Estimates Cormack-Jolly-Seber (CJS) capture-recapture models with individual, time, and individual-time varying covariates using the "regression" parametrization of Amstrup et al (2005, Ch 9). For live recaptures only. Losses on capture allowed. Uses a logistic link function to relate probability of capture and survival to external covariates.
F.cjs.estim(capture, survival, histories, cap.init, sur.init, group, nhat.v.meth = 1,
c.hat = -1, df = NA, intervals=rep(1,ncol(histories)-1), conf=0.95,
link="logit", control=mra.control())Formula specifying the capture probability model. Must be a formula object with
no response. I.e., "~" followed by the names of 2-D arrays of covariates to fit in the capture model.
For example: 'capture = ~ age + sex', where age and sex are matrices of size NAN X NS
containing the age and sex covariate values. NAN = number of animals = number of rows in
histories matrix (see below). NS = number of samples = number of columns in histories
matrix (see below). Number of matrices specified in the capture model is assumed to be NX. Time
varying and individual varying vectors are fitted using ivar() and tvar() (see Details).
Factors are allowed within ivar() and tvar().
Formula specifying the survival probability model. Must be a formula object with
no response. I.e., "~" followed by the names of 2-D arrays of covariates to fit in the survival model.
For example: 'survival = ~ year + ageclass' where year and ageclass are matrices of size NAN X NS
containing year and ageclass covariate values. Number of matrices specified in the survival
model is assumed to be NY.
Time varying and individual varying vectors are fitted using ivar() and tvar()
(see Details). Factors are allowed within ivar() and tvar().
A NAN X NS = (number of animals) X (number of capture occasions) matrix containing capture histories. Capture histories are comprised of 0's, 1',s and 2's. 0 in cell (i,j) means animal i was not captured on occasion j, 1 in cell (i,j) means animal i was captured on occasion j and released live back into the population, 2 in cell (i,j) means animal i was captured on occasion j and was not released back into the population (e.g., it died). Animals with '2' as the last non-zero entry of their history are considered 'censored'. Their lack of capture information is removed from the likelihood after the occasion with the 2. Rows of all zeros (i.e., no captures) are allowed in the history matrix, but do not affect coefficient or population size estimates. A warning is thrown if rows of all zeros exist. Capture and survival probabilities are computed for animals with all zero histories. In this way, it is possible to have the routine compute capture or survival estimates for combinations of covariates that do not exist in the data by associating the covariate combinations with histories that have all zero entries.
(optional) Vector of initial values for coefficients in the capture model. One element
per covariate in capture. The default value usually works.
(optional) Vector or initial values for coefficients in the survival model. One element
per covariate in survival. The default value usually works.
(optional) A vector of length NAN giving the (non-changing) group membership of every
captured animal (e.g., sex). Group is used only for computing TEST 2 and TEST 3.
TEST 2 and TEST 3 are computed separately for each group. E.g., if group=sex,
TEST 2 and TEST 3 are computed for each sex. TEST 2 and TEST3 are used only
to estimate C-hat. See c.hat for pooling rules for these
test components to estimate C-hat.
Integer specifying method for computing variance estimates
of population size estimates. nhat.v.meth = 1 uses the variance estimator of
Taylor et al. 2002, Ursus, p. 188 which is the so-called Huggins variance estimator, and incorporates
covariances. nhat.v.meth = 2 uses the variance estimator of Amstrup et al.
2005 (p. 244, Eqn. 9.10), which is the same variance estimator as nhat.v.meth = 1
with more 2nd order approximation terms included. Method 2 should provide better variances
than method 1, especially if the coefficient of variation of capture probabilities
are >1.0, but method 2 has not been studied as much as method 1.
nhat.v.meth = 3 uses the variance estimator of McDonald and Amstrup, 1999, JABES, which is a 1st order approximation
that does not incorporate covariances. Method 3 is much faster than methods 1 and 2
and could be easily calculated by hand, but
should only be used when there is little capture heterogeneity.
