Environ Ecol Stat (2008) 15:39–47 DOI 10.1007/s10651-007-0031-2
Modeling resource selection using polytomous logistic regression and kernel density estimates Chadwick D. Rittenhouse · Joshua J. Millspaugh · Andrew B. Cooper · Michael W. Hubbard · Steven L. Sheriff · Robert A. Gitzen
Received: 1 August 2005 / Revised: 7 July 2006 / Published online: 15 September 2007 © Springer Science+Business Media, LLC 2007
Abstract Wildlife resource selection studies typically compare used to available resources; selection or avoidance occurs when use is disproportionately greater or less than availability. Comparing used to available resources is problematic because results are often greatly influenced by what is considered available to the animal. Moreover, placing relocation points within resource units is often difficult due to radiotelemetry and mapping errors. Given these problems, we suggest that an animal’s resource use be summarized at the scale of the home range (i.e., the spatial distribution of all point locations of an animal) rather than by individual points that are considered used or available. To account for differences in use-intensity throughout an animal’s home range, we model resource selection using kernel density estimates and polytomous logistic regression. We present a case study of elk (Cervus elaphus) resource selection in South Dakota to illustrate the procedure. There are several advantages of our proposed approach. First, resource availability goes undefined by the investigator, which is a difficult and often arbitrary decision. Instead, the technique compares the intensity of animal use throughout the home range. This technique also avoids problems with classifying locations rigidly as used or unused. Second, location coordinates do not need to C. D. Rittenhouse · J. J. Millspaugh (B) Department of Fisheries and Wildlife Sciences, University of Missouri, 302 Natural Resources Building, Columbia, MO 65211, USA e-mail:
[email protected] A. B. Cooper Department of Natural Resources, University of New Hampshire, 208A Nesmith Hall, Durham, NH 03824, USA M. W. Hubbard Missouri Department of Conservation, 2901 W. Truman Boulevard, Jefferson City, MO 65109, USA S. L. Sheriff Missouri Department of Conservation, Conservation Research Center, 1110 S. College Avenue, Columbia, MO 65201, USA R. A. Gitzen College of Forest Resources, University of Washington, Box 352100, Seattle, WA 98195, USA
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be placed within mapped resource units, which is problematic given mapping and telemetry error. Finally, resource use is considered at an appropriate scale for management because most wildlife resource decisions are made at the level of the patch. Despite the advantages of this use-intensity procedure, future research should address spatial autocorrelation and develop spatial models for ordered categorical variables. Keywords Fixed kernel · Habitat use · Polytomous logistic regression
1 Introduction Resource selection is a hierarchical process of behavioral responses that often results in unequal use of resources. By quantifying resource use over time and space, resource selection models provide insight into how a species meets its life history needs. The information gathered from the modeling process may be used to identify a species’ biological requirements or estimate the effects of habitat modification or management. The typical resource selection study compares used to available resources (Manly et al. 2002). Used resources are often defined by the resource attributes at telemetry relocation points (Erickson et al. 2001). One concern with defining used resources based on relocation points is the potential for misclassification of resource use (Erickson et al. 2001). For example, use of a habitat patch might be misclassified depending on radio telemetry and mapping (i.e., geographic information system) error. The potential for misclassification of habitat use when using relocation points is greater when error is large relative to the habitat patch size (Nams 1989; White and Garrott 1986). This problem frequently occurs in study areas with rugged topography and very patchy habitats (e.g., Fig. 1). Available resources are quantified within a spatially- and temporally-defined area that the researcher considers “available” to the animal. Manly et al. (2002, p. 1) defined the availability of a resource as “the quantity accessible to the animal (or population of animals) during that same period of time.” Using this commonly followed definition, defining available resources becomes difficult because we must understand how an animal perceives its environment (Marzluff et al. 2001). In addition, resource availability is often defined at large spatial scales (e.g., study area) and some resources within that area might not be available to the animal. Further, study animals rarely can be monitored continuously in very high frequency (VHF) telemetry studies. Instead, locations may be collected at intervals >1 day. “Available” locations may have been used between monitoring times, and the sample of available locations may contain observations of both used and unused sites, causing difficulty in interpreting results of use-availability analysis (Keating and Cherry 2004). Consequently, when used resources are compared with available ones, the importance of certain resource features might be erroneous. Recent developments such as discrete choice analysis incorporate a matched-case pairing of used and available points, thus refining the extent of area considered available. Cooper and Millspaugh (1999) considered distances moved by elk (Cervus elaphus) to define the extent of resource availability. However, the relocation frequency in most standard very high frequency (VHF) telemetry studies (e.g., 1–3 locations per week) results in a large area considered available between successive observations. While such techniques narrow the spatial boundaries around what is considered available, the researcher must still make a somewhat subjective decision regarding the extent of resource availability. Additionally, VHF studies often have too coarse a temporal grain to conclude absolutely whether a patch was used or unused. However, with adequate sample size, the relative intensity of use throughout the home range can be estimated.
