Introduction

Understanding species distributions is essential to ecology, evolution, and conservation biology (Guisan and Thuiller 2005; Elith and Leathwick 2009). Species distributions models (SDMs) are often used to address ecological and conservation issues, such as quantifying habitat/environmental relationships, evaluating potential management actions, predicting the effects of land-use and climate change, and identifying priority areas for conservation planning (Loiselle et al. 2003; Evans et al. 2010; Lawler et al. 2010). In addition, these models have been frequently used for understanding the characteristic scales of environmental relationships (i.e., ‘scales of effects’; Urban et al. 1987; Smith et al. 2011; Jackson and Fahrig 2015) and the multi-level effects of landscapes on organisms (Fletcher and Hutto 2008; Thornton et al. 2011). However, the usefulness of such models is limited by the data used in model building, which often contain relatively limited information and sample bias (Norris 2004; Phillips et al. 2009; McCarthy et al. 2012).

Models of species distributions vary widely in spatial scale. Franklin (2009) categorizes species distribution models based on two contrasting spatial extents and how species data tend to vary with extent (cf. Wiens 1989). While there are exceptions to this categorization of distribution models (e.g., Albright et al. 2010), this tendency for data type to covary with spatial extent is common (e.g., Jones et al. 2010). One general category of models occurs at large extents, typically at the geographic range of species. In these situations, presence-only data are frequently used (Brotons et al. 2004; Elith et al. 2006), which often come from museum specimens or through citizen science programs (Dickinson et al. 2010). Such data are useful because of the large spatial extent, but they often suffer from sample selection bias (Phillips et al. 2009; Fourcade et al. 2014), such as presence records being more likely in areas that are easily accessible. This bias can limit the ability of using such data to identify environmental relationships (Kadmon et al. 2004) and make accurate predictions to new areas or times (Brotons et al. 2004). In addition, because absences are not available, only measures of relative suitability/probability can typically be modeled (but see Dorazio 2012; Royle et al. 2012).

In contrast, a second general category of models come from planned surveys typically collected at extents smaller than the geographic range, where interest lies in understanding local to landscape-scale distributions (Franklin 2009). In this category of models, presence–absence data (or more appropriately ‘detection–nondetection’ data; presence–absence hereafter) are frequently used (Rota et al. 2011; Lawson et al. 2014). These data are useful in that they typically have minimal sample selection bias regarding occurrence because at any survey location, presence or absence can be quantified (Dorazio 2014). This category of models contains richer information regarding the prevalence of species and may potentially allow less biased quantification of environmental relationships (Fithian et al. 2015); however, this category of models frequently suffers from small amounts of data and geographic extent. Consequently, such models may be limited in identifying the characteristic spatial scale of environmental relationships in situations where samples do not cover enough large-scale variation in landscape conditions (Oneill et al. 1996).

Given the strengths and limitations of both presence-only and presence–absence data, models that leverage both types of information may be valuable for quantifying environmental relationships and the scale of such relationships, as well as predicting species distributions across regions. Recent advances in species distribution modeling have focused on how to integrate these different sources of data to make more reliable predictions (Dorazio 2014; Keil et al. 2014; Fithian et al. 2015). We refer to this general group of models as integrated distribution models, borrowing this term from recent developments in population ecology where integrated population models link multiple data sources to estimate population dynamics (Schaub et al. 2007; Abadi et al. 2010). These approaches simultaneously use different sources of data to develop statistical models of species distribution. These advances are in contrast to increasingly used ensemble modeling consensus techniques (Araujo and New 2007; Marmion et al. 2009), which focus on summarizing predictions (e.g., average predictions) from separate models.

