Using occupancy models to understand the

Ecological Applications, 20(1), 2010, pp. 289–302 Ó 2010 by the Ecological Society of America

Using occupancy models to understand the distribution of an amphibian pathogen, Batrachochytrium dendrobatidis MICHAEL J. ADAMS,1,7 NATHAN D. CHELGREN,1 DAVID REINITZ,2 REBECCA A. COLE,2 LARA J. RACHOWICZ,3 STEPHANIE GALVAN,1 BROME MCCREARY,1 CHRISTOPHER A. PEARL,1 LARISSA L. BAILEY,4 JAMIE BETTASO,5 EVELYN L. BULL,6 AND MATTHIAS LEU1 1

U.S. Geological Survey, Forest and Rangeland Ecosystem Science Center, 3200 SW Jefferson Way, Corvallis, Oregon 97331 USA 2 U.S. Geological Survey, National Wildlife Disease Center, 6006 Schroeder Road, Madison, Wisconsin 53711 USA 3 National Park Service, Golden Gate National Recreation Area, Fort Mason, Building 201, San Francisco, California 94123 USA 4 U.S. Geological Survey, Patuxent Wildlife Research Center, 12100 Beech Forest Road, Laurel, Maryland 20708 USA 5 U.S. Fish and Wildlife Service, Arcata Field Office, 1655 Heindon Road, Arcata, California 95521 USA 6 U.S. Forest Service, Pacific Northwest Research Station, 1401 Gekeler Lane, La Grande, Oregon 97850 USA

Abstract. Batrachochytrium dendrobatidis is a fungal pathogen that is receiving attention around the world for its role in amphibian declines. Study of its occurrence patterns is hampered by false negatives: the failure to detect the pathogen when it is present. Occupancy models are a useful but currently underutilized tool for analyzing detection data when the probability of detecting a species is ,1. We use occupancy models to evaluate hypotheses concerning the occurrence and prevalence of B. dendrobatidis and discuss how this application differs from a conventional occupancy approach. We found that the probability of detecting the pathogen, conditional on presence of the pathogen in the anuran population, was related to amphibian development stage, day of the year, elevation, and human activities. Batrachochytrium dendrobatidis was found throughout our study area but was only estimated to occur in 53.4% of 78 populations of native amphibians and 66.4% of 40 populations of nonnative Rana catesbeiana tested. We found little evidence to support any spatial hypotheses concerning the probability that the pathogen occurs in a population, but did find evidence of some taxonomic variation. We discuss the interpretation of occupancy model parameters, when, unlike a conventional occupancy application, the number of potential samples or observations is finite. Key words: amphibian declines; Batrachochytrium dendrobatidis; California, USA; disease; occupancy models; Oregon, USA; Rana spp.

INTRODUCTION Batrachochytrium dendrobatidis (Bd) is a fungal pathogen that has been associated with amphibian mortality and declines in several regions around the world (Skerratt et al. 2007). This chytridiomycete was only recently identified as a disease agent (Longcore et al. 1999), but examination of museum specimens has revealed instances of chytridiomycosis as early as 1938 in southern Africa and from the 1960s in Canada (Ouellet et al. 2005, Skerratt et al. 2007). Chytridiomycosis has been diagnosed for both live and dead amphibians during mass mortality events (Berger et al. 1998, Bosch et al. 2001, Lips et al. 2006) and isolates of Bd cultured from sick anurans can induce disease experimentally in healthy anurans (Berger et al. 2004, Carey et al. 2006). While much recent research has addressed the environmental tolerance, lethality, and transmission of Manuscript received 16 December 2008; revised 29 April 2009; accepted 11 May 2009. Corresponding Editor: J. Van Buskirk. 7 E-mail: [email protected]

Bd (Carey et al. 2003, Johnson et al. 2003, Rachowicz and Briggs 2007), there remains little information on the factors associated with the distribution of Bd within (prevalence patterns) and among (occurrence patterns) amphibian populations. In particular, there is a need to better document the association of occurrence and prevalence of Bd with environmental gradients. Doing so might help us understand and predict the dynamics of infection over different geographic scales. A commonly used approach to examine the occurrence or prevalence of Bd in populations involves testing individual amphibians for the presence of the pathogen using a nonlethal test (Hyatt et al. 2007, Skerratt et al. 2008). Each individual is only tested once but multiple individuals from a population are tested. While straightforward, this approach has problems related to the possibility of false negatives at both the population and individual scales. A related issue is that we can expect the process that results in the presence of Bd at a site to differ from the process that results in the presence of Bd on an amphibian within a site. The former involves vectors capable of dispersing Bd over long distances and the environmental tolerance of the pathogen, while the 289

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latter involves epidemiological factors within a population in which Bd occurs. Because of this, we may wish to separate the influence of environmental gradients on occurrence among sites from their influence on prevalence within sites but must rely on the detection data collected from individual amphibians. Kriger et al. (2007) recognized the problem with false negatives when, as a part of their analysis, they conducted two regressions of climate variables on the prevalence of Bd: one using all the data and one using only the subset of sites where Bd was found. Muths et al. (2008) conducted a similar study and included all sites in their analysis but used a bootstrapping method to assess the sensitivity of their conclusions to the false negatives in their data. Both of these approaches appropriately work around the problem but do not explicitly model the two-step process: occurrence of the pathogen in populations and prevalence of the pathogen within infected populations. A new class of models, called occupancy models (MacKenzie et al. 2006), has been developed to address these types of problems but has seen limited use thus far in studies of pathogens or disease (but see Thompson 2007, Kendall 2008). The purpose of this paper is to explore the use of occupancy models to analyze data from studies of disease occurrence. We discuss how these data differ from typical occupancy data and the implications of those differences. We then use occupancy models to assess the strength of evidence for multiple hypotheses concerning the distribution and prevalence of Bd in Oregon and northern California, USA. The use of occupancy models to analyze observations of disease occurrence Occurrence data for disease or pathogens can take several forms, leading to different potential applications of occupancy models. Thompson (2007) used occupancy models to analyze data that consist of multiple tests for the presence of a pathogen on individual trout. The aim was to estimate prevalence of the pathogen within populations while taking into account the possibility that some tests might miss the pathogen in an infected fish. Our scenario differs in that a subset of individuals are sampled from multiple populations and tested for the presence of a pathogen, but individual animals are tested only once. If the aim is to understand the factors that relate to variation in the occurrence of the pathogen among populations, then the analytical methods will need to address the possibility of false negatives at the population scale. For example, if all samples from a population are negative, this does not mean that the population is negative, unless all resident individuals were tested and tests for the pathogen were completely reliable. In most field settings it is infeasible to census the population and test every individual (Williams et al. 2002). Moreover, if the test sometimes fails to detect a pathogen that is present in or on an individual, then it would be desirable to use analytical methods that

