) may be due to unmeasured environmental variables 13
or population processes, but cannot conclusively distinguish these factors. The pattern of spatial 14
effect for the fisher (Figure 3a) closely resembles that for a sympatric raptor of conservation 15
concern, the California Spotted Owl (Strix occidentalis occidentalis), suggesting a common 16
unmeasured covariate (C. Carroll, unpubl. data). 17
Evaluating predicted occurrence without the rho term allowed us to assess the effects of 18
predicted species-habitat relationships without the influence of unmeasured environmental 19
variables or population processes. The environmental component from the best model for M. p. 20
pacifica identifies a large area in the northern Sierra Nevada as fisher habitat (Figure 3b). If the 21
spatial random effects for M. p. pacifica can be attributed to population processes (home range 22
clustering or dispersal limitation), this result may suggest potential reintroduction areas. 23
Carroll et al. 19
EXTRACTING ECOLOGICAL HYPOTHESES FROM SPATIAL STRUCTURE 1
Dormann (2007b) found that adding spatial effect to distribution models improved model 2
fit in most contexts. Our results suggest that spatial models are most useful when either previous 3
studies have suggested strong spatial patterns in residuals from non-spatial models, as for M. p. 4
pacifica (Carroll et al. 1999; Davis et al. 2007), or a species occupies a small proportion of the 5
analysis region, and its range boundary is complex, poorly-known, or potentially constrained by 6
past geologic or climatic events as in the mollusk analyzed here (Dunk et al. 2004). Spatial 7
models are less useful where the analysis or survey extent or survey data has been limited a 8
priori to an area of relatively uniform spatial random effects such as well-described and cohesive 9
range boundary. 10
Our results suggest that despite their greater computational cost, Bayesian spatial models 11
provide benefits when compared to alternative spatial modeling approaches. This is most evident 12
when we compare them to non-autoregressive approaches such as trend surface models. 13
Although trend surface variables can represent broad-scale spatial trends, their flexibility is 14
limited to quadratic or occasionally cubic surfaces. Thus they are unable to represent complex 15
spatial effect surfaces such as shown in our analyses by A. longicaudus (Figure 2c). More 16
complex CAR models based on distance weighting can also be employed where this improves 17
fit. In addition, CAR models, unlike trend surface or generalized additive models, allow 18
quantitative estimates of the strength of the spatial association using a single parameter (τ). 19
Although the survey data used in this study was collected based on a systematic sampling design, 20
spatial autoregressive models may also aid analyses of data with geographically-uneven levels of 21
survey effort, because such bias can be incorporated within the spatial random effect term, thus 22
Carroll et al. 20
reducing its influence on estimation of the effects of environmental variables (Carroll and 1
Johnson 2008). 2
By treating spatial effects as a variable of interest rather than a nuisance, hierarchical 3
Bayesian spatial models can suggest the identity of additional environmental covariates that may 4
improve model fit, or the existence of area- and isolation effects that may limit population 5
viability. Our results suggest that spatial models developed at coarse scales (such as the FIA 6
hexagons considered here) are statistically and biologically relevant to processes influencing the 7
distribution of a broad range of taxa. Standardizing modeling efforts for multiple taxa to such a 8
common spatial lattice can help link regional-scale monitoring efforts and increase the relevance 9
of habitat models to multi-species conservation planning. 10
ACKNOWLEDGMENTS 11
We thank the many U.S. Forest Service biologists who assisted in collection of species 12
distribution data. T. Martin and two anonymous reviewers provided helpful comments. 13
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Carroll et al. 24
TABLES
Table 1. Abbreviations of environmental variables referenced in text.