Advances in integrated distribution models have focused on two issues. First, models have integrated coarse-grain data, such as atlas data, with fine-grain data to downscale predictions of models (Keil et al. 2014). This approach is akin to a multi-level model (sensu Cushman and McGarigal 2002), in the sense that predictions at fine-grains are conditioned on predictions at coarse grains. One potential benefit of such models is that multi-level processes could be captured, yet these models could be limited when mis-matches occur in data (e.g., where atlas data suggest absence in a region where fine-scale data provides presence information). In addition, most distribution modeling data derive from point locations rather than coarse-grain grids, such that the spatial grain of different data sets (presence-only, presence–absence) are comparable (i.e., response data are single-level, rather than multi-level) while the spatial extents might vary considerably. A more general approach to integrate presence-only and presence–absence data has recently been independently described by Dorazio (2014) and Fithian et al. (2015). This approach is driven by viewing species distributions as spatial point processes (see below), and integrates presence-only data with planned surveys that include either presence–absence or detection–nondetection data (i.e., data that include imperfect detection). Here we focus on this latter approach. These models emphasize accounting for sample selection bias and leveraging planned surveys to improve presence-only modeling. While these models have been developed, our understanding of benefits and limitations of these models in real-world applications is still limited, with only one empirical example on Eucalpytus in Australia (Fithian et al. 2015). Furthermore, it is unclear if such models will provide insights to the characteristic scaling of environmental relationships (Thompson and McGarigal 2002; Holland et al. 2004; Jackson and Fahrig 2015). Yet given that such data integration may offer the potential to capture greater large-scale variability of the environment, integrated distribution models could provide better insight to the spatial scales of environmental relationships.

Here, we apply and compare recent statistical advances in integrated modeling of species distribution data to test if, and the extent to which, the inclusion of broad-scale presence-only data can improve modeling efforts, in terms of model predictions, estimation of environmental relationships, and the identification of the characteristic scales of environmental relationships. We first review recent modeling frameworks regarding unified models of species distribution data and how such frameworks can be applied to understand environmental relationships. We then apply this framework for interpreting the distribution of Sherman’s fox squirrels (Sciurus niger shermani) in north-central Florida. We contrast inferences from integrated distribution models with more conventional methods for species distribution, including conventional presence–absence models and ensemble models (Araujo and New 2007; Marmion et al. 2009), using a multi-scale analysis (Holland et al. 2004). We conclude by discussing the current limitations of integrated distribution models and potential extensions to improve such modeling efforts.

Methods

Uniting presence-only and presence–absence data

To integrate presence-only and presence–absence data, it is useful to view species distributions as being derived from inhomogeneous point processes (Warton and Shepherd 2010; Renner et al. 2015). Indeed, many species distribution modeling algorithms, such as MAXENT (Phillips et al. 2006), can be derived as spatial point process models (Renner and Warton 2013; Renner et al. 2015). Given that the application of such models to species distributions is recent, we briefly review and describe the relationship of inhomogeneous point processes to species distributions and its relevance for integrated distribution modeling based on recent model developments (Renner and Warton 2013; Dorazio 2014; Fithian et al. 2015; Renner et al. 2015). See Dorazio (2014) and Fithian et al. (2015) for derivation of these types of models and more details.

Point processes and thinned point processes

In spatial statistics, there has been a long history of viewing locations of species occurrence as point processes (Thompson 1955; Getis and Franklin 1987). In this context the focus is on species location, s, within a specified study region D. Point process models (PPM) focus on understanding the intensity (~density) of species, λ, in a bounded area or region across D. A point process is ‘inhomogeneous’ when intensity varies across D. This variation can be captured by spatially-explicit covariates by modeling intensity based on a log-linear relationship:

$$\log \lambda (s) = \alpha + \beta x(s)$$
(1)

Consequently, PPMs are similar to generalized linear models (GLMs), but the focus is on spatial locations of point occurrences rather than focus being on the point occurrences themselves (Renner et al. 2015).

Thinned point process models are an extension of PPMs that acknowledge that observed point locations are a sample of all locations, and possibly a highly biased sample. For example, presence-only sampling is often biased near roads and such bias can influence predictions of models (Phillips et al. 2009). Fithian et al. (2015) extend the PPM to account for sample selection bias by adjusting the regression model to include covariates that help explain sampling bias:

$$\log \lambda (s) = \alpha + \beta x(s) + \gamma + \delta z(s)$$
(2)

where z is a bias covariate, such as distance from road (see also Warton et al. 2013). Note that such bias adjustment may only be identifiable when z(s) is not highly correlated with x(s). The log-likelihood of the inhomogeneous point process model that accounts for sample bias is then (Fithian et al. 2015):

$$L_{po} (\alpha ,\beta ,\delta ,\gamma ) = \sum\limits_{{}} {(\alpha + \beta x_{i} + \gamma + \delta z_{i} ) - } \int\limits_{D} {\exp (\alpha + \beta x_{i} + \gamma + \delta z_{i} )ds}$$
(3)

where each observation i of a species is associated with a location s i and the covariates at that location (x i  = x(s i ) and z i  = z(s i )). The first component of this likelihood focuses on the information from the presence locations whereas the second component focuses on the covariate values across the region. The integral in this latter component cannot be directly estimated and is therefore approximated, typically from selecting background data points (sometimes called pseudo-absences). Background points are frequently used in presence-only modeling (Elith et al. 2006). See Warton and Shepherd (2010) and Renner et al. (2015) for interesting discussions on how PPMs help clarify the role of background points and the number background points that should be included in analyses of presence-only data. In general, Renner et al. (2015) emphasize that more background points should be used to estimate point processes than what is typically done in species distribution modeling (see below).