address detectability at the scale of both individuals and populations. Kendall (2008) conceptually outlined this hierarchy of the detection process at multiple scales. We discuss the situation in which the pathogen might be missed at either scale but focus on the situation in which the pathogen might be missed only at the scale of the population. Occupancy models use a maximum likelihood approach to evaluate information from multiple observations of a site to estimate p, the probability that a species is detected during a single observation of an occupied site, and w, the probability a site is occupied (i.e., the probability that the species is present at a site; MacKenzie et al. 2002). Modeling assumptions include: (1) the occupancy status of a site does not change over the course of the sampling; (2) the probability of occupancy is the same for all sites or any heterogeneity is related to covariates; (3) the probability of detection is the same across all sites and observations or any heterogeneity is related to covariates; and (4) detection histories at each location are independent (MacKenzie et al. 2002). In standard applications of occupancy models there is theoretically no limit to the number of times each site can be surveyed; the result of each site survey is a realization of an infinite pool of potential detection/nondetection observations (MacKenzie et al. 2006). Studies designed to assess occurrence patterns of a pathogen among sites may conform to this model if the pathogen is detected independent of the host. For example, Kendall (2008) considered the use of occupancy models to analyze the presence of a pathogen in water containing waterfowl fecal matter. However, pathogens are often detected by testing potential hosts only once (Zarnke et al. 2006, Pearl et al. 2007). In such cases, the number of independent observations will be limited by the number of individuals in the population. Indeed, the number of potential observations in a population (the population size) can be viewed as the number of opportunities that exist for the disease to occur. In this situation, the interpretation of the parameters in an occupancy model will differ from applications in which the potential number of observations is infinite. Under the scenario in which individuals are tested once for the presence of a pathogen, detection probability has two components: p ¼ p0 3 p00 where p 0 is the probability the pathogen is present on an individual in a population in which the pathogen is present, and p00 is the probability of detecting the pathogen on an individual when the pathogen is present on that animal. Understanding factors affecting p 0 is of the most biological interest because it involves epidemiological factors within populations in which the pathogen is present. The p00 component, on the other hand, is affected by a combination of swabbing and laboratory procedures. The real-time Taqman polymer-

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ase chain reaction (PCR) test for presence of Bd is quite sensitive relative to histology (Kriger et al. 2006), but p00 can be ,1.0 and variable (Retallick et al. 2006). Consequently, we used an intense and standardized swabbing method. We then assumed p00 in our study is close to 1.0 and that the majority of variation in p is attributed to variation in the biological component p 0 . This simplifying assumption allowed us to use existing software to estimate p (the product of p 0 p00 ) along with w. We included covariates of p and w in the models to explain any heterogeneity. Any gross lack of model fit that could be related to un-modeled variation in p00 should be detectable in the procedure for assessing model fit (MacKenzie and Bailey 2004) as it would be manifest as variation in p. The interpretation of w in our application also differs from the traditional interpretation. An occupancy model of species occurrence can be thought of as a mixture model for detection probability at a collection of sites. At any one site, detection probability is either p or, when the species is not present at the site, it is 0. The mixture proportion w is the proportion of sites where detection probability is p rather than 0, and w is typically interpreted as the probability that a site is occupied by the target species. However, this interpretation of w is incorrect in studies such as ours in which individuals are sampled from finite populations and tested only once for the presence of a pathogen. Consider a population of two individuals in which both are sampled and both are found to be negative for Bd using a reliable test. We know that this population is negative for Bd because we sampled both members and found them to be negative. Occupancy models typically do not use information on population size (but see Royle and Nichols 2003, Royle 2004) or on the potential number of observations, even if such information is known. Instead it is assumed that the potential number of observations is infinite. Because the size of a population is finite, the potential number of observations is also finite, and the true probability that at least one animal in a population has Bd will tend to be smaller than w. When all modeling assumptions are met, the estimate wˆ is still an unbiased estimate of the mixture proportion w. However, for a population to have at least one animal with Bd, it is insufficient that the random process involving w results in a success for the given population (that is, results in p 6¼ 0 as in a typical occupancy application). It is also necessary that not all of the random processes (Bernoulli trials) involving p 0 fail. If Ni represents the number of individuals in the population at site i, then the probability that all Ni Bernoulli trials fail is ð1  p 0 ÞNi , and the probability that not all Ni trial fail is [1  ð1  p 0 ÞNi ]. Thus, the probability that at least one animal is infected is lower than w by a factor of [1  ð1  p 0 ÞNi ]. For p . 0.05 and Ni . 50, [1  ð1  p 0 ÞNi ] will be near 1.0 and the difference between w and the proportion of populations having at least one infected animal will be small (Appendix A).

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To summarize our use of occupancy models in this study, the observations are individual anurans that were tested for the presence of Bd on their skin. Multiple individuals of a single species were sampled from each site in the study and thus constitute multiple observations, each having the possibility of detecting the presence of Bd. We refer to w as the probability of occurrence of Bd in populations but recognize, as discussed above, that it is actually the mixture proportion that is poorly defined when the numbers of individuals at the sites are small. While the sizes of the populations of anurans that we sampled are not known, we assume that the populations were large enough (i.e., Ni . 50 for all i ) that the difference between our estimated w and the probability a population has at least one infected animal will be negligible (Appendix A). Our assumption about population size is justified by the fact that we sampled both larvae and post-metamorphic anurans. Even a single small egg mass from any of the species we examined would contain hundreds of eggs. The parameter p is less than the prevalence of Bd in a population due to the fact that Bd can be missed on individual anurans on which it is present (Hyatt et al. 2007). Despite evidence that the effectiveness of swabbing tadpoles can be quite variable among different practitioners (Retallick et al. 2006), we will interpret heterogeneity in p as largely reflecting heterogeneity in prevalence for adults. We justify this interpretation due to our intensive swabbing protocol and with goodnessof-fit tests that evaluate the evidence for unexplained heterogeneity. Hypotheses concerning heterogeneity in the occurrence and prevalence of Bd Our objective was to study spatial patterns in the occurrence and prevalence (conditional on occurrence) of Bd in Oregon and northern California, USA. We investigated 13 taxonomic, spatial, climatic, and local environmental factors that were hypothesized to relate to p or w (Table 1). There is evidence of taxonomic variation in susceptibility to chytridiomycosis, and this may translate into taxonomic variation in occurrence or prevalence of Bd (Daszak et al. 2004, Blaustein et al. 2005, Rollins-Smith et al. 2005). We analyzed bullfrogs and native species separately as one way to account for this variation. We also singled out two native species in our data that are thought to be declining and that were common enough for analysis (Rana luteiventris [Columbia spotted frog] and Bufo boreas [western toad]) by using indicator variables to distinguish each of them from the other species in the native species analysis. When we initiated this study, Bd had only been detected in three populations in this region (Pearl and Green 2005) but was known to be common further south (Rachowicz et al. 2006). This suggested a hypothesis that Bd might be spreading through the region in a wave-like fashion. Such a pattern has been postulated in Central America (Lips et al. 2006, 2008)

