We proposed (Methods in Ecology and Evolution, 2013, 4) a model for combining telemetry data with spatial capture–recapture (SCR) data that was vigorously criticized by Efford (Methods in Ecology and Evolution, 2014, 000, 000). Efford's main claim was that our encounter probability model was incorrect, and therefore our R code and simulation results were wrong.

In fact, our encounter probability model is correct under the Poisson point process model that we used as a basis for our integrated model. On the other hand, the basis for Efford's claims clearly rest on the assumption of an alternative model which, while possibly useful, is distinct from that analysed in Royle et al. (Methods in Ecology and Evolution, 2013, 4).

A key point of Royle et al. (Methods in Ecology and Evolution, 2013, 4) was that active resource selection induces heterogeneity in encounter probability which, if unaccounted for, should bias estimates of population size or density. The models of Royle et al. (Methods in Ecology and Evolution, 2013, 4) and Efford (Methods in Ecology and Evolution, 2014, 000, 000) merely amount to alternative models of resource selection, and hence varying amounts of heterogeneity in encounter probability.

We proposed (Royle et al. 2013) a model for combining telemetry data with spatial capture–recapture (SCR) data that was criticized by Efford (2014). Efford wrote (point 1 in the abstract):

Royle et al. (2013) proposed a spatially explicit capture–recapture (SECR) model in which an animal's usage of a site, hence its probability of detection, depends on a function of site-specific covariates normalized using a weighted sum of such values across the animal's home range.

This statement is inaccurate as we did not propose such a model. As a result, the claims made by Efford related to errors in the simulation study of Royle et al. (2013) are incorrect. The specific model for probability of detection that Efford (2014) refers to is this model:

where C = ∑_{x}λ(x|s) (see below). This model is the direct implication of eqns 1 and 2 in Efford (2014). However, this model appears nowhere in our paper either directly or implied. Our key disagreement with Efford (2014) is his attribution of this encounter probability model to us and his subsequent claim that our analyses were in error because the model was not used in our R code.

Our paper developed a model for combining telemetry and SCR data based on a Poisson point process to describe space usage in the vicinity of an animal's home range centre (s). We assumed a model for space usage in which the Poisson intensity of space usage has the form:

λ(x|s)=exp(α0−α1dist(x,s)2+α2z(x))(eqn 2)

where x∈X indexes the two-dimensional region around s, and z(x) is some covariate measurable at any point in the two-dimensional region. When α_{2}=0, this is the kernel of a bivariate Gaussian probability density function with α_{1}=1/(2σ^{2}). We considered a discrete formulation of the model such as relevant to raster covariates. In discrete space, over any prescribed period of time, the frequency of use of some individual with activity centre s is well approximated (to within numerical error induced by discretizing space) by a Poisson random variable with mean proportional to λ(x|s), as long as the raster cell size is small relative to σ 1. The assumption of the Poisson model for space usage is noted on page 523 of Royle et al. (2013) under the heading ‘Poisson Use Model’ where we noted also that the classical multinomial RSF model for telemetry data is naturally consistent with the Poisson model of space use if one conditions on the total sample size, which is a design feature of telemetry studies (see below).

The Poisson use model implies a model for telemetry and SCR data

The Poisson point process model of space use is the fundamental model underlying the likelihood we proposed for combining telemetry and SCR data. From the Poisson point process model, one can immediately deduce the implied models for both telemetry and SCR data. In particular:

For SCR data, if we imagine that individual encounters in traps are a binomial thinning of the underlying Poisson point process, with thinning rate λ_{0}, then the probability that an individual is encountered in a trap located at x is:

p(x|s)=1−exp(−λ0λ(x|s)).eqn 3(eqn 3)

Clearly, in this case of random thinning, the parameters α_{0} and λ_{0} are confounded, and we denote their product by p_{0}=λ_{0}α_{0}. It is the parameter p_{0} which is estimated by SCR models. Thus, under this Poisson use model, p_{0} represents the compounding of the intensity of space usage with a trap efficiency process (being the binomial thinning). Equation 3 resembles the model proposed by Efford (2014) (eqn 1) except for the term involving log(C). Our model, that without log(C), is justified under the Poisson point process.

For spatial location data arising by telemetry, under the Poisson space usage model, conditional on a fixed number of use observations, the conditional probability mass function of use frequency in some pixel x is multinomial with probability:

π(x|s)=λ(x|s)∑xλ(x|s).

In the case of telemetry data, it is natural (even necessary) to express the model conditional on the total sample size, because the number of telemetry locations is fixed by design, and not the result of resource selection, movement, space usage or other biological processes.

The relationship between the underlying Poisson point process model and the two observation models used by Royle et al. (2013) to form the combined likelihood is summarized in Fig. 1.