Joint likelihoods for integrating distribution data

Given the above point process model, a natural extension is to unite a presence-only point process model with a similar point process model based on presence–absence survey data. Joint analysis is commonly used in statistics and is increasingly used in population ecology, where likelihoods are developed for two (or more) independent data sets and a joint likelihood is formulated as the product of the two likelihoods (Schaub et al. 2007; Abadi et al. 2010). Because data from presence–absence and presence-only data are collected independently, the data can be analyzed together based on this joint likelihood. Joint analysis is thought to decrease bias and increase precision of estimates than modeling each data set in isolation (Dorazio 2014).

To appropriately link a presence-only PPM with planned survey data based on presence–absence information, Fithian et al. (2015) suggest that a model formulation for presence–absence data by relating counts at sample locations, A i (where |A i | is the area of A i ) to a point process:

$$Ni \approx Pois(|Ai|\lambda (si)) = Pois(|Ai|\exp (\alpha + \beta xi))$$
(4)

The probability of occurrence can be derived based on using a complementary log–log link function:

$$\Pr (Ni > 0) = 1 - \exp ( - \exp (\alpha + \beta xi + \log |Ai|))$$
(5)

See Royle and Dorazio (2008, pp. 150–151) for further discussion regarding the use of the complementary log–log link function for relating abundance to occupancy. Taken together, the log-likelihood of the presence–absence data can be described as (Fithian et al. 2015):

$$L_{pa} (\alpha ,\beta ) = \sum\limits_{{}} { - y_{i} \log (1 - \exp ( - \exp (\alpha + \beta x_{i} )) + (1 - y_{i} )\exp (\alpha + \beta x_{i} )}$$
(6)

The joint log-likelihood is then the sum of these two log-likelihoods (Eqs. 3, 6). This modeling approach can be implemented with the multispeciesPP package in R (Fithian et al. 2015). To demonstrate the utility of this approach, we used presence-only and presence–absence (i.e., detection–nondetection; see below) data collected on Sherman’s fox squirrels.

Study species and study area

Fox squirrels (S. niger) are a common, widely-distributed tree squirrel native to North America and found east of the Rocky Mountains (Koprowski 1994). Of the 10 subspecies of fox squirrel, four subspecies occur in Florida. Our analysis focuses on Sherman’s fox squirrel, which primarily occurs in central and north Florida, but also ranges into southern Georgia. Sherman’s fox squirrels are sparsely distributed and in low densities and are listed as a State Species of Special Concern by Florida’s Endangered and Threatened Species Rule (Loeb and Moncrief 1993). Sherman’s fox squirrels and are known to use most major vegetative communities throughout their range; however, they are most commonly associated with inland pine-dominated vegetation found throughout most of peninsular Florida (Moore 1956, 1957).

We conducted our surveys within the core range of Sherman’s fox squirrels in central and northern Florida on public and private lands. The vegetative communities at our sites were highly variable, and included open grasslands, pine-dominated forests, hardwood-dominated forests, mixed pine–hardwoods, bottomland forest, and clear cuts. The canopy trees varied between sites, but the dominant pine trees included longleaf (Pinus palustris), slash (P. elliottii) and loblolly (P. taeda) pines, and the dominant oaks were turkey (Quercus laevis), live (Q. virginiana), laurel (Q. laurifolia), and water (Q. nigra) oaks. The sites varied in their vegetation management practices, which included cattle grazing, mowing, prescribed burning, and no active management. Nearly all sites where pines outnumbered hardwoods were managed for timber.

Presence-only data

We developed and promoted a publically available web-based tool (webpage) to generate presence-only locations of Sherman’s fox squirrel throughout their geographic range in Florida. The web-based tool recorded georeferenced locations (latitude and longitude in decimal degrees) using a Google map application to record sightings in the database. For this analysis, we extracted data from the website from August 20 2011 until May 1 2012 to coincide with the peak activity in fox squirrel activity typically related to mast availability (Weigl et al. 1989). We also subset these data further for the time period January–May 2012 to reflect only the time period of camera-trapping (see below). Results from this subset of data were similar, in terms of environmental relationships and metrics of predictive performance (Unpublished results).