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TABLE 1. Covariates of the probability of occurrence (w) and detectability conditional on occurrence (p) used in the analyses of detection data for the amphibian pathogen Batrachochytrium dendrobatidis (Bd) in Oregon and northern California, USA. Covariate Design date  larvae crew Taxonomic ralu bubo

Use

Range

p p p

96 to 298 1 to 1 1 to 1

day of the year that the specimen was sampled ¼1 if specimen was larval ¼1 if the swab was collected by a USGS crew

1 to 1 1 to 1

¼1 if specimen was Rana luteiventris ¼1 if specimen was Bufo boreas

p, w p, w

Spatial lat long lat 3 long

w w w

38.41 to 46.17 124.53 to 117.05 5722.66 to 4708.80

Climate tday

w

0.17 to 24.52

w p, w

0.04 to 14.99 4 to 2222

pday elev  Local bull hf stag

w p, w p, w

1 to 1 1.53 to 9.99 1 to 1

Description

decimal degrees of north latitude decimal degrees of longitude the interaction between latitude and longitude estimated average annual number of days with maximum temperature 288C from 1980 to 2003 estimated average annual number of days with rain from 1980 to 2003 meters above sea level ¼1 if site is within the approximated range of the bullfrog as mapped in Jones et al. (2005) human footprint score: an index of human activity (Leu et al. 2008) ¼1 if the site lacks any inlets or outlets

Notes: Design covariates were measured at the scale of observations (individual animals tested for Bd). The other covariates were characteristics of the populations sampled or their location.   Conditional detectability was also modeled as an interaction between date and elev.

and Australia (Laurance et al. 1996). While subsequent testing unrelated to the present study revealed Bd to be more common in our study area than was initially known (Pearl et al. 2007), assessing the evidence that Bd was in the process of spreading through the region remained of interest. To represent this hypothesis, we used models that relate w to latitude, longitude, and the interaction of the two. Laboratory experiments suggest that Bd may be sensitive to warm temperatures, dying at temperatures around 298C (Piotrowski et al. 2004). Consequently some authors hypothesized that Bd occurrence might be lower for populations that experience a relatively high number of days when the maximum air temperature reaches 288C (Drew et al. 2006, Bosch et al. 2007). We hypothesized that air temperature and elevation could relate to w. Despite some information to the contrary (Drew et al. 2006), we also hypothesized that precipitation could relate to w because it can affect movements of anurans among wetlands (Baldwin et al. 2006), which could affect the spread of Bd. Finally, we evaluated three hypotheses involving local factors that might contribute either to the occurrence or spread of Bd. One local factor is the distribution of bullfrogs. Bullfrogs can carry Bd but are less likely to die from chytridiomycosis than some other species (Daszak et al. 2004). If bullfrogs have contributed to the spread of Bd throughout the region, then populations of native amphibians within the range of bullfrogs should be more likely to have Bd than populations outside the range of

bullfrogs. However, we note that this pattern is only predicted if the arrival of Bd is recent relative to the timescale required to spread throughout our study area by other vectors. Another local factor we considered was the extent of human activities around a site. Human activities could serve as stressors or vectors. We represented human activities with a factor called the human footprint (HF) index. This index ranges between 1 and 10 and increases with human activities based on seven input models that delineate a variety of anthropogenic effects, such as roads, agriculture, synanthropic predators, exotic plants, habitat fragmentation, human-induced fires, and energy extraction (Leu et al. 2008). We hypothesized that both w and p might increase with the human footprint score. The final local factor we considered was whether a site was stagnant (i.e., lacking inlets and outlets). We hypothesized that stagnant waters might have higher p if spore concentration in the water affects spread within populations. We hypothesized that stagnant waters might have lower w because overland movement would be required for Bd to be distributed to an isolated site. In addition to these environmental and spatial hypotheses, we evaluated a hypothesis that the process that results in Bd being present on an individual anuran has two steps. The first step is the arrival of Bd in a population (the pattern modeled with w). The second step is the transmission of Bd within a population (the pattern modeled with p).

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METHODS Data collection We sampled populations of native and introduced anurans throughout Oregon and northern California (Fig. 1). We used a county-based sampling frame because we felt this would improve our ability to communicate sampling needs with collaborators. Our objective was to sample one native ranid, one native bufonid, and one bullfrog population in every county where any of these taxonomic groups occur. In the case of Lake, Harney, and Malheur counties (Oregon), we sought to sample two populations of each taxonomic group because these counties are much larger than the other counties in our sampling frame. Sampling sites were any habitat in which the focal amphibians occurred. Pond-breeding amphibians were the focus of our study but two species we sampled (R. catesbeiana and R. luteiventris) were sometimes found in slowmoving stream habitats while a third species (R. boylii ) breeds exclusively in such habitats. Within counties, sites were chosen opportunistically. Typically, we went to known populations and favored sites that were relatively easy to access. In many cases, the easiest site to reach was still remote and only accessible by foot; so, despite its opportunistic nature, our sampling scheme still captured a broad range of potential human influences (see human footprint covariate, Table 1). Our approach emphasized sampling as many populations as possible with the available funds and was used at the expense of achieving broad statistical inference beyond the specific populations sampled. At each site, we captured tadpoles (.30 mm total length), juveniles, or adults by hand or dipnet and kept them in separate containers. We wore new gloves to capture each anuran except in rare cases when multiple individuals were captured together. We rubbed a rayon swab (Medical Wire, MW113; Medical Wire and Equipment, Corsham, UK) over the entire surface of each captured animal. For larvae, the swab was also spun 30 times against the mouth, rubbed 30 times over the ventral surface, and rubbed thoroughly and repeatedly around the hind limbs or limb buds (see Plate 1). For post-metamorphic individuals, the swab was rubbed 30 times over the drink patch and mouth and was also spun between each toe and against any tubercles. We then air dried the swab out of direct sunlight for 30 min and stored it in a pre-labeled, sterile, screw cap 1.5-mL microcentrifuge tube (Fisher Scientific International, Pittsburgh, Pennsylvania, USA) with cryo-babies labels (Diversified Biotech, Boston, Massachusetts, USA). Vials were kept cool and shipped to the laboratory for analysis within two weeks of collection. Each swab was analyzed individually for the presence of Bd using a realtime Taqman PCR assay (Applied Biosystems, Foster City, California, USA) at the U.S. Geological Survey, National Wildlife Health Center (Boyle et al. 2004). Samples were run on an ABI 7000 machine (Applied

