For spatially explicit capture--recapture estimation of a closed population, we model the detection of individual \(i\) on occasion \(s\) at detector \(k\). Given \(n\) observed individuals on \(S\) occasions at \(K\) detectors there are therefore \(n.S.K\) detection probabilities of interest. We can think of these as elements of a 3-dimensional array. Strictly, we are also interested in the detection probabilities of unobserved individuals, but these are estimated only by extrapolation from those observed so we do not consider them in the array.
In a null (constant) model, all \(n.S.K\) detection probabilities are the same. The conventional sources of variation in capture probability (Otis et al. 1978) appear as variation in the \(n\) dimension (`individual heterogeneity' h), in the \(S\) dimension (`time variation' t) or as a particular interaction in these two dimensions (`behavioural response to capture' b). Combined effects are possible.
Spatially explicit capture--recapture introduces two sorts of additional complexity. Firstly, detection probability is no longer a scalar (even for a particular animal, occasion and detector combination); it is described by the detection function, which may have two parameters (e.g. g0, sigma for half-normal), three parameters (e.g. g0, sigma, z for the hazard-rate function), or potentially more.
Secondly, many more types of variation are possible. Any of the
parameters of the detection function may vary with respect to individual
(\(i\)), occasion (\(s\)) or detector (\(k\)). For example, there
may be a covariate associated with trap location that influences
detection probability, and this effect may vary between occasions (see
timevaryingcov.
The full design matrix for each detection submodel has one row for each combination of \(i\), \(s\) and \(k\) (animal, occasion and trap). Allowing a distinct probability for each animal (the `\(n\)' dimension) may seem excessive, as continuous individual-specific covariates are feasible only when a model is fitted by maximizing the conditional likelihood (cf Huggins 1989). However, the full \(n.S.K\) array is convenient for coding both group membership (Lebreton et al. 1992, Cooch and White 2008) and experience of capture, even when individual-level heterogeneity cannot be modelled.
Variation between `sessions' and between latent classes in a finite mixture adds two further dimensions: in principle there is an \(n.S.K\) array for each latent class (classes are numbered 1..\(M\)), and an \(n.S.K.M\) array for each session (sessions are numbered 1..\(R\)). The full design matrix has \(n.S.K.M.R\) rows. We do not expand on this here.
Effects on parameters of detection probability are specified with R
formulae using standard variable names or named covariates supplied by
the user. The formula for each detection parameter (g0, sigma, z) may be
constant (\(\sim{~}\)1, the default) or some combination of terms
in standard R formula notation (see formula).
| Variable | Description | Data source | Dim |
| t | time factor (one level for each occasion) | automatic | \(S\) |
| T | time trend (integer covariate 0:(S-1)) | automatic | \(S\) |
| tcov | default time covariate | timecov[,1] | \(S\) |
| kcov | default trap covariate | covariates (traps)[,1] | \(K\) |
| b | learned response | capthist |
\(n.S\) |
| B | transient (Markovian) response | capthist |
\(n.S\) |
| bk | animal x site learned response | capthist |
\(n.S.K\) |
| Bk | animal x site transient response | capthist |
\(n.S.K\) |
| k | site learned response | capthist |
\(S.K\) |
| K | site transient response | capthist |
\(S.K\) |
| g | group | see below | \(n\) |
| h2 | 2-class mixture | -- | 2 |
| h3 | 3-class mixture | -- | 3 |
| session | session factor (one level for each session) | automatic | \(R\) |
| Session | session number 0:(R-1) | automatic | \(R\) |
| ts | marking vs sighting occasions | automatic | \(S\) |
| tt | capture--recapture vs telemetry occasions | automatic | \(S\) |
| [user] | individual covariate | covariates (capthist) |
\(n\) |
| [user] | session covariate | sessioncov |
\(R\) |
| [user] | time covariate | timecov |
\(S\) |
| [user] | detector covariate | covariates (traps) |
\(K\) |
The classic `learned response' is a step change following first detection; this is implemented with the predictor variable `b' which is FALSE up to and including the time of first capture and TRUE afterwards. An alternative is a response that depends only on detection at the last opportunity (`B').
The site-specific learned and transient responses `bk' and `Bk' imply that an individual becomes trap happy or trap shy in relation to a particular detector, as in the wolverine example of Royle et al. (2011).
Groups (`g') are defined by the interaction of the capthist
categorical (factor) individual covariates identified in secr.fit
argument `groups'. Groups are redundant with conditional likelihood
because individual covariates of whatever sort (continuous or
categorical) may be included freely in the model.
Individual heterogeneity (`h' in the notation of Otis et al. 1978) may modelled by treating any detection parameter as a 2-part or 3-part finite mixture e.g. g0 \(\sim{~}\) h2. See http://www.otago.ac.nz/density/pdfs/secr-finitemixtures.pdf.
Any other variable name appearing in a formula is assumed to refer to a user-defined predictor. These will be interpreted by searching for name matches in the dataframes of individual, session, time and trap covariates, in that order (remembering that individual covariates other than groups are allowed only when the model is fitted by maximizing the conditional likelihood). The type of the predictor is inferred from the data frame in which it first occurs. Thus if the model included the formula `g0 ~ wetness', and `wetness' was a column in the data frame of time covariates (timecov), then `wetness' would be interpreted as a time covariate, and a column of the same name in covariates(traps) would be ignored. In this case, renaming the column in timecov would expose the traps covariate, and `wetness' would be interpreted as an attribute of detectors, rather than sample intervals. This is a good reason to give covariates distinctive names!
The design matrix for detection parameters may also be provided manually
in the argument dframe. This feature requires some care and is
better avoided.
The submodels for `g0' (`lambda0'), `sigma' and `z' are named components of the
model argument of secr.fit. They are expressed in R
formula notation by appending terms to \(\sim{~}\).
The name of the response may optionally appear on the left hand side of
the formula (e.g. g0\(\sim{~}\)b).
Cooch, E. and White, G. (eds) (2008) Program MARK: A Gentle Introduction. 6th edition. Available online at http://www.phidot.org.
Hayes, R. J. and Buckland, S. T. (1983) Radial-distance models for the line-transect method. Biometrics 39, 29--42.
Huggins, R. M. (1989) On the statistical analysis of capture experiments. Biometrika 76, 133--140.
Lebreton, J.-D., Burnham, K. P., Clobert, J. and Anderson, D. R. (1992) Modeling survival and testing biological hypotheses using marked animals: a unified approach with case studies. Ecological Monographs 62, 67--118.
Royle, J. A., Magoun, A. J., Gardner, B., Valkenburg, P. and Lowell, R. E. (2011) Density estimation in a wolverine population using spatial capture--reecapture models. Journal of Wildlife Management 75, 604--611.
secr models, secr density models, secr.fit
## constant (null) model
list(g0 = ~1, sigma = ~1)
## both detection parameters change after first capture
list(g0 = ~b, sigma = ~b)
## group-specific parameters; additive time effect on g0
## groups are defined via the '`groups' argument of secr.fit
list(g0 = ~ g + t, sigma = ~ g)
## g0 depends on trap-specific covariate
list(g0 = ~ kcov)
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