We reviewed each presence-only data point from the online survey. We quarantined location points which appeared to erroneous due to user error of mapping software (i.e. locations in the middle of a water body). To increase the validity of these suspicious occurrence data, we attempted to verify the locations and receive more specific location information from the participant that submitted the data. We removed all erroneous points which could not be verified. This resulted in 2785 presence-only points used for modeling (871 points from January to May 2012; Fig. 1).

Fig. 1
figure 1

Study area in Florida, USA, where presence–absence and presence-only data collection occurred for Sherman’s fox squirrels. Presence-only (citizen science) locations shown as small, grey dots. Landscapes for presence–absence sampling (camera trapping) shown in red. Major roads included as grey lines. (Color figure online)

Full size image

Presence–absence data

We conducted intensive field surveys for Sherman’s fox squirrels using passive camera traps. We sampled across the study region using a hierarchal approach. We selected 40, 7.65 km2 landscapes across the study region using a stratified random design to capture major vegetative communities used by fox squirrels (Fig. 1). We sampled 10 landscapes in upland pine habitats, 10 in mesic flatwoods habitats and the remaining 20 without regard to a vegetative community. Within each landscape we random placed five 5.3 ha survey grids within each landscape. Each grid consisted of 9 sampling points in a 3 × 3-grid arrangement with 115 m spacing. For this analysis, we randomly selected a subset of points (n = 252 across the 40 landscapes) to use in models to reduce potential impacts of spatial autocorrelation influencing results (see “Discussion” section). Between January and June 2012 (a period of high activity for fox squirrels in Florida; Moore 1957), each sampling point was surveyed for 7 consecutive nights with a passive digital camera (Bushnell Trophy Cam model 119436c, Bushnell Outdoor Products, Overland Park, KS) baited with pecans (Carya illinoinensis) and cracked corn (Zea mays) placed at the base of a tree 1.5 m from the camera. When no photo of a fox squirrel was taken during the 7-night period, it was considered an absence (see below).

Analysis

To illustrate the approach and potential utility of integrated models, we contrasted conventional GLM models for presence–absence (detection–nondetection) data with integrated models and ensemble models that averaged predictions from separate presence-only and presence–absence models to make consensus predictions. In this comparison, integrated models use more data than conventional models, though such data contain less information (and possibly include sample selection bias; see below), while ensemble models use the same data used in integrated models, yet models are developed separately for presence–absence and presence-only data.

We considered three covariates for modeling the distribution of fox squirrels: canopy cover, distance from edge, and distance from roads. For canopy cover, we used circular buffers of different size radii (100 m, 500 m, 1 km, 2 km, 3 km, 4 km, and 5 km) to determine the characteristic scale of the environmental relationship of fox squirrel occurrence (Thornton and Fletcher 2014), and considered non-linear relationships with canopy cover via inclusion of quadratic terms. The scales we consider reflect area less than the average home range of fox squirrels (~25 ha on average) and up to approximately twice the average dispersal distance of fox squirrels (Kantola and Humphrey 1990). Canopy cover (percent tree cover) was taken from the 2011 National Land Cover Database and was re-scaled using a moving-window analysis. We did not adjust for overlapping landscapes being used in these buffers (sensu Holland et al. 2004) because we found no significant evidence for spatial autocorrelation in the residuals of models (Fig. S1; Zuckerberg et al. 2012). We did not consider buffers of different sizes in the same model to reduce multi-collinearity in model fitting. Because fox squirrels can concentrate their activities near habitat edges (Koprowski 1994), we considered the natural log of distance to edge of different vegetative communities using land-cover classes defined by the Florida Natural Areas Inventory (2010). We also considered distance to roads as a covariate (z(s) in Eq. 3 above) in integrated models to account for potential presence-only sample bias (Kadmon et al. 2004; McCarthy et al. 2012). While other factors may also influence fox squirrel distributions, these covariates were selected to illustrate the different ways in which covariates can be relevant for integrated models and the identification of characteristic scales of species–environment relationships.