FIG. 1. Detection/non-detection of the amphibian pathogen Batrachochytrium dendrobatidis (Bd) at 118 anuran populations in Oregon and northern California, USA, in 2005 and 2006. County boundaries are shown within the region sampled.

Biosystems). All field gear was scrubbed and rinsed to remove any mud or debris and then dipped in or sprayed with a 1% bleach solution between sites. Covariate information All of the covariates used in our analysis are listed in Table 1. We determined latitude and longitude at each site using a Garmin GPS12XL (Garmin, Olathe, Kansas, USA). Elevation was extracted from a 30arcsecond digital elevation model (PRISM Group, Oregon State University, available online)8 using the GPS coordinates. We acquired model-generated, daily climate data from DayMet (available online)9 (Thorntom et al. 1997) from 1980 to 2003 for each survey site and for four additional points offset by 1 km in each cardinal direction from each survey site. Climate data from all five points, excluding any points that overlaid either the Pacific Ocean or the Columbia River, were averaged to get a single value for each climate variable at each site. We calculated the average annual number of days in which the maximum temperature was .288C and the average annual number of days with measurable 8 9

hhttp://www.prismclimate.orgi hhttp://www.daymet.orgi

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rain (i.e., days when the daily mean air temperature was .78C and precipitation was .0.2 cm). We designated sites as being within or outside of the mapped distribution of bullfrogs (Jones et al. 2005) but we did not determine the presence of bullfrogs at the specific sites we sampled. Because of this, the bullfrog covariate simply defines a subset of the region we examined where bullfrogs could have hastened the spread of Bd, if they act as a vector. This covariate does not provide any information about direct contact of native anurans with bullfrogs. To assign a quantitative measure of human activity around our study sites, we created a 5-km buffer around each survey site, ‘‘clipped’’ the buffers so that coastal site buffers included only the terrestrial portion of the buffer, and overlaid these buffers on the human footprint raster. We used the Tabulate Area function in ArcToolbox (ArcGIS 9.2; Environmental Systems Research Institute, Redlands, California, USA) to calculate the proportion of the buffer attributable to each human impact category present within that buffer. We then calculated an average of all human impact scores weighted by the proportion of the buffer they occupy and used this weighted average as a human footprint covariate in our analysis. The Pearson correlations among covariates for both the native and bullfrog data sets are given in Appendix B. Statistical analysis We conducted separate analyses for native anurans and bullfrogs, in each case relating hypothesized covariates to p and w with the logit link function. We separated bullfrogs because the range of bullfrogs was a hypothesized predictor for Bd occurrence in native amphibian populations and because bullfrogs were the only species with enough samples to analyze separately (N ¼ 40 populations). We fit each hypothesized model to the data sets using Program MARK 5.1 (White and Burnham 1999). Previous studies documented that Bd prevalence varies among life stages (Lamirande and Nichols 2002) and seasons (Retallick et al. 2004, Kriger and Hero 2006a, Longcore et al. 2007). We included two design covariates of p in all models to account for heterogeneity associated with these effects: larvae and date (Table 1). Because our native species samples spanned a wide elevation range, we assumed that the effect of date on p would depend on elevation; thus a date 3 elev interaction was also included. We did not use elev in the bullfrog analysis due to its more limited elevation range. We standardized all continuous covariates to X¯ ¼ 0 and standard deviation ¼ 1. For both the native frog and the bullfrog analyses, we began by fitting all of the hypothesized models for p while using a global model structure for w that included as many covariates as possible provided the model converged. The global model structure for the native species included all of the potential covariates of w (Table 1). The bullfrog analysis had convergence

problems due to the smaller sample size. We dropped tday and pday from the global model for bullfrogs because of their correlation with elev. We then arbitrarily dropped lat to arrive at a global structure for w that would converge: w (long, elev, hf, and stag). After ranking the various models for p with the global structure for w, we used the best p structure to fit different hypothesized models for w. We evaluated the strength of evidence for each of our hypotheses based on Akaike’s Information Criteria modified for small samples (AICc) and the resulting Akaike weights (wi ) for each model (Burnham and Anderson 1998). We then conducted a post hoc analysis that involved adding terms to or deleting terms from the best a priori model. This was done to further evaluate patterns that emerged from the a priori analysis and to determine the robustness of the model selection process to variation in the model set. Because unexplained heterogeneity can bias parameter estimates, we conducted a boot-strapped goodness-of-fit test following MacKenzie and Bailey (2004) by using the software Presence 2.2 (available online).10 A significant goodness-of-fit test (P , 0.05) or cˆ . 2.0 are evidence of excessive heterogeneity in p or w not captured by the hypothesized covariates. We also retrospectively examined whether p varied among sampling crews by adding a factor that indicated whether the sample came from the primary crew that did most of the sampling or from a collaborator. The analysis described in the previous paragraphs attempts to model the probability that an individual frog has Bd as a two-step process: the probability of occurrence of Bd among populations and the prevalence of Bd in populations where it occurs. An alternative hypothesis is that the individual frog probability of Bd is variable among populations but that this variation can be attributed to a one-step process, given the collection of covariates. This might occur if Bd moves frequently among populations or if it is present in the environment in forms not detectable by our methods (e.g., if Bd lives independently of amphibians). To determine whether the data better support a one-step or two-step process, we refit all post hoc models for p keeping w fixed at 1. We then recalculated the wi’s for all one-step and twostep models and compared the cumulative wi for each type. The resulting weight for each approach is interpreted as the probability that the given approach (one-step vs. two-step) is the best fit for the data given the collection of covariates considered. RESULTS We obtained between 1 and 50 (X¯ ¼ 16.0) swabs at 118 sites in Oregon and northern California and detected Bd in at least one individual from 48 (40.7%) of those sites (Fig. 1). Of 1933 swabs collected, 220 (11.4%) were positive (Table 2). We obtained positive swabs from 10

hhttp://www.mbr-pwrc.usgs.gov/softwarei

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TABLE 2. Taxonomic composition of samples for the detection of Batrachochytrium dendrobatidis (Bd) in Oregon and northern California, 2005–2006. Sites Species

