For each collared fisher and available camera, we extracted the number of telemetry locations within 250 and 500 m circular neighborhoods centered on camera trap locations (NLocs). We limited the extent to 500 m to ensure that each telemetry location was only counted for one camera trap; cameras were located 1000–1400 m from each other (e.g., within 1-km2 grids); thus, the maximum of 500 m ensured that each telemetry location could only be counted once. We used variable Season to test for differences in detection probability between periods with high (Summer: June–September) and low (Fall/Winter: October–March) food availability.
We used generalized linear mixed-effects models (GLMM; McCulloch et al. 2008) to investigate whether the number of known telemetry locations within 250 and 500 m from camera traps can be used to predict the camera capture probability of collared animals (Year 1: 35 fishers, 208 camera locations, and 547 fisher/camera combinations; Year 2: 35 fishers, 212 camera locations, and 371 fisher/camera combinations). We used a binary response variable indicating detection  (e.g., fisher was photographed at least once at a given camera trap) or nondetection , and ran models with the number of telemetry locations (NLocs), sampling season (Season), and Sex as fixed effects, and individual Fisher as a random effect. Along with additive models, the candidate model set contained the interaction terms Sex × Season and NLocs × Sex. We used Akaike Information Criterion corrected (AICc for small sample size) for model selection and likelihood ratio tests to examine a priori hypotheses (Royle and Dorazio 2008).
Home range analysis
According to the theory behind kernel density estimation of home ranges, the utilization distribution describes the estimated frequency of space use at any location. The probability that an animal is found in a small area is proportional to that area times the utilization density (UD) at that location, and the entire surface of utilization densities is the utilization distribution. Therefore, one would predict that probability of detection at a camera trap should be proportional to the UD at the camera location, which we call the simple model. We expect that the proportionality constant could differ between males and females for behavioral reasons, with males moving across larger areas and thus more likely to find cameras, but potentially spending less time in areas surrounding any one camera.
Alternatively, if camera trap probabilities are not predicted solely by UD at camera locations, then additional variables and/or model forms will provide better predictions. For example, highly heterogeneous home range sizes would yield utilization densities not perfectly correlated to the isopleth percentile. First, if including isopleth percentiles themselves (Isopleth, to 1% accuracy) or a categorical variable for the core versus noncore parts of the home range (Core), separated by the 50% isopleth, provides a better model, it would mean animals tend to find cameras in the central versus peripheral parts of their home range more or less often relative to the time they spend in those areas. Second, we considered that probability of finding a camera could vary nonlinearly with UD, meaning that larger or smaller isopleth values lead to different photo probabilities beyond just the effect of UD. Third, we considered interactions between these variables and Sex.
We considered two types of response variables: (1) a binary response indicating whether each available camera ever captured a photo of each animal and (2) a count response indicating the number of times each camera detected each animal. The binary variable allows modeling of camera trap probabilities without complications due to behaviors induced by camera traps themselves (e.g., “trap-happiness” due to baiting or “trap-shyness”), or other latent factors. The count variable uses more information from the cameras, but at the cost of these additional complications for interpretation.
A set of candidate models was represented by generalized linear (possibly mixed-effects) models (GLM or GLMM). For the binary response, we used a complementary log-log (cloglog) link and binomial or quasibinomial variation. The cloglog link is more appropriate for modeling detection/nondetection as a spatial process, as opposed to the more traditional logit link approach (Baddeley et al. 2010). For count responses, we used a log link with Poisson or quasipoisson variation. Consider a GLM with the linear part describing the log rate of camera captures:
Here, βSex,i takes one value if the animal in observation i is male and another if it is female; log(UDi) is the natural log of the utilization density of the camera for observation i; and βUD is a coefficient for log(UDi). The right-hand side can be extended to other combinations of fixed and random effects.
The rate of camera captures is:
Thus, exp[βSex,i] is the slope for the utilization density. And if the simple model is correct, βUD should be 1. In some candidate models, we estimate βUD to see whether it deviates from 1, while in others, we set it to 1. Setting it to 1 means that the value of 1*log(UDi) is forced into each linear predictor (equation (1)), which is called an offset. Thus, our simple model is denoted as (Sex + offset[log(UD)]).
For count responses, equation (2) gives the expected value. For binary responses, the cloglog link gives the probability of at least one camera capture over a fixed time interval as
The cloglog link itself is ηi = log(−log[1−πi]).
For each type of response variable, we compared a predefined set of hypotheses that included the simple model (Sex + offset[log(UD)]), as well as nonlinear effects of UD (i.e., βUD estimated) and additive and interactive effects of Isopleth or Core, with or without UD. We evaluated these models using model selection with AIC (GLMMs), AICc (GLMs), or QAICc (GLMs with quasilikelihoods; Burnham and Anderson 2002). The GLMMs and quasilikelihoods represent two different ways to accommodate overdispersion (Fieberg et al. 2009). For the GLMMs, we first selected random effects using models with saturated fixed effects and then used the chosen random effect structure to compare different fixed effects (Zuur et al. 2009). We used Fisher as a random effect (n = 26 fishers in Year 1 and n = 18 fishers in Year 2) to account for behavioral heterogeneity between animals [1 | Fisher]. We also examined the use of random effects for Camera [1 | Camera], and Camera × Fisher combinations [1 | Fisher/Camera], to account for unexplained heterogeneity related to camera location only, and camera location within a fisher home range, respectively (Year 1: 131 camera trap locations, 402 fisher/camera combinations; Year 2: 123 camera trap locations, 330 fisher/camera combinations). GLMM fitting was performed using package lme4 (Bates et al. 2012) for R 3.0.1 (R Core Team 2013), and we used package AICcmodavg (Mazerolle 2012) for model selection.