SCR: Bayesian vs frequentist models
Spatial capture-recapture (SCR) - also known as
spatially-explicit capture recapture (SECR) - is the accepted
way to estimate density of individually-identifiable animals.
The analysis can be done with maximum likelihood (ML) methods or
with Bayesian (usually MCMC) methods, but these two methods use
very different models.|
By "model" I mean the set of assumptions and equations which allow us to calculate the likelihood - the probability of obtaining the observed data given specific values of certain parameters. Likelihood is central to both ML and Bayesian methods.
Probability of detection
The basic insight of SCR is that each animal has an activity centre (AC) and the probability of capture in a specific trap depends on the distance between the AC and the trap, being highest when the distance is zero and then declining according to a specific detection function. The simplest models have two detection parameters: the probability of detection at distance 0 (g0 or p0) and the scale parameter of the detection function (\(\sigma\)). This much is common to both approaches.
## maybe note this assumes isotropy
Both approaches limit the spatial region, termed the "state space", were ACs can occur and the animal has a non-negligible probability of capture. This will include locations within some suitably large distance of any trap, and will exclude areas of unsuitable habitat.
## "suitably large" ??
The state space thus defined is a polygon (or multiple polygons), in the simplest case, a rectangle. More complex state spaces are represented in practice by either a raster or a grid of points.
Modelling activity centres
The Bayesian approach is to model the locations (x and y coordinates) of all the ACs inside the state space, and these are parameters of the model together with p0 and \(\sigma\).
For the animals captured, we obtain MCMC chains for the locations similar to the plot on the left, where 4 animals were captured.
For the uncaught animals in the population but, as shown in the right plot above, their probable activity centres lie away from the traps. Of course, we don't know how many uncaught animals are in the state space, but this becomes another parameter in the model.
Because we are explicitly modelling the ACs, we can specify a polygon within the state space and obtain a posterior distribution (ie, an MCMC chain) for the number of ACs within the polygon.
Notice that density is not a parameter in the model and is derived from the estimate of the number of ACs (in the state space or a smaller polygon) divided by the relevant area. Thus, no assumptions about density affect the likelihood calculation.
We use uniform priors for the AC locations - all locations in the state space have equal probability. But that is only a prior and, as the plots above show, has little effect on the posterior when adequate data are available. On the other hand, it has proved to be difficult to implement informative priors for AC locations.
Maximum likelihood methods do not attempt to model individual ACs, but instead integrate across the whole state space. In practice, the state space is represented as a grid of points that are treated as plausible ACs, and the likelihood of the capture history is summed across all points in the grid. The model parameters here are the density, D, together with p0 and \(\sigma\). See Efford, Borchers & Byrom (2009) for details.
The simplest model, D ~ 1, assumes constant density, and if this assumption is violated (ie, we have unmodelled heterogeneity in density) the likelihood calculation will be affected.
On the other hand, it's relatively easy to model D at each point of the grid as a function of habitat covariates.
|Updated 9 Feb 2019 by Mike Meredith|