No. sampled

Bufo boreas Bufo woodhousii Rana aurora Rana boylii Rana cascadae Rana catesbeiana Rana draytoni Rana luteiventris Rana muscosa Rana pretiosa

27 2 16 6 9 40 1 14 1 2

Total

118

Swabs of adults and juveniles Bd positive 6 0 1 3 3 21 1 12 0 1

No. tested

(22.2)

Bd positive

178 12 103 47 41 153 10 198 20 30

(6.3) (50.0) (33.3) (52.5) (85.7)

48 (40.7)

792

Swabs of larvae

15 0 1 5 15 16 2 71 0 8

(8.4)

No. tested

(26.7)

268 5 207 20 101 492 0 37 0 11

133 (16.8)

1141

(1.0) (10.6) (36.6) (10.5) (20.0) (35.9)

Bd positive 1 0 0 0 0 85

(0.4) (0.0) (0.0) (0.0) (0.0) (17.3) ... 1 (2.7) ... 0 (0.0)

87 (7.6)

Note: The proportion of positive samples is given in parentheses when at least five were sampled (n . 4 sites or swabs).

every species except R. muscosa, which was only sampled at one site, and B. woodhousei, which was sampled at two sites. Native anurans We detected Bd in at least one individual from 25 (32%) of the 78 native populations sampled. We did not analyze data for species with five or fewer populations in our sample. Of the 72 populations analyzed, we detected Bd in 23 (31.9%) but estimated that it occurred in wˆ ¼ 53.4% of the populations (95% CI ¼ 36.6–69.4%) based on the top-ranked model. We found evidence that the

probability of Bd occurrence differed between R. luteiventris and other species (Table 3). Based on the populations that we sampled, we estimated that the odds of Bd occurrence was 4.2 (95% CI ¼ 1.4–12.4) times greater for R. luteiventris populations than for other native anuran populations (Table 4). The cumulative weight for models that included taxonomic covariates was wcum ¼ 0.816, which offers moderate support for the hypothesis that differences among species best explain the heterogeneity in w. It does not rule out the possibility of a wave-like pattern of spread corresponding with one of the spatial models but there was little support for this

TABLE 3. Model selection statistics for a priori models relating the conditional detectability of Batrachochytrium dendrobatidis ( p) and probability of occurrence in populations (w) to environmental covariates for native anurans in Oregon and northern California, USA.

Model

AICc

DAICc

AICc weights

Model likelihood 

No. parameters

Deviance

487.328 489.754 492.417 492.445 494.063 494.136 494.788 495.841 496.195 496.491 497.110 497.598 498.663 605.296

0.000 2.426 5.089 5.117 6.735 6.808 7.460 8.512 8.867 9.163 9.782 10.270 11.334 117.968

0.63 0.19 0.05 0.05 0.02 0.02 0.02 0.01 0.01 0.007 0.00 0.00 0.00 0.00

1.000 0.297 0.079 0.077 0.035 0.033 0.024 0.014 0.012 0.010 0.008 0.006 0.004 0.000

8 9 10 8 8 8 8 8 7 8 8 8 8 2

469.241 469.107 469.134 474.358 475.976 476.049 476.701 477.754 480.595 478.404 479.023 479.511 480.576 601.136

504.472 521.381 522.871 524.142 525.301 600.409 605.296

0.000 16.909 18.399 19.670 20.829 95.937 100.824

1.00 0.00 0.00 0.00 0.00 0.00 0.00

1.000 0.000 0.000 0.000 0.000 0.000 0.000

17 17 16 17 17 12 2

460.272 477.181 481.953 479.942 481.101 571.609 601.136

w p(best), w(ralu) p(best), w(ralu, bubo) p(best), w(lat 3 long) p(best), w(long) p(best), w(pday) p(best), w(bull) p(best), w(stag) p(best), w(bubo) p(best), w() p(best), w(tday) p(best), w(elev) p(best), w(hf ) p(best), w(lat) p(), w() p p(larv, date, elev, p(larv, date, elev, p(larv, date, elev, p(larv, date, elev, p(larv, date, elev, p(), w(global) p(), w()

date date date date date

3 3 3 3 3

elev, hf ), w(global) elev, ralu), w(global) elev), w(global) elev, stag), w(global) elev, bubo), w(global)

Notes: Models listed under w all used the best model for p [p(larv, date, elev, date 3 elev, hf )]. Models listed under p all used the global model for w [w(ralu, bubo, lat, long, elev, tday, pday, bf, hf, stag)]. For both model sets, a model with no covariates, p(), w(), is shown for comparison. See Table 1 for explanations of covariate abbreviations.   Model likelihood is exp[1/2(DAIC)].

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TABLE 4. Beta estimates from the best a priori model for native anurans. Parameter

Covariate

Beta

SE

LCI

UCI

Scale

p p p p p p w w

intercept larv date elev date 3 elev hf intercept ralu

3.107 2.432 0.663 1.230 2.366 0.872 1.053 1.430

0.402 0.388 0.257 0.773 0.853 0.189 0.555 0.554

3.895 3.193 1.167 2.744 0.695 0.501 0.035 0.343

2.318 1.672 0.158 0.285 4.037 1.243 2.141 2.516

NA NA 29.838 710.102 145 125.449 2.023 NA NA

Notes: LCI and UCI are the lower and upper 95% confidence limits on beta. Because continuous covariates were standardized to X¯ ¼ 0.0, SD ¼ 1.0, it is necessary to divide beta by the standard deviation of the original covariate (Scale) to produce a beta estimate on the scale of the original (unstandardized) covariate. See Table 1 for explanations of covariate abbreviations. The abbreviation ‘‘NA’’ stands for ‘‘not applicable.’’