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Other Seedling (22.86 cm dbh, 22.86 cm dbh, 40-60% cc) Pine (>22.86 cm dbh, >60% cc) Fire killed Meadow range Custer State Park boundary
N
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Fig. 1 Habitat patches within an elk subherd home range in Custer State Park, South Dakota, spring 1994. Ponderosa pine is composed of two size classes (1: 2.54–22.86 cm dbh; 2: >22.86 cm dbh) and includes percent canopy closure (A: 60% canopy closure)
An alternative to the use versus availability approach is to consider the intensity of resource use within an animal’s home range. For example, Cross and Petersen (2002) modeled resource selection using categorized relative use intensity (e.g., low, medium, or high) based on the percentage of total telemetry locations contained within grid cells and used ordinal polytomous logistic regression to associate patch resource attributes to the different use-intensity levels. Their approach requires a subjective approach to defining the number of grids and grid cell size. North and Reynolds (1996) used a similar approach, but categorized use intensity based on the number of radiotelemetry locations within a resource patch. However, their approach still requires that relocations be placed within resource units which might be problematic (Erickson et al. 2001). Moreover, both approaches only consider locations where the animal was observed, which might be problematic because the entire area of use (i.e., home range) is not considered in the analysis. Instead, we should consider resource use as a continuum of intensity throughout the entire use area of an animal. The logical scale to delimit areas potentially used by an animal is the home range, which is the extent of area with a defined intensity of use during a specified time period (Kernohan et al. 2001). Recently utilization distribution (UD) approaches (Marzluff et al. 2004) have been used to summarize use along a continuum throughout an animal’s range. Although numerous UD models exist (e.g., bivariate normal), kernel-based estimators are currently most popular in the wildlife literature (Kernohan et al. 2001). Using the UD to describe resource use offers several advantages over use of individual relocation points. First, misclassifications due to radiotelemetry or mapping errors are reduced because the UD is a function of all telemetry
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locations and does not require that individual locations be placed in specific habitat patches. Second, instead of a binary classification system of used or not used, a refined measure of use can be computed based on the estimated probability density function (pdf) (Marzluff et al. 2001). The integrated pdf over an area indicates the relative amount of use in the patch. By quantifying the amount of use and assigning that use to resources, we have a relative measure of the use of each resource (Marzluff et al. 2001). Furthermore, with use defined along a continuum throughout an animal’s range, there is no need to define available points. Instead, we model resource selection by comparing the attributes of resources used at different intensities. In this paper, we use kernel-based UDs to define use-intensity levels for resource patches within an animal’s home range and use polytomous logistic regression to estimate resource selection of elk (Cervus elaphus) in South Dakota. Because this approach requires use-intensity levels to be defined, we begin by discussing factors that must be considered to estimate UDs.