The presence–absence GLM was based on a complementary log–log link function (see Eq. 6); note that a conventional logistic regression using a logit link function provided identical results. For the presence-only component of the integrated model, we selected background points based on a regular 2 × 2 km grid across the study area (Renner et al. 2015), which resulted in 13,713 points. Presence-only models were also used to create a consensus ensemble (Marmion et al. 2009), which were the same as the integrated model but with the presence–absence data removed (Eq. 3). Ensemble predictions were derived using a weighted average of model predictions (based on the area under the curve (AUC) statistic for each model) from the separate presence-only and presence–absence models (Marmion et al. 2009). Note that presence–absence and integrated models used here ignore the problem of imperfect detection. Currently, imperfect detection for occupancy has yet to be accounted for in integrated distribution models (see “Discussion” section). Nonetheless, based on an occupancy model using the camera-trapping data (MacKenzie et al. 2002), the estimated probability of detecting fox squirrels at a sample point at least one night of the seven night sample, given that the species occurred at the point, was 0.82.

To determine the characteristic scales of environmental relationships, we used a model selection approach to identify the most parsimonious scale for interpreting the relationship of fox squirrel distribution with canopy cover separately for the presence–absence GLM and the integrated GLM. In this assessment, we contrasted models using Akaike’s information criterion (AIC) that varied in the spatial scale of canopy cover (allowing for non-linearity in canopy cover relationships) while forcing distance from edge into the model and distance to roads in the integrated GLM to account for sample selection bias. We then took the selected model and further attempted to reduce model complexity by contrasting the model with a model that only included distance from edge and an intercept-only model. To determine effects on environmental relationships, we contrasted estimated β values, associated SEs, and predicted partial relationships.

To determine predictive performance, we used block validation (Wenger and Olden 2012). To do so, we split our presence–absence data into fourfolds at the scale of landscapes (7.65 km2) rather than points. Presence-only data were only used in model training (building) and were not used for validation. Block validation is helpful because creating folds at the sample unit (point) level results in validation data that can be more correlated with training data due to closer geographic proximity. We then assessed model predictions using two threshold independent measures—the AUC and a cross-validated log-likelihood (LL; Lawson et al. 2014). AUC is based on rank differences and is not impacted by predicted prevalence, whereas cross-validated log-likelihoods are measures of fit to the new data and incorporate predicted prevalence into assessments. We also used two threshold-dependent measures—the true skill statistic (TSS) and the kappa statistic (Fielding and Bell 1997; Liu et al. 2011). For TSS and kappa, we set a threshold cutoff based on maximizing the sum of the specificity and sensitivity (Liu et al. 2013).

Finally, we determined how the amount of presence–absence and presence-only data could alter the potential benefits of integrated models. To do so, we performed a similar block validation scheme as that described above; however, we altered model training in two ways. First, we sequentially reduced the amount of presence–absence data by reducing the number of landscapes used in model training from 30 (three of four folds above) down to 5 landscapes, using 10 landscapes for validation. For each sequential reduction, we calculated AUC, LL, TSS, and kappa based on integrated distribution models versus GLM on presence–absence data. We expected that GLMs would decline in predictive performance as the amount of training data declined, whereas integrated models to retain similar predictive performance. Second, we reduced the amount of presence-only data used in model training while keeping the amount of presence–absence data constant. In this scenario, we randomly removed 5–95 % of the original presence-only data. We expected that integrated models would decline in performance as the amount of presence-only data declined.

Results

The most parsimonious GLM model describing presence–absence data included a non-linear relationship of canopy cover at the 1-km scale and no significant relationship with distance from edge based on model selection criteria and parameter estimates (βedge = 0.99 ± 0.85 SE, P = 0.247; Table S1; Figs. 2, 3). For integrated models, the most parsimonious model identified a non-linear relationship of canopy cover at the 4-km scale and a positive, significant relationship with distance from edge (βedge = 0.19 ± 0.02, P < 0.0001; Table S2; Figs. 2, 3). For the integrated model, there was strong evidence for road-based sampling bias in presence-only locations identified through this model (δ road = −1.15 ± 0.04; z = −23.10, P < 0.0001). As expected, data for the integrated model spanned a larger environmental gradient than for the presence–absence GLM (Fig. S2). For example, the range of percent canopy cover at the 4 km scale for the presence–absence data was 0–82 %, while for the integrated data it was 0–100 %. In general, estimates from the integrated model were weaker (in terms of smaller β parameters) but also had greater precision (smaller SEs), such that the coefficient of variation of parameter estimates from integrated models was smaller and there was less uncertainty in partial relationships (Fig. 3).