alternative (wcum ¼ 0.10 for models that included lat or long). Of the three spatial models, the longitudinal model, w(long), and the interaction model w(lat 3 long) had nearly identical rankings. Because R. luteiventris only occurs in the northeastern half of our study area, it is difficult to differentiate any spatial trends from the taxonomic pattern but our analysis gives much more support to the taxonomic variation hypothesis than to the spatial spread hypothesis. Contrary to the hypothesis about bullfrogs spreading Bd, the model containing w(bull) estimated that the odds of a native anuran population having Bd present, if anything, decreased within the range of bullfrogs by a factor of 0.49 (95% CI ¼ 0.25–0.96) and had a low model weight (w ¼ 0.021). Other native species models had little support, as indicated by low model weights and confidence intervals for odds ratios that included 1.0 (Table 3). Models involving air temperature, human activities, elevation, and latitude had weaker support than a model with no covariates of w (Table 3). Our analysis strongly supported the collective inclusion of larv, date 3 elev, and hf as covariates of p and gave no support for the inclusion of additional covariates that we hypothesized might relate to conditional prevalence (Table 3). The influence of elevation on p was not consistent within our sampling season. There was no elevation trend or a slightly declining trend in conditional prevalence earlier in the year, but an increasing elevation trend was evident later in the year (Fig. 2). Stated differently, there was a declining seasonal trend at lower elevations and an increasing seasonal trend at higher elevations. We note that some of the higher elevations were only available for sampling later in the year. We interpret this interaction to indicate that we encountered different stages of the previously documented seasonal cycle in prevalence at different elevations (Retallick et al. 2004, Kriger and Hero 2006b). The conditional individual animal odds of Bd decreased by a factor of 0.10 (95% CI ¼ 0.04–0.20) for larvae compared to post-metamorphic native anurans. The conditional individual animal odds of Bd increased by a factor of 2.37 (95% CI ¼ 1.63–3.43) for each additional point on the 10-point human footprint index.

The post hoc model comparison did not give any reason to question the ranking of the a priori models or suggest any modifications to those models. In particular, the continued inclusion of HF as a predictor of p was strongly supported (w ¼ 1.00). We did not find evidence of excessive unexplained heterogeneity in the model (P ¼ 0.10, cˆ ¼ 1.38) and adding crew as a covariate of p did not improve the model (DAICc ¼ 1.43). When all of the two-step models that estimated w were compared to the one-step models in which w was fixed at 1, the cumulative weight of two-step models was wcum ¼ 1.00 and the cumulative weight of the one-step models was wcum ¼ 0. Given the collection of covariates, the pattern of positive Bd swabs was clearly better explained as a two-step process.

FIG. 2. The effect of date and elevation on estimated Batrachochytrium dendrobatidis (Bd) detectability ( p) for adult native anurans at sites where Bd occurs in Oregon and northern California, USA. Estimates are based on the best occupancy model (Table 4). We interpret trends in detectability to largely reflect changes in prevalence, but p is also influenced by the probability of detecting Bd on an individual that has Bd. True prevalence is likely to be greater than estimated detectability.

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297

TABLE 5. Model selection statistics for a priori models relating the conditional detectability of Batrachochytrium dendrobatidis ( p) and probability of occurrence in populations (w) to environmental covariates for bullfrogs (Rana catesbeiana) in Oregon and northern California, USA.

AICc

DAICc

AICc weights

Model likelihood 

p(best), w(lat) p(best), w() p(best), w(elev) p(best), w(pday) p(best), w(tday) p(best), w(stag) p(best), w(long) p(best), w(hf ) p(), w()

493.576 494.038 495.612 496.200 496.658 496.680 496.680 496.816 509.223

0.000 0.462 2.035 2.624 3.082 3.103 3.104 3.240 15.647

0.31 0.24 0.11 0.08 0.07 0.06 0.06 0.06 0.00

1.000 0.794 0.361 0.269 0.214 0.212 0.212 0.198 0.000

6 5 6 6 6 6 6 6 2

479.031 482.273 481.066 481.655 482.113 482.134 482.134 482.270 504.899

p(larv, date, stag), w(global) p(larv, date), w(global) p(larv, date, hf ), w(global) p(larv, date, elev), w(global) p(), w() p(), w(global) p(larv), w(global)

501.305 506.768 508.861 509.099 509.223 514.094 515.578

0.000 5.463 7.556 7.794 7.918 12.789 14.272

0.89 0.06 0.02 0.02 0.02 0.00 0.00

1.000 0.065 0.023 0.020 0.019 0.002 0.001

9 8 9 9 2 6 7

477.305 486.123 484.861 485.099 504.899 499.549 498.078

Model

No. parameters Deviance

w

p

Notes: Models listed under w all used the best model for p [p(larv, date, stag)]. Models listed under p all used the global model for w [w(long, elev, hf, stag)]. For both model sets, a model with no covariates, p(), w(), is shown for comparison. See Table 1 for explanations of covariate abbreviations.   Model likelihood is exp[1/2(DAIC)].

Bullfrogs We detected Bd in 21 (53%) of the 40 bullfrog populations that we sampled but estimated that Bd occurred in wˆ ¼ 66.4% of the populations (95% CI ¼ 42.9–83.8%) based on the top-ranked model. Model comparison did not strongly support adding any covariates of w but allowed a weak possibility that latitude should be included (Table 5). This model estimated that the odds of occurrence increased by a factor of 1.61 (95% CI ¼ 0.94–2.78) for each degree of north latitude (Table 6). All other models that included covariates of w had lower AICc and explained little deviance relative to the model that only included covariates of p. The lack of support for covariates of w is consistent with the analysis for native species that, at the scale of populations, only gave strong support for the hypothesis that occupancy varies among taxa.

Model comparison strongly supported the collective inclusion of date and larv as covariates of p and gave moderately strong support for also including stag (Table 5). The best model estimated that the conditional individual animal odds of Bd decreased by a factor of 0.75 (95% CI ¼ 0.61–0.93) every 30 days and decreased by a factor of 0.59 (95% CI ¼ 0.41–0.85) for stagnant ponds compared to water bodies with an inlet or outlet. The best model also estimated that the conditional individual animal odds of Bd increased by a factor of 1.09 (95% CI ¼ 0.77–1.53) for larvae compared to postmetamorphic anurans, though the confidence interval included 1.0. The post hoc data exploration favored dropping larvae as a covariate of p. Dropping larvae increased the support for latitude as a covariate of w but the effect was still weak (w ¼ 0.517). We did not find evidence of excessive unexplained heterogeneity in the best model (P

TABLE 6. Beta estimates from the best a priori model for bullfrogs. Parameter

Covariate

Beta

SE

LCI

UCI

Scale

p p p p w w

intercept larv date stag intercept lat

1.788 0.085 0.381 0.522 0.680 1.007

0.224 0.173 0.142 0.186 0.493 0.585

2.228 0.255 0.660 0.886 0.287 0.139

1.348 0.425 0.102 0.157 1.647 2.152

NA NA 39.572 NA NA 2.107

Notes: LCI and UCI are the lower and upper 95% confidence limits on beta. Because continuous covariates were standardized to X¯ ¼ 0.0, SD ¼ 1.0, it is necessary to divide beta by the standard deviation of the original covariate (Scale) to produce a beta estimate on the scale of the original (unstandardized) covariate. See Table 1 for explanations of covariate abbreviations. The abbreviation ‘‘NA’’ stands for ‘‘not applicable.’’