2 Utilization distribution estimates Several methods for estimating UDs have been developed in the last several decades. Simulation research (e.g., Seaman et al. 1999) and critical reviews indicate that kernel approaches are superior to previously used methods in most contexts (Kernohan et al. 2001). The first choice in kernel UD estimation is whether the bandwidth will remain constant (fixed) or vary (adaptive) over the area of interest. Statistical research (Terrell and Scott 1992) and homerange simulations (Seaman et al. 1999) have noted that the adaptive kernel approach used in wildlife space use studies often has higher bias than fixed kernel approaches in the distribution tails. For this reason, fixed kernel approaches are commonly favored (Kernohan et al. 2001), although continued developments of adaptive kernel approach should be considered. Regardless of the kernel form, an automatic bandwidth selection method is desired. In ecological studies the reference or normal method (REF) and least squares cross-validation (LSCV) have been used most frequently (Gitzen and Millspaugh 2003). Although not currently used widely in ecological studies, plug-in (PI) and solve-the-equation methods (STE; Wand and Jones 1995) have been promoted in the statistical literature. Kernohan et al. (2001, pp. 144–145) reviewed general properties of these three classes of methods in the context of home-range estimation. The LSCV approach outperforms REF, but maintains high sampling variability and tends to select small bandwidth values when location points are clustered (Silverman 1986). The PI/STE approaches have been used in ecological studies, but tend to over-smooth real peaks in the underlying density (Loader 1999, pp. 191–192). Sample size guidelines for kernel based UD’s are developed from simulations or detailed case studies of real data. In the most in-depth investigation of sample size in ecological studies, Seaman et al. (1999) recommended a minimum sample size of 30 locations when estimating animal home-range size for fixed kernel methods using LSCV. For any bandwidth method, performance will improve as sample size increases, up to some threshold (Seaman et al. 1999).
3 Use of polytomous logistic regression Ordinal polytomous logistic regression, in particular the proportional odds logistic regression model (Hosmer and Lemeshow 2000), has been used in studies of resource selection by considering the frequency of individual relocation points assigned to resource units or
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arbitrarily selected grid cells. However, polytomous logistic regression has not integrated the UD to define use intensity throughout an animal’s range. When used in resource selection, polytomous logistic regression models the cumulative probability that a patch belongs to a use category i(i = 1, . . . , k), (e.g., low, medium, or high use-intensity) as Fhigh (X ) = 1 − Fmed (X ) =
high
(X ) = 1 −
k−1 i=1
i
(X ).
(1)
A link function ensures that a function of the dependent variables is a linear function of the independent variables (North and Reynolds 1996; Hosmer and Lemeshow 2000). In logistic regression, the link function is the natural logarithm of the cumulative odds (Hosmer and Lemeshow 2000). Polytomous logistic regression models the logit as a linear function of the patch resource attributes L i (X ) = αi + β1 x1 + β2 x2 + · · · + βn xn
(2)
L k (X ) = − (αk−1 + β1 x1 + β2 x2 + · · · + βn xn ) ,
(3)
where i = 1, . . . , k − 1, Eq. 2; and i = k, Eq. 3. The equations produced by the model for the different use-intensity categories differ only by their associated intercepts, not by the slopes of the resource attributes.
4 Case study We modeled seasonal resource selection of cow elk (Cervus elaphus) in Custer State Park, South Dakota (Custer). We modeled only diurnal resource selection of one subherd for the spring season for this case study. We defined habitat patches using Custer’s Geographic Information System (GIS). We divided Custer into discrete habitat patches by grouping areas of similar habitat characteristics based on dominant vegetation type (TYPE) (Fig. 1). Based on 116 locations, we calculated fixed kernel density estimates for the subherd using the ‘Kde folder’ (Beardah and Baxter 1995) and “plug-in” methods for bandwidth selection (Wand and Jones 1995), which provided density estimates in the subherd’s range. To reduce potential bias from resource patches with extremely low use on the tails of the UD, we included only the minimum area containing 95% of the probability mass of the estimated UD. We classified use-intensity into a four-level, ordered factor based on the quantiles of the UD and fit seven a priori models. We choose a random