Fig. 2
figure 2

Model selection indicates a larger characteristic scaling relationship of fox squirrel distribution with canopy cover using an integrated distribution model. Shown are changes in Akaike’s information criterion (AIC) as a function of the scale (radius) at which canopy cover is quantified for conventional (presence–absence) GLMs and integrated GLMs. In both sets of models, non-linear relationships with canopy cover were considered (via a quadratic term) and distance from edge was included

Full size image
Fig. 3
figure 3

Predicted environmental relationships of Sherman fox squirrels with a percent canopy cover at the 1-km scale, b percent canopy cover at a 4-km scale; and c distance to edge based on a generalized linear model (GLM) with presence–absence data only (red/dashed line) and integrated distribution model (blue). (Color figure online)

Full size image

Based on block validation, the most parsimonious integrated model tended to predict fox squirrel distribution better than the GLM model that used presence–absence data alone or an ensemble model that averaged predictions from separate presence–absence and presence-only models, based on measures of AUC (presence–absence = 0.74; ensemble = 0.76, integrated = 0.79), TSS (presence–absence = 0.51; ensemble = 0.53, integrated = 0.57), and kappa (presence–absence = 0.31; ensemble = 0.32, integrated = 0.41). For cross-validated log-likelihoods, performance was similar for presence–absence and integrated models and slightly less for ensemble models (presence–absence = −25.2; ensemble = −26.6, integrated = −25.3). In addition, predictions from the integrated models were less sensitive to the amount of presence–absence data used in model building, whereas predictive accuracy from the GLM and the ensemble model decreased as the amount of presence–absence data decreased in model building, particularly when less than 11 landscapes were used in model training (Fig. 4a–d). The integrated model also showed little sensitivity to the amount of presence-only data used in model building (Fig. 4e–h).

Fig. 4
figure 4

Effects of (a–d) the amount of presence–absence data (number of landscapes) and (e–h) amount of presence-only data used in distribution modeling on measures of predictive performance for presence–absence models, ensemble models, and integrated distribution models. Shown are results based on (a, e) AUC and the True Skill Statistic; other metrics (r and kappa) showed similar patterns

Full size image

Discussion

Accurately predicting species distributions is essential for many ecological and conservation problems, yet the data used for building predicted models are often limited (Pearce and Boyce 2006). We found that by uniting both broad-scale presence-only data with finer-scale presence–absence data, we identified a larger characteristic scale for the relationship of fox squirrel distribution with canopy cover. In addition, metrics regarding predictive performance were generally greater. While each metric showed modest improvements (Fig. 4), this pattern is notable for two reasons: (1) the presence–absence data were used for model evaluation, such that presence–absence models could be expected to perform better due to similar sampling methods, and (2) benefits were more robust to sample size. Importantly, these benefits tended to be greater than from using common post hoc approaches of ensemble techniques (Araujo and New 2007; Marmion et al. 2009).

The value of uniting data in distribution models

The integrated distribution modeling framework we used was recently developed independently by Dorazio (2014) and Fithian et al. (2015). In both situations, the rationale for integrating data was to improve presence-only modeling. Presence-only modeling is commonly used, in a large part because of widely available data and because such data are frequently available across broad scales. However, presence-only data have limited information content (e.g., prevalence of the species is unknown) and such data often suffer from sample selection bias, where presence locations are frequently collected near roads or other easily accessible areas (Kadmon et al. 2004; Loiselle et al. 2008; Phillips et al. 2009). By integrating presence–absence information, this bias potentially can be ameliorated, even with relatively little amounts of presence–absence data. In addition, integrating presence–absence data provides information on species prevalence that is not available with presence-only data (Hastie and Fithian 2013), which can be helpful when the goal is to model the probability of occurrence.