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FIG. 3. Probably of detecting Batrachochytrium dendrobatidis (Bd) in a population in which it occurs when the sample consists of: all adults, at 100 m elevation, sampled in April (early); all adults, at 100 m elevation, sampled in August (late); all adults, at 1000 m elevation, sampled in August; or all larvae, at 100 m elevation, sampled in April. The estimated probability of detecting Bd where it occurs is 1  (1  p)n, where n is the number of individuals tested and p comes from the best model for native species (Table 4). We used a human footprint score of 5 for these estimates. These projections are averages based on our data for native ranids and bufonids in Oregon and northern California. Probabilities may differ for other species and regions. Moreover, there may be considerable variation in these probabilities among sites related to other factors, such as human activities.

¼ 0.48, cˆ ¼ 0.37), and adding crew as a covariate of p only caused a slight decrease in AICc (0.19). When all of the post hoc models that estimate w (two-step models) were compared to the same set of models but with w fixed at 1 (one-step models), the cumulative weight of the two-step models was wcum ¼ 0.985 compared to wcum ¼ 0.015 for the one-step models. This is consistent with the previous conclusion that the two-step process for the spread of Bd is better favored. DISCUSSION Analyzing hypotheses concerning the occurrence and prevalence of diseases and pathogens has historically been problematic. At the population or site scale, the likelihood of false negatives (e.g., failing to detect a pathogen that is present) can significantly bias parameter estimates and conclusions (Muths et al. 2008). At the scale of individuals that may or may not be infected, there is a problem with zero inflation in what would otherwise be a logistic regression problem: some animals don’t have the pathogen due to an individual-scale process while other animals don’t have the pathogen because they were simply never exposed. Efforts to minimize the rate of false negatives at the population scale results in large numbers of swabs that are expensive to collect and analyze (Skerratt et al. 2008).

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Occupancy models offer at least a partial solution to all of these problems with the caveat that the interpretation of parameter estimates can differ somewhat from a conventional occupancy study (see Introduction: The use of occupancy models . . .). An occupancy analysis is useful for evaluating patterns of occurrence but is not as useful when it is desirable to know the true state of each population. For the later objective, it is necessary to determine a sample size that is large enough to convincingly conclude that Bd is rarely missed (e.g., Skerratt et al. 2008). There is likely to always be some rate of false negatives, and occupancy models can be used to measure that rate and, if necessary, to offset the bias that false negatives cause to other parameter estimates. Occupancy models can also be used to calculate the number of samples necessary to achieve some probability that Bd is detected given certain characteristics of the sample. In our study, p depended in part on a strong interaction between elevation and date (Fig. 2). The number of samples needed to achieve some threshold probability of a false negative at a site will depend on this interaction as well as other features of the sample and the site. Based on our data, if the objective was to determine whether Bd occurs at a particular site or population, a sample of 20 adults might be deemed sufficient at a low-elevation site in April or at a midelevation site in August but would probably be considered inadequate for a low-elevation site in August (Fig. 3). Moreover, sampling any reasonable number of larvae appears inadequate for native anurans in our area but note that sampling larval bullfrogs was slightly more productive than sampling post-metamorphic bullfrogs (Table 6). Bullfrog larvae tend to be older and larger, which could increase prevalence or detectability. The use of occupancy models might allow the target number of swabs to decrease in studies designed to document regional patterns of occurrence (e.g., Skerratt et al. 2008) if it is not essential that the probability of false negatives at each site or population be minimized. Because of the recent discovery of Bd and the role of this pathogen in amphibian declines (Skerratt et al. 2007), there is great interest in understanding and predicting its distribution and prevalence (Ron 2005, Drew et al. 2006, Kriger and Hero 2007). The pathogen is thought to be spreading and does not generally appear to occupy every available amphibian population in regions where it occurs (Longcore et al. 2007, Pearl et al. 2007, Skerratt et al. 2007). Our data support the hypothesis that this pathogen, though widely distributed, does not occur in every potential population. This is important because it emphasizes the need to understand heterogeneity in its occurrence as well as the need to take steps to prevent its spread, even within regions where it is widely distributed. This finding emphasizes the value of occupancy models or perhaps other methods to account for what is likely to be a two-step process: step 1 is the spread of Bd to a site or amphibian population

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299

PLATE 1. Collecting a sample from a Cascades frog (Rana cascadae) to test for the presence of the fungal pathogen, Batrachochytrium dendrobatidis. Photo credit: S. Unger, USGS.

and step 2 is the spread of Bd within an amphibian population in which Bd occurs. Despite the appeal of occupancy models for analyzing occurrence and conditional prevalence of a disease or pathogen, there are several complications that should be considered. First, the finite N problem exists in any real study to some extent and adds uncertainty to the interpretation of w. It is exacerbated by a low p 0 (the probability that the pathogen occurs on an individual in a population in which it is present) and the value of this parameter will generally be unknown a priori. Under a scenario in which the pathogen can be present at a site (e.g., in the water) yet no individuals infected, then the finite N issue is no longer a problem. The sampling scenario becomes closely aligned with the original occupancy model application but, unlike in our study, it must be understood that sites with zero individuals with the pathogen could still have the pathogen present. A second potential complication is that the pathogen may sometimes be missed on individual animals on which it is present ( p00 , 1, where p00 is the probability that the pathogen is detected on an individual conditional on presence). Kendall (2008) considers methods that could estimate both p00 and p 0 that rely on collecting repeated samples at both the scale of individuals and the scale of populations. When p 0 can be estimated separately from p00 , or using supplemental data, then corrections for the finite population problem can be made using w[1  ð1  p 0 ÞNi ]. Using the likelihood in Kendall (2008), covariates could be applied to each of the three parameter types, enabling elucidation of factors affecting p00 , p 0 , and w. When p00 is close to 1,