External (override) estimate of variance inflation factor (c.hat) to use
during estimation. If input value of c.hat is <= 0, MRA computes
an estimate of variance inflation based on TEST 2 and TEST 3
applied to groups (if called for, see group above)
using Manly, McDonald, and McDonald, 1993, rules for pooling. I.e.,
all cells in each TEST 2 or TEST 3 Chi-square component table must be
>= 5 before that component contributes to the estimate of C-hat. This rules is
slightly different than program MARK's pooling rules, so MRA's and MARK's
estimates of c.hat will generally be different. If the input c.hat > 0,
MRA does not estimate C.hat, and uses the supplied value.
External (override) model degrees of freedom to use during estimation.
If df == NA, the number of parameters is estimated from the rank of the
matrix of 2nd derivatives or Hessian, depending on cov.meth parameter.
If df <= 0, the number of parameters will be
set to NX+NY = the number of estimated coefficients. Otherwise, if df > 0,
the supplied value is used. Only AIC, QAIC, AICc, and QAICc are dependent on
this value (in their penalty terms).
Time intervals. This is a vector of length ncol(histories)-1 (i.e.,
number of capture occasions minus 1) specifying relative time intervals between occasions.
For example, if capture occasions occurred in 1999, 2000, 2005, and 2007 intervals would
be set to c(1,5,2). Estimates of survival are adjusted for time intervals between
occasions assuming an exponential lifetime model, i.e., probability of surviving
from occasion j to occasion j+1 is Phi(j)^(jth interval length), and
it is the Phi(j)'s that are related to covariates through the survival model. In
other words, all survival estimates are for an interval of length 1. If an interval
of 1 is one year, then all survival estimates will be annual survival, with probability
of surviving 2 years equal to annual survival squared, probability of surviving 3 years
equal to annual survival cubed, etc.
Confidence level for the confidence intervals placed around estimates of population size. Default 95% confidence.
The link function to be used. The link function converts linear predictors in the range (-infinity, infinity) to probabilities in the range (0,1). Valid values for the link function are "logit" (default), "sine", and "hazard". (see Examples for a plot of the link functions)
The "logit" link is \(\eta = log( \frac{p}{1 - p} )\) with inverse \(p = \frac{1}{1 + exp(-\eta)}\).
The "sine" link is \(\eta = \frac{8asin( 2p - 1 )}{\pi}\), which ranges from -4 to 4. The inverse "sine" link is \(p = \frac{1 + sin( \eta\pi/8 )}{2}\) for values of \(\eta\) between -4 and 4. For values of \(\eta\) < -4, \(p\) = 0. For values of \(\eta\) > 4, \(p\) = 1. Scaling of the sine link was chosen to yield coefficients roughly the same magnitude as the logit link.
The "hazard" link is \(\eta = log( -log( 1 - p ))\), with inverse \(1 - exp( -exp( \eta ))\). The value of \(p\) from the inverse hazard link approaches 0 as \(\eta\) decreases. For values of \(\eta\) > 3, \(p\) = 1 for all intents and purposes.
A list containing named control parameters for the minimization and estimation process.
Control parameters include number of iterations, covariance estimation method, etc.
Although the default values work in the vast majority of cases, changes to these
variables can effect speed and performance for ill-behaved models. See
mra.control() for a description of the individual control parameters.
An object (list) of class c("cjs","cr") with many components. Use print.cr to print
it nicely. Use names(fit), where the call was fit <- F.cr.estim(...),
to see names of all returned components. To see values of individual components,
issue commands like fit$s.hat, fit$se.s.hat, fit$n.hat, etc.
Components of the returned object are as follows:
The input capture history matrix.
Auxiliary information about the fit, mostly stored input values. This is a list containing the following components:
call = original call
nan = number of animals
ns = number of samples = number of capture occasions
nx = number of coefficients in capture model
ny = number of coefficients in survival model
cov.name = names of all coefficients
ic.name = name of capture history matrix.
mra.version = version number of MRA package used to estimate the model
R.version = R version used for during estimation
run.date = date the model was estimated.
Maximized CJS likelihood value for the model
Model deviance = -2*loglik. This is relative deviance, see help for
F.sat.lik.