VARIABLES MEASURED AT THE PLOT OR SITE SCALE REFERENCE
BAREA - basal area of trees with dbh between 45 and 90 cm Dunk and Hawley 2009
CDBH - mean dbh conifers Dunk et al. 2004
HC - Percentage canopy cover of hardwoods Dunk et al. 2004
MAXDBH - maximum tree dbh Dunk and Hawley 2009
SDDBH - standard deviation of conifer dbh Dunk and Hawley 2009
SLOPE - percent slope Dunk and Hawley 2009
CLIMATE VARIABLES DERIVED IN GIS FOR SITE-SCALE MODELS
CVPRE - coefficient of variation of December and July precipitation Dunk et al. 2004
SMRT - Moisture stress index relating summer temperature and precipitation Dunk et al. 2004
VARIABLES MEASURED OVER EXTENT OF FIA CELL FOR LANDSCAPE-SCALE MODELS
Carroll et al. 25
DEN - density or tree canopy cover Carroll et al. 1999
GRN - tasselled-cap greenness Carroll et al. 2001
SIZ - tree size class Carroll et al. 1999, Carroll 2005
SNOW - annual snowfall Krohn et al. 1997
TRI - terrain ruggedness index Davis et al. 2007
WHR - California Wildlife Habitat Relationships index Carroll 1999, Davis et al. 2007
Carroll et al. 26
Table 2. Comparison across species of model performance and contrast between spatial and non-
spatial models. The contrast between spatial and non-spatial models is evaluated by means of the
the PCC and PPPV statistics, the Spearman rank correlation between predictions from the best
spatial model and its non-spatial analogue, and the percent reduction in MSPE from non-spatial
to spatial models. Model performance was also evaluated using the AUC metric for the full data
set and for data withheld during ten-fold cross-validation for non-spatial GLM, spatial CAR, and
Maxent models, and compared with AUC reported for previous studies using GAM models with
spatial components. Number of sampled sites or hexagons, n = 365, 993, and 308 for A.
longicaudus, M. p. pacifica, and M. churchi, respectively.
Species A. longicaudus M. p. pacifica M. churchi
Model performance
non-spatial GLM
PCC 0.704 0.848 0.736
PPPV 0.432 0.494 0.448
spatial CAR
PCC 0.813 0.918 0.883
PPPV 0.529 0.558 0.518
Correlation 0.843 0.596 0.528
MSPE reduction 35.26 54.70 53.95
Comparison of AUC between methods
Non-spatial GLM
Full 0.814 0.720 0.746
Carroll et al. 27
CV 0.783 0.724 0.690
Spatial CAR
Full 0.985 0.940 0.994
CV 0.807 0.851 0.863
Maximum entropy (Maxent)
Full 0.852 0.769 0.832
CV 0.783 0.741 0.774
GAM with spatial term
Full 0.866a 0.96b 0.844c
CV 0.798a 0.86b 0.795c
a Dunk and Hawley 2009.
b Davis et al. 2007.
c Dunk et al. 2004.
Abbreviations: AUC: area under the receiver operating characteristic curve); CAR: conditional
autorgressive model; GAM: generalized additive model; GLM: generalized linear model; MSPE:
mean squared predictive error; PCC: proportion correctly classified; PPPV: posterior predictive
p-value.
Carroll et al. 28
FIGURES
Figure 1. Improvement in model fit, as measured by percentage reduction in the MSPE (mean
square predictive error) diagnostic, conferred by spatial models over their non-spatial analogues,
for three species in the Pacific states of the USA (Arborimus longicaudus, Martes pennanti
pacifica, and Monadenia churchi). Curves are left-truncated at 6 km, the approximate mean
distance between survey locations.
Figure 2. Pattern of (a-b) spatial random effect and (c-d) predicted probability of occurrence,
from best spatial conditional autogressive (CAR) models for two species (Church's sideband
snail (Monadenia churchi) and red tree vole (Arborimus longicaudus)) modeled using plot-scale
covariates in the Pacific states (USA).
Figure 3. Pattern of a) spatial random effect, b) predicted probability of occurrence without the
spatial random effect term, and c) predicted probability, from best spatial intrinsic conditional
autogressive (ICAR) models for the Pacific fisher (Martes pennanti pacifica), a wide-ranging
species modeled using landscape-scale covariates in the Pacific states (USA).
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#SS SSS SS #S SS#S #S S#S #SSS SSS #S #S#S#S SSS S
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S S#S #S#S #S#S S#S#SSS #S
S #S#S #S#S S#SS#SS #SS#S#S #SSS
S SS#S#SSS
0 - 0.1
0.1 - 0.5
0.5 - 1#
Predictedprobabilityof occurrence
#S
#S
(a) (b)
(c) (d)
M. churchi A. longicaudus
-10 - -2-2 - -0.5-0.5 - 0.50.5 - 22 - 10
Spatial randomeffect value
0 - 0.005
0.005 - 0.01
0.01 - 0.02
0.02 - 0.03
> 0.03
Predicted probabilitywithout rho term
0 - 0.05
0.05 - 0.15
0.15 - 0.3
0.3- 0.5
> 0.5
Predicted probability
(a) (c)(b)