then the MacKenzie-style occupancy model is sufficient and software is available to facilitate the analysis. This is especially reasonable when there is little concern that N or p 0 is low enough to cause substantial bias. We did not collect multiple swabs per individual but we did retrospectively consider whether variation in swabbing efficiency among investigators contributed to heterogeneity in p. There was not strong support for including crew as a covariate of p but the results did not rule out the hypothesis that such heterogeneity exists. Goodnessof-fit tests suggested that any heterogeneity in p00 was minor, but it would be prudent for future studies to design surveys in a manner that either controls for variation among investigators or allows a more efficient comparison. Development of likelihood models and software that would enable separate estimation of p 0 and p00 should also consider the potential for false positives. This can occur due to contamination of PCR tests for the presence of an organism including tests for Bd (Skerratt et al. 2008). Royle and Link (2006) suggest a possible approach to accommodate false positives if the relative magnitude of false positive and false negative rates is known, but this was not done in a hierarchical manner that could separate p 0 and p00 . A common problem with wildlife disease studies is the possibility that infected individuals have a capture probability that is different from uninfected individuals due to changes in behavior. This type of nonrandom process would likely bias estimates of p and thus w (Jennelle et al. 2007) and was not dealt with in our study.

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We found strong support for the hypothesis that human activities on the landscape might be related to the conditional detectability of Bd in the native anurans we examined. Assuming that nonrandom heterogeneity in p00 was low, this pattern likely reflects an association between prevalence and human activities. We did not find strong support for this pattern in bullfrogs. The index of human activities that we used in our models incorporates many forms of human influences on the landscape but is strongly influenced by populated areas (1 person/ha), secondary roads, and agricultural land (Leu et al. 2008). Thus, much of the variation in the human footprint score reflects a gradient from wild and inaccessible areas to areas with either high agricultural use or high human densities. Our hypothesis was based on the idea that various human activities might stress anuran populations in a way that makes them more vulnerable to chytridiomycosis. There is evidence for such a mechanism in one of our sampled amphibians, Rana boylii (Davidson et al. 2007). For that species, pesticides used in urban and agricultural areas may inhibit immune defenses and increase susceptibility to disease (Davidson et al. 2007). We found little support for any of the hypotheses concerning the probability of occurrence for Bd other than a difference in occurrence rates between R. luteiventris populations and other species of native anuran. There is some possibility that this apparent taxonomic difference is actually a longitudinal trend but our analysis gives little support to this interpretation (wcum ¼ 0.099; Table 3). Moreover, a separate analysis of Bd in bullfrog populations, which are widely distributed in our study area, gave only weak support (w ¼ 0.063; Table 5) for a longitudinal trend. The lack of a temperature effect for both bullfrogs and native anurans may be explained by the relatively moderate climate of our study area. A threshold temperature of ;288C has been found to be important for Bd survival (Piotrowski et al. 2004), and our sites averaged only 6.3 days above that threshold for the native species populations and 12.3 days for the bullfrog populations. While Bosch et al. (2007) found some evidence of a temperature effect in a climate roughly similar to our study area, Drew et al. (2006) suggested stronger negative effects on Bd when the average summer temperature is .308C, much warmer than our area. Batrachochytrium dendrobatidis is widely distributed across all parts of our study area at all elevations examined. Our study estimated that Bd occurred in 53.4% of the native anuran populations examined, but we note that the proportion of sites, as opposed to populations, with Bd may be higher because we cannot rule out that Bd may be present on other, unsampled species of amphibian that were sometimes present at our sites or perhaps on other organic or inorganic substrates. This later possibility has not been confirmed in nature but Bd can survive temporarily on non-amphib-

ian substrates in the laboratory (Johnson and Speare 2005). While there is evidence that Bd is a spreading pathogen and that, in some cases, its arrival has negative consequences for amphibian populations (Lips et al. 2006, Skerratt et al. 2007), there is also information that Bd is widespread in some areas where there is currently little evidence of harm (Longcore et al. 2007, Pearl et al. 2007) or where Bd has become endemic in apparently stabilized populations (Retallick et al. 2004, McDonald et al. 2005). Documenting the consequences of Bd’s arrival in a naı¨ ve population is rare, and often little information exists to estimate how long Bd has been in a particular region (Ouellet et al. 2005). Few of the populations we examined had long-term population data and none had data on the arrival of Bd, so it is impossible to know how uninfected populations responded following Bd introduction/colonization. Despite detecting Bd in many of our sampled populations, we did not encounter any animals with outward signs of infection. While this pattern appears at odds with a hypothesis that Bd has severe negative consequences for the anurans in our area, we note that it would be unlikely to observe symptomatic individuals without more intensive monitoring. Moreover, we cannot rule out the possibility of past or future declines related to Bd. Other than R. pretiosa and R. boylii, there is little evidence of major declines in the amphibians we sampled (Wente et al. 2005). We know that Bd is geographically and taxonomically widespread in our study area but was only estimated to occur in about half of the populations we examined. The pattern of occurrence did not correlate with any hypothesized environmental factors but Bd detectability increased markedly with human influence on the landscape, and this may reflect an increase in prevalence. This correlation of p with the human footprint index merits further investigation and occupancy models are a useful tool for examining such patterns in the field. While currently available software does not allow two scales of detectability, future studies should consider collecting multiple swabs from at least some individuals and explore the use of Kendall’s (2008) likelihood model to separate occurrence and prevalence from the detectability of Bd on individuals. ACKNOWLEDGMENTS We thank Jim Nichols and Darryl MacKenzie for advice on this application of occupancy models. We thank P. Stephen Corn, Blake Hossack, and Hardin Waddle for providing critical reviews of this paper. We thank Cynthia Tait for providing assistance with locating sites and collecting swabs. This work was funded by the U.S. Geological Survey’s Amphibian Research and Monitoring Initiative. Use of trade names is for informational purposes only and does not constitute endorsement. LITERATURE CITED Baldwin, R. F., A. J. K. Calhoun, and P. G. DeMaynadier. 2006. Conservation planning for amphibian species with

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APPENDIX A Ratio of the probability of disease occurrence in a finite population, w[1  (1  p 0 )N ], to the probability of disease occurrence in an infinite population, w (Ecological Archives A020-006-A1).

APPENDIX B The Pearson correlation between covariates used in the analysis of occupancy patterns for Batrachochytrium dendrobatidis (Ecological Archives A020-006-A2).