subsample of 45 patches, without replacement, from our 402 patch dataset, with each patch maintaining an equal probability of selection. We tested for spatial autocorrelation in the UD values of these 45 patches using Geary’s C with weights equal to the inverse of the distance between the centroids of each patch using the S+ SpatialStats module version 1.5 (Insightful Corp., Seattle, Washington, USA, 2003). We based the significance of spatial correlation on the one-sided P-value, with an α = 0.05, which was computed using the permutation distribution. If the permutation P-value was greater than 0.05, we then ran each of the seven a priori polytomous logistic regression models with the four-level ordered factors as the dependent variable, using the polr() command in the MASS library (Venables and Ripley 2002) of SPlus 6.2 for Windows (Insightful Corp., Seattle, Washington, USA, 2003). We tested for spatial autocorrelation because we expected the level of use among patches to be correlated because kernel methods inherently are spatially autocorrelated (Marzluff et al. 2004) and polytomous logistic
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regression assumes independent observations. The a priori models were associated with specific hypotheses of elk resource selection, including thermal cover, hiding cover, and human disturbance. We saved AIC values (Burnham and Anderson 2002) from each model applied to each spatially uncorrelated subsample of the data. We repeated these steps until we fit our models to 1,250 spatially uncorrelated subsamples. Last, using Tukey multiple comparison test with simultaneous 95% confidence intervals, we compared the fit of all models. The human disturbance model, which considered roads and trails and other factors affecting human disturbance was the most supported model (Table 1). In addition to AIC scores being >3 AIC units from the next nearest model, our multiple comparison test indicated that it was more supported than any other model (Table 2). The second most supported model contained variables associated with hiding cover factors (Table 1); based on multiple comparison results this model was more supported than all but the first thermal cover model.
Table 1 Description and representation of a priori models concerning the effects of forage, thermal and hiding cover, and human disturbance on seasonal resource selection of cow elk subherds in Custer State Park, South Dakota, USA, spring 1994 Hypothesis
Model structurea
1. Thermal 1
β0 + β2−7 (type) + β8 (aspect)
2. Thermal 2
β0 + β2−7 (type) + β8 (aspect) + β9 (slope)
3. Forage 1
β0 + β2−7 (type) + β8 (area) + β9 (aspect)
4. Hiding 1
β0 + β2−7 (type) + β8 (tpa)
5. Hiding 2
β0 + β2−7 (type) + β8 (tpa) + β9 (slope)
6. Human 1
β0 + β2 (roadden) + β3 (trailden)
7. Human 2
β0 + β2−7 (type) + β8 (roadden) + β9 (trailden)
a Type = habitat type (defined by dominant cover type and structural stage), aspect = patch aspect (cosine trans-
formation of average patch aspect in degrees), slope = percent slope, area = patch size (ha), tpa = trees per acre, trails = trail density (km/ha), roadden = road density (roads/km2 ), trailden = trail density (trails/km2 )
Table 2 Ranking of a priori models of cow elk subherd resource selection in Custer State Park, South Dakota, USA, spring 1994. Models were ranked by mean AIC and mean AICi values Model
Mean AIC
Standard
Mean
Standard
Lower CI
Upper CI
error AIC
AICi
error AICi
AICi
AICi
Human 1
128.93
0.10
1.35
0.08
1.19
1.51
Hiding 1
132.33
0.14
4.76
0.10
4.57
4.94
Thermal 1
132.67
0.13
5.09
0.09
4.91
5.28
Hiding 2
132.89
0.15
5.32
0.10
5.12
5.51
Human 2
132.94
0.15
5.37
0.08
5.21
5.52
Forage 1
133.11
0.14
5.54
0.10
5.34
5.74
Thermal 2
133.22
0.14
5.65
0.10
5.46
5.84
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5 Discussion Resource selection studies are conducted to assist managers in making habitat management decisions that favor certain species. In traditional resource use versus availability studies, how and what the researcher defines as “used” and “available” is critically important. Depending on the degree of telemetry and GIS mapping error, use points could be misclassified, potentially biasing results and management prescriptions. Our proposed approach helps overcome this issue by first generating a UD from relocation points, which is then overlaid on resource maps. Thus, the relocation points contribute to UD estimation, but do not require placement in a resource unit. In contrast to use versus available studies, the emphasis instead is placed on which patches receive more or less use instead of defining some patches as used and some available. In use versus availability studies, defining resource availability is inherently difficult (Marzluff et al. 2001; Cooper and Millspaugh 1999) and is problematic when available locations contain both used and unused sites (Keating and Cherry 2004). Given we compare differential intensities of use, we do not define locations that are assumed “available,” which has plagued other resource selection analyses (Marzluff et al. 2001). Instead, the home-range boundary defines the spatial boundary of use, thus eliminating the need to define locations that are assumed available. Despite these advantages, future research should address spatial autocorrelation and develop spatial models for ordered categorical variables. Spatial models are a natural choice for resource selection studies, but have not yet been implemented. As with all analytical techniques, appropriate modeling of resource selection requires certain assumptions to be met. First, we assume that the UD accurately represents use where we did not locate animals. Our ability to meet this assumption increases when we obtain unbiased samples of animal movements throughout the range of space use. To do so requires systematic sampling of animals throughout the range of times, seasons, and activity periods likely to affect their space use (Garton et al. 2001). Second, we must have enough data to compute an unbiased estimate of the UD. Simulation work by Seaman et al. (1999) suggests that accurate kernel-based estimates of home-range size need a minimum of 30, and preferably >50, locations per animal. Examination of relative precision of point-specific density estimates has been examined briefly in the statistical literature (Silverman 1986). Further investigation is needed about the accuracy and precision of use-estimates for specific locations or patches within the home range. Third, we assume that habitat patches are biologically relevant to the species under study. Several approaches to define habitat patches might be used. Habitat patches might be defined based on the smallest ecological unit at which management occurs (e.g., a timber stand) or combinations of multiple map layers (e.g., vegetation type and soils). Difficulties might arise in highly fragmented and linear environments, particularly when use-intensity varies across the habitat patch. We encourage exploration of alternative habitat patch definitions in this situation. Finally, we assume that intensity of use represents the relative importance of habitat patches to the study animal. However, habitat patches containing resources that are essential to survival may be visited only briefly. Conversely, an animal may use a common patch type proportional to its availability, but obtain most of its food from this habitat. Careful interpretation of the ecological context and results are needed in all resource selection studies, regardless of the analysis method. Acknowledgements We thank the Missouri Department of Conservation, Prairie Fork Trust Fund, U.S.D.A. Forest Service North Central Research Station, and the University of Missouri for financial and logistical support. We thank Z. He and T. McCoy for constructive comments.
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References Beardah CC, Baxter MJ (1995) MATLAB routines for kernel density estimation and the graphical representation of archaeological data. Anelecta Prehistorica Leidensia 28. Leiden University, Rapenburg, The Netherlands Burnham KP, Anderson DR (2002) Model selection and inference: a practical information theoretic approach, 2nd edn. Springer-Verlag, New York Cooper AB, Millspaugh JJ (1999) The application of discrete choice models to wildlife resource selection studies. Ecology 80:566–575 Cross CL, Petersen CE (2002) Modeling snake microhabitat from radiotelemetry studies using polytomous logistic regression. J Herptel 35:590–597 Erickson WP, McDonald TL, Gerow KG, Howlin S, Kern JW (2001) Statistical issues in resource selection studies with radio-marked animals. In: Millspaugh JJ, Marzluff JM (eds) Radio tracking and animal populations. Academic Press, Inc., San Diego, pp 209–242 Garton EO, Wisdom MJ, Leban FA, Johnson BK (2001) Experimental design for radiotelemetry studies. In: Millspaugh JJ, Marzluff JM (eds) Radio tracking and animal populations. Academic Press, Inc., San Diego, pp 15–42 Gitzen RA, Millspaugh JJ (2003) Evaluation of least squares cross validation bandwidth selection options for kernel estimation. Wildlife Soc Bull 31:823–831 Hosmer DW, Lemeshow S (2000) Applied logistic regression, 2nd edn. John Wiley and Sons, New York Keating KA, Cherry S (2004) Use and interpretation of logistic regression in habitat-selection studies. J Wildlife Manage 68:774–789 Kernohan BJ, Gitzen RA, Millspaugh JJ (2001) Analysis of animal space use and movements. In: Millspaugh