While presence–absence data can improve presence-only modeling efforts, we argue the corollary that integration of presence-only data can improve presence–absence modeling efforts. There are at least three reasons why integration of presence-only data may prove helpful for presence–absence modeling efforts. First, presence–absence data are often more limited in amount, which can lead to greater uncertainty in environmental relationships. For fox squirrels, there was greater uncertainty in predicted environmental relationships when using presence–absence data than when using an integrated distribution model and significant relationships with distance from edge were only revealed with the integrated model (Fig. 3). Given the widespread influence of edge effects in shaping distributions across landscapes (Ries et al. 2004), this pattern illustrates how important landscape effects can potentially be better revealed through data integration. Second, presence-only data are often available across broader extents than presence–absence data. When there is interest in extrapolating predictions of presence–absence models into new areas (Miller et al. 2004), integrating presence-only information that may be available across a broader region may improve modeling efforts. Third, integrating presence-only data may provide more information regarding wider environmental gradients than those sampled with presence–absence information, particularly for large-scale gradients and conditions not captured in planned surveys (Smith et al. 2011). Such integration could improve the ability of presence–absence modeling to capture potential non-linearities of environmental relationships and ultimately environmental spaces of niches (Austin 2002; Soberon 2007).

Furthermore, this example illustrates how characteristic scales identified in species-environment relationships can be sensitive to the data used and highlights that by increasing the spatial extent considered, larger characteristic scales may emerge. Such issues may help explain variability observed in characteristic scales identified in species in different regions (Jackson and Fahrig 2015). Given the long-standing and widespread interest in understanding the importance of spatial scale in ecology and evolution (Wiens 1989; Horne and Schneider 1995; Fletcher et al. 2013; Jackson and Fahrig 2014), models that leverage different sources of data across scales may prove useful in many situations.

Current limitations and future extensions

Integrating multiple data sources into models of species distribution is a new perspective for distribution modeling and many advancements should occur. While the integrated distribution model improved predictions, there were some limitations of this modeling approach that could be improved.

First, the model we used only included simple non-linearities in environmental relationships that were accounted for by adding polynomial terms. Many other SDMs can capture highly non-linear relationships (e.g., Elith and Graham 2009). Fithian et al. (2015) argue that Eq. 3 could be extended to include non-linear basis functions and models could be extended to include splines, akin to generalized additive models. In addition, Renner et al. (2015) show how point process models can be implemented with MAXENT, an algorithm that allows for highly non-linear relationships to be estimated. Yet further developments are needed for formally integrating MAXENT with planned survey data.

Second, this approach ignored the problem of imperfect detection due to observation errors. Imperfect detection is common (Kellner and Swihart 2014), particularly in animal populations, and heterogeneity in imperfect detection can influence the predictive performance of models (Rota et al. 2011; Lahoz-Monfort et al. 2014). Dorazio (2014) formulated an integrated distribution model that linked presence-only data with abundance data using an N-mixture model formulation (Royle 2004; Kery et al. 2005), but analogous formulations that focus on occupancy, rather than abundance, have yet to be developed.

Third, this modeling approach does not formally account for spatial autocorrelation. While we found no significant autocorrelation in the residuals of models, there was a non-significant tendency for autocorrelation in the integrated model (Fig. S1). Renner et al. (2015) review some point process models that acknowledge the potential for spatial autocorrelation, which could be extended into an integrated distribution modeling framework. In addition, it may be feasible to extend this approach to other common regression-based approaches for dealing with spatial dependence (Beale et al. 2010; de Knegt et al. 2010).

Fourth, these frameworks could be extended to better accommodate some aspects of multi-scale environmental relationships. While the approach by Keil et al. (2014) naturally accounts for multi-level effects (e.g., different effects at the patch or landscape scale), the approach we used here did not accommodate covariates influencing distributions at different grains or levels for the independent data sets (Fletcher and Hutto 2008). The framework we used assumed that environmental covariates relevant to presence-only locations are the same as for presence–absence and occur at the same scales for the two data sources. Yet it would be possible to link differential covariate effects operating across regions (e.g., watersheds) that could influence distribution based on both presence–absence and presence-only data. Nonetheless, integrated models easily accommodate data arising from different spatial extents, which can help improve extrapolation and transferring model predictions across space (Miller et al. 2004; Wenger and Olden 2012).

Conclusions

Integrating multiple data sources into models of species distribution is a new frontier for distribution modeling and provides a means to reliably tackle some current limitations in our understanding of characteristic scales (Smith et al. 2011; Jackson and Fahrig 2015; Miguet et al. in press). Here we provide an example of integrating presence–absence and presence-only data, but other types of formal data integration could occur (e.g., information on breeding, dispersal, etc.). By integrating multiple sources of information into the modeling process, greater insights into environmental relationships and more accurate predictions can occur. Consequently, we expect such models will provide a powerful approach for addressing problems of species distributions and ongoing landscape change.