AIC for the model = deviance + 2*(df). df is either the estimated number of independent
parameters (by default), or NX+NY, or a specified value, depending on the input value of DF parameter.
QAIC (quasi-AIC) = (deviance / vif) + 2(df)
AIC with small sample correction = AIC + (2*df*(df+1)) / (nan - df - 1)
QAIC with small sample correction = QAIC + (2*df*(df+1))/(nan - df - 1)
Variance inflation factor used = estimate of c.hat = chisq.vif / chisq.df
Composite Chi-square statistic from Test 2 and Test 3 used to compute vif, based
on pooling rules.
Degrees of freedom for composite chi-square statistic from Test 2 and Test 3, based on pooling rules.
Vector of all coefficient estimates, NX capture probability coefficients first, then NY survival coefficients. This vector is length NX+NY regardless of estimated DF.
Standard error estimates for all coefficients. Length NX+NY.
Vector of coefficients in the capture model. Length NX.
Vector of standard errors for coefficients in capture model. Length NX.
Vector of coefficients in the survival model. Length NY.
Vector of standard errors for coefficients in survival model. Length NY.
Variance-covariance matrix for the estimated model coefficients. Size (NX+NY) X (NX+NY).
Matrix of estimated capture probabilities computed from the model. One for each animal each occasion. Cell (i,j) is estimated capture probability for animal i during capture occasion j. Size NAN X NS. First column corresponding to first capture probability is NA because cannot estimate P1 in a CJS model.
Matrix of standard errors for estimated capture probabilities. One for each animal each occasion. Size NAN X NS. First column is NA.
Matrix of estimated survival probabilities computed from the model. One for each animal each occasion. Size NAN X NS. Cell (i,j) is estimated probability animal i survives from occasion j to j+1. There are only NS-1 intervals between occasions. Last column corresponding to survival between occasion NS and NS+1 is NA.
Matrix of standard errors for estimated survival probabilities. Size NAN X NS. Last column is NA.
The number of parameters assumed in the model. This value was used in the penalty term of AIC, AICc, QAIC, and QAICc.
This value is either the number of independent
parameters estimated from the rank of the variance-covariance matrix (by default),
or NX+NY, or a specified value, depending on the input value of DF parameter. See F.update.df to update this value after
the model is fitted.
The number of parameters estimated from the rank of the variance-covariance matrix. This
is stored so that df can be updated using F.update.df.
A list containing the input maximization and estimation control parameters.
A vector of strings interpreting various codes about the estimation.
The messages interpret, in this order, the codes for (1) maximization algorithm used,
(2) exit code from the maximization algorithm (interprets exit.code), and
(3) covariance matrix code (interprets cov.code).
Exit code from the maximization routine. Interpretation for exit.code is in message.
Exit codes are as follows:
exit.code = 0: FAILURE: Initial Hessian not positive definite.
exit.code = 1: SUCCESS: Convergence criterion met.
exit.code = 2: FAILURE: G'dX > 0, rounding error.
exit.code = 3: FAILURE: Likelihood evaluated too many times.
exit.code =-1: FAILURE: Unknown optimization algorithm."
A code indicating the method used to compute the covariance matrix.
The number of times the likelihood was evaluated prior to exit from the minimization routine.
If exit.code = 3, fn.evals equals the maximum set in mra.control. This, in combination
with the exit codes and execution time, can help detect non-convergence or bad behavior.
Execution time for the maximization routine, in minutes. This is returned for
2 reasons. First, this is useful for benchmarking. Second, in conjunction with
exit.code, cov.code, and fn.evals, this could be used to detect ill-behaved
or marginally unstable problems, if you know what you are doing. Assuming maxfn is set high
in mra.control() (e.g., 1000),
if exit.code = 1 but the model takes a long time to execute relative to similarly sized problems,
it could indicate unstable or marginally ill-behaved models.