A special case of the general model arises when there is no resource selection, that is when α_{2}=0. In this case, the two data sources share a single parameter (α_{1}), and both models are consistent with a bivariate Gaussian model of space usage. Therefore, telemetry data can be used to provide direct information about the scale parameter of the SCR encounter probability model (Sollmann et al. 2013). More generally, the choice of any model of encounter probability can be regarded as being proportional to the intensity function of some Poisson point process; therefore, a stationary and isotropic model of home range is implied (Royle et al. 2014, sec. 5·4).

Efford's model

A referee of this Rejoinder noted that if one assumes individual space usage is governed by a Poisson point process with intensity:

log(λ(x|s))=α0−α1dist(x|s)2+α2z(x)−log(C)

then this is also consistent with the multinomial resource selection model shared by both Royle et al. (2013) and Efford (2014) while at the same time providing a justification for Efford's encounter probability model (eqn 1). Thus, one technical view of the distinction between the models of Royle et al. and Efford hinges on whether the Poisson point process governing space usage has the additive offset log(C). As noted by Efford (2014), this is not constant but, rather, depends in a complicated way on the parameters α_{0}, α_{1} and α_{2}, and the variable z(x) for all values of x in the vicinity of s. This ‘makes the SECR detection model unwieldy, and substantial additional processing is required for each evaluation of the likelihood.’ (Efford 2014, p. 2). For a given data set, we believe that one could formally compare and evaluate these directly using AIC and formal methods of evaluating model goodness-of-fit.

A key feature of Efford's model is that the amount of space used by an individual will depend on the landscape in the vicinity of its home range, so that animals with more available resources should use less space and vice versa, and this affects the SECR encounter probability. Indeed, the ‘effective intercept’, say α0∗, of the Efford model is α0∗=α0−log(C) which indicates as much. Note that

C=∑xexp(α0−α1d(x|s)2+α2z(x))(eqn 4)

and therefore, in homogeneous space, that is when α_{2}=0, then the Efford model and that of Royle et al. (2013) are equivalent in the sense that the baseline SECR encounter rate is constant under both models. In this case, C is no longer a function of z(x), but rather just adds a constant offset on the log-encounter rate scale: C≈2πσ^{2} (replace ≈ with = in continuous space) where σ2=1/(2α1) is it is the normalizing constant of a bivariate normal density.

Summary

The likelihood proposed by Royle et al. (2013) for combining telemetry data and SCR data is based on a Poisson point process model for space usage. That assumption implies a multinomial observation model for telemetry data, by conditioning on the total telemetry sample size, and, by random thinning, it implies a Bernoulli SCR encounter model with probability of encounter p(x|s) = 1− exp (−λ_{0}λ(x|s)) for some trap located at x. The combined likelihood in Royle et al. (2013) correctly uses both of these component models. Therefore, we stand by the analyses we reported there. In particular, (1) the R functions provided in appendix 2 of Royle et al. (2013) for carrying out the likelihood analysis for a combined likelihood are correct under the Poisson point process intensity model (eqn (eqn 2)); and (2) We stand by our simulation study which illustrates that failure to account for resource selection can induce bias in the estimator of population size N and density. We do, however, caution that our simulation results were very clearly restricted to a limited set of parameter values, and we did not and do not mean to imply that bias of that level is expected to hold uniformly across any model of resource selection. When resource selection is less intense, we expect that less bias in estimating N should arise and, under alternative models of resource selection such as that considered by Efford (2014), the simulation results of Royle et al. (2013) should not expect to hold.

As Royle et al. (2013) noted, active resource selection induces individual heterogeneity in encounter probability, and, as in classical capture–recapture, failure to account for this heterogeneity produces bias in estimators of population size depending on the amount and nature of heterogeneity (Dorazio & Royle 2003). Efford (2014) did not refute this key point. Instead, he suggested an alternative model for that resource selection-induced heterogeneity. His simulation results do not contradict those of Royle et al. (2013), but rather they are based on models that imply different levels of induced heterogeneity in encounter probability. We believe that both Efford's model (eqn 1)) and that of Royle et al. (2013) (eqn 3) are viable models of resource selection, and hence SECR encounter probability, and that investigators may use existing methodologies to evaluate these different models and potentially others that have not yet been described.

Acknowledgements

We thank Bob O’Hara and an anonymous referee for helpful comments and discussion of specific technical issues.

Note

1

This is because, strictly speaking, the use frequency for pixel x is Poisson with mean based on integrating the Poisson intensity function over the raster grid cell centred at x, and this integral is approximately A*λ(x|s) if λ(x|s) is locally constant, where A is the raster grid cell area.