JJ, Marzluff JM (eds) Radio tracking and animal populations. Academic Press, Inc., San Diego, pp 104–106 Loader CR (1999) Local regression and likelihood. Springer-Verlag, New York Manly BFJ, McDonald LL, Thomas DL, McDonald TL, Erickson WP (2002) Resource selection by animals: statistical design and analysis for field studies, 2nd edn. Kluwer Academic Publishers, Dordrecht, The Netherlands Marzluff JM, Knick S, Millspaugh JJ (2001) High-tech behavioral ecology. In: Millspaugh JJ, Marzluff JM (eds) Radio tracking and animal populations. Academic Press, Inc., San Diego, pp 309–326 Marzluff JM, Millspaugh JJ, Hurvitz P, Handcock MS (2004) Resource utilization by an avian nest predator: relating resources to a probabilistic measure of animal space use. Ecology 85:1411–1427 Nams VO (1989) Effects of radiotelemetry error on sample size and bias when testing for habitat selection. Can J Zool 67:1631–1636 North MP, Reynolds JH (1996) Microhabitat analysis using radiotelemetry locations and polytomous logistic regression. J Wildlife Manage 60:639–653 Seaman DE, Millspaugh JJ, Kernohan BJ, Brundige GC, Raedeke KJ, Gitzen RA (1999) Effects of sample size on kernel home range estimates. J Wildlife Manage 63:739–747 Silverman BW (1986) Density estimation for statistics and data analysis. Chapman and Hall, London Terrell GR, Scott DW (1992) Variable kernel density estimation. Annal Stat 20:1236–1265 Venables WN, Ripley BD (2002) Modern applied statistics with S, 4th edn. Springer-Verlag, New York Wand MP, Jones MC (1995) Kernel smoothing. Chapman and Hall, London White GC, Garrott RA (1986) Effects of biotelemetry triangulation error on detecting habitat selection. J Wildlife Manage 50:509–513 Author biographies Chadwick (Chad) D. Rittenhouse is a Ph.D. student at the University of Missouri (MU). He received his B.S. from the University of Wisconsin–Madison and his M.S. from MU, where he studied resource selection of three-toed box turtles. Chad’s dissertation research examines large-scale habitat and resource selection modeling for terrestrial vertebrates. Joshua (Josh) J. Millspaugh is an Associate Professor of Wildlife Conservation and Adjunct Assistant Professor of Statistics at MU. Prior to joining the MU faculty, Josh was a post-doctoral researcher in the School of Fisheries at the University of Washington (UW). Josh received his Ph.D. in Wildlife Science from the UW, his M.S. from South Dakota State University, and his B.S. from the State University of New York’s College of Environmental Science and Forestry. Josh’s research currently focuses on design and analysis of radio tracking studies, and population ecology of mammals.
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Andrew (Andy) B. Cooper is a research assistant professor with a joint appointment in the Department of Natural Resources and the Institute for the Study of Earth, Oceans and Space at the University of New Hampshire. He earned his B.A. in Economics and Mathematical Methods in the Social Sciences from Northwestern University, his Master of Forest Science from Yale University, and his Ph.D. in Quantitative Ecology and Resource Management from the University of Washington. His research interests center on the development and application of statistical and mathematical methods to explore ecological processes and natural resource management. Michael (Mike) W. Hubbard is a resource science supervisor for the Missouri Department of Conservation (MDC). Prior to his current position, Mike was a research biologist for the MDC. He received his B.S. from the University of Missouri, and his M.S. and Ph.D. from Iowa State University. Mike’s research interests include population dynamics, dispersal, and habitat use analysis. Steven (Steve) L. Sheriff is a biometrician with the Missouri Department of Conservation. During his 28 years with the Department of Conservation, he has conducted numerous surveys of constituent groups and worked with resource scientists in their design and analysis of population and habitat studies. He received his B.S. and M.S. in wildlife biology from Colorado State University. Steve is actively involved in the Biometrics Working Group of The Wildlife Society. Robert (Bob) A. Gitzen received his B.S. in fisheries and wildlife from the University of Minnesota and his M.S. and Ph.D. in Wildlife Science from the University of Washington (UW). Currently he is a post-doctoral fellow with the School of Natural Resource at the University of Missouri, working with the National Park Service Northern Great Plains Inventory and Monitoring Network. His interests include quantitative methods in wildlife studies and the ecology and conservation of mammals.
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