Vector of Horvitz-Thompson estimates of population size. The Horvitz-Thompson
estimator of size is,
$$\hat{N}_{ij} = \sum_{i=1}^{NAN} \frac{h_{ij}}{\hat{p}_{ij}}$$
Length of n.hat = NS. No estimate for
first occasion.
Estimated standard errors for n.hat estimates. Computed using method
specified in nhat.v.meth.
Lower limit of n.hat.conf percent on n.hat. Length NS.
Upper limit of n.hat.conf percent on n.hat. Length NS.
Confidence level of intervals on n.hat
Code for method used to compute variance of n.hat
Vector of observed number of animals captured each occasion. Length NS.
Matrix of fitted values for the capture histories. Size NAN X NS.
Cell (i,j)
is expected value of capture indicator in cell (i,j) of histories matrix.
Matrix of Pearson residuals defined as,
$$r_{ij} = \frac{(h_{ij} - \Psi_{ij})^2}{\Psi_{ij}}$$,
where \(\Psi_{ij}\) is the expected (or fitted) value for cell
(i,j) and \(h_{ij}\) is the capture indicator for
animal i at occasion j. This matrix has size NAN X NS. See parts pertaining to the "overall test" in
documentation for F.cjs.gof for a description of \(\Psi_{ij}\).
String describing the type of residuals computed. Currently, only Pearson residuals are returned.
This is the work-horse routine for estimating CJS models. It compiles all the covariate matrices, then calls a Fortran routine to maximize the CJS likelihood and perform goodness-of-fit tests. Horvitz-Thompson-type population size estimates are also computed by default.
If control=mra.control(trace=1), a log file, named mra.log, is written to the current directory. This file contains
additional details, such as individual Test 2 and Test 3 components, in a semi-friendly
format. This file is overwritten each run. See help(mra.control) for more details.
Model Specification: Both the capture and survival model can be
specified as any combination of 2-d matrices (time and individual varying covariates),
1-d time varying vectors, 1-d individual
varying vectors, 1-d time varying factors, and 1-d individual varying factors.
Specification of time or individual varying effects uses the
tvar (for 'time varying') and ivar (for 'individual varying') functions.
These functions expand covariate vectors along the appropriate dimension to
be 2-d matrices suitable for fitting in the model. ivar expands
an individual varying vector to all occasions. tvar expands a time
varying covariate to all individuals. To do the expansion, both tvar
and ivar need to know the size of the 'other' dimension. Thus, tvar(x,100)
specifies a 2-d matrix with size 100 by length(x).
ivar(x,100)
specifies a 2-d matrix with size length(x) by 100.
For convenience, the 'other' dimension of time or individual varying covariates
can be specified as an attribute of the vector. Assuming x
is a NS vector and the 'nan' attribute of x has been set as
attr(x,"nan") <- NAN, tvar(x,NAN) and
tvar(x) are equivalent. Same, but vise-versa, for individual varying
covariates (i.e.,
assign the number of occasions using attr(x,"ns")<-NS). This saves
some typing in model specification.
Factors are allowed in ivar and tvar. When a factor is specified,
the contr.treatment coding is used. By default, an intercept
is assumed and the first level of all
factors are dropped from the model (i.e., first levels are the reference levels,
the default R action). However, there are applications where more than one level
will need to be dropped, and the user has control over this via the drop.levels
argument to ivar and tvar. For example,
tvar(x,drop.levels=c(1,2)) drops the first 2 levels of factor x.
tvar(x,drop.levels=length(levels(x))) does the SAS thing and drops
the last level of factor x. If drop.levels is outside the range
[1,length(levels(x))] (e.g., negative or 0), no levels of the factor
are dropped. If no intercept is fitted in the model, this results in the
so-called cell means coding for factors.
Example model specifications: Assume 'age' is a NAN x NS 2-d matrix of ages, 'effort' is a size NS 1-d vector of efforts, and 'sex' is a size NAN 1-d factor of sex designations ('M' and 'F').
capture= ~ 1 : constant effect over all individuals and time (intercept only model)
capture= ~ age : Intercept plus age
capture= ~ age + tvar(effort,NAN) : Intercept plus age plus effort
capture= ~ age + tvar(effort,NAN) + ivar(sex,NS) : Intercept plus age plus effort plus sex. Females (1st level) are the reference.
capture= ~ -1 + ivar(sex,NS,0) : sex as a factor, cell means coding
capture= ~ tvar(as.factor(1:ncol(histories)),nrow(histories),c(1,2)) : time varying effects
Values in 2-d Matrix Covariates: Even though covariate matrices are required to be NAN x NS (same size as capture histories), there are not that many parameters. The first capture probability cannot be estimated in CJS models, and the NS-th survival parameter does not exist. When a covariate matrix appears in the capture model, only values in columns 2:ncol(histories) are used. When a covariate matrix appears in the survival model, only values in columns 1:(ncol(histories)-1) are used. See examples for demonstration.
Taylor, M. K., J. Laake, H. D. Cluff, M. Ramsay, and F. Messier. 2002. Managing the risk from hunting for the Viscount Melville Sound polar bear population. Ursus 13:185-202.
Manly, B. F. J., L. L. McDonald, and T. L. McDonald. 1999. The robustness of mark-recapture methods: a case study for the northern spotted owl. Journal of Agricultural, Biological, and Environmental Statistics 4:78-101.
Huggins, R. M. 1989. On the statistical analysis of capture experiments. Biometrika 76:133-140.
Amstrup, S. C., T. L. McDonald, and B. F. J. Manly (editors). 2005. Handbook of Capture-Recapture Analysis. Princeton University Press.
Peterson. 1986. Statistics and Probability Letters. p.227.
McDonald, T. L., and S. C. Amstrup. 2001. Estimation of population size using open capture-recapture models. Journal of Agricultural, Biological, and Environmental Statistics 6:206-220.
tvar, ivar, print.cjs, residuals.cjs, plot.cjs,
F.cjs.covars, F.cjs.gof, mra.control, F.update.df
# NOT RUN {
## Fit CJS model to dipper data, time-varying capture and survivals.
## Method 1 : using factors
data(dipper.histories)
ct <- as.factor( paste("T",1:ncol(dipper.histories), sep=""))
attr(ct,"nan")<-nrow(dipper.histories)
dipper.cjs <- F.cjs.estim( ~tvar(ct,drop=c(1,2)), ~tvar(ct,drop=c(1,6,7)), dipper.histories )
## Method 2 : same thing using 2-d matrices
xy <- F.cjs.covars( nrow(dipper.histories), ncol(dipper.histories) )
# The following extracts 2-D matrices of 0s and 1s
for(j in 1:ncol(dipper.histories)){ assign(paste("x",j,sep=""), xy$x[,,j]) }
dipper.cjs <- F.cjs.estim( ~x3+x4+x5+x6+x7, ~x2+x3+x4+x5, dipper.histories )
## Values in the 1st column of capture covariates do not matter
x3.a <- x3
x3.a[,1] <- 999
dipper.cjs2 <- F.cjs.estim( ~x3.a+x4+x5+x6+x7, ~x2+x3+x4+x5, dipper.histories )
# compare dipper.cjs2 to dipper.cjs
## Values in the last column of survival covariates do not matter
x3.a <- x3
x3.a[,ncol(dipper.histories)] <- 999
dipper.cjs2 <- F.cjs.estim( ~x3+x4+x5+x6+x7, ~x2+x3.a+x4+x5, dipper.histories )
# compare dipper.cjs2 to dipper.cjs
## A plot to compare the link functions
sine.link <- function(eta){ ifelse( eta < -4, 0, ifelse( eta > 4, 1, .5*(1+sin(eta*pi/8)))) }
eta <- seq(-5,5, length=40)
p1 <- 1 / (1 + exp(-eta))
p2 <- sine.link(eta)
p3 <- 1.0 - exp( -exp( eta ))
plot(eta, p1, type="l" )
lines(eta, p2, col="red" )
lines(eta, p3, col="blue" )
legend( "topleft", legend=c("logit", "sine", "hazard"), col=c("black", "red", "blue"), lty=1)
# }
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