F.cjs.estim {mra}R Documentation

F.cjs.estim - Cormack-Jolly-Seber estimation

Description

Estimates Cormack-Jolly-Seber (CJS) capture-recapture models with individual, time, and individual-time varying covariates using the "regression" parametrization of Amstrup et al (2006, 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.

Usage

F.cjs.estim(capture, survival, histories, cap.init, sur.init, group,  
        algorithm = 1, cov.meth = 1, nhat.v.meth = 1, c.hat = -1, df = NA,
        intervals=rep(1,ncol(histories)-1), conf=0.95)

Arguments

capture 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().
survival 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().
histories 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 in their histories are 'censored'. I.e., Animals with 2's are 'removed' from the likelihood after the occasion with the 2.
cap.init (optional) Vector of initial values for coefficients in the capture model. One element per covariate in capture. The default value usually works.
sur.init (optional) Vector or initial values for coefficients in the survival model. One element per covariate in survival. The default value usually works.
group (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. Results of TEST 2 and TEST 3 are output only to the log file ('mrawin.log'). See c.hat for pooling rules for these test components to estimate C-hat.
algorithm Integer specifying the maximization algorithm to use. If algorithm = 1, the VA09AD algorithm from the HSL library is used. The VA09AD algorithm is very reliable, has been tested extensively (same algorithm as Program MARK), and will almost always find the maximum likelihood estimates. This parameter was added to allow easy addition of other algorithms in the future, but no other options are currently implemented because VA09AD works so well. Anything other than 1 is ignored.
cov.meth Integer specifying the covariance estimation method. cov.meth = 1 takes numerical 2nd partial derivatives. cov.meth = 2 inverts the Hessian of the maximization. Method 1 (numeric 2nd derivatives) is the preferred method, but is very computationally expensive and therefore much slower than method 2. Difference in speed of methods should be minimal when number of parameters < 15. Method 2 variances are generally very close to method 1 variances, and could be used during initial model fitting stages when exact estimation of variance may not be necessary.
nhat.v.meth 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.
c.hat External (override) estimate of variance inflation factor (c.hat) to use during estimation. If input value of c.hat is <= 0, MRAWIN 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, MRAWIN does not estimate C.hat, and uses the supplied value.
df 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).
intervals 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.
conf Confidence level for the confidence intervals placed around estimates of population size. Default 95% confidence.

Details

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.

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.

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').

  1. capture= ~ 1 : constant effect over all individuals and time (intercept only model)
  2. capture= ~ age : Intercept plus age
  3. capture= ~ age + tvar(effort,NAN) : Intercept plus age plus effort
  4. capture= ~ age + tvar(effort,NAN) + ivar(sex,NS) : Intercept plus age plus effort plus sex. Females (1st level) are the reference.
  5. capture= ~ -1 + ivar(sex,NS,0) : sex as a factor, cell means coding
  6. 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.

Value

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:

aux Auxiliary information, mostly stored input values. This is a list containing: $call, $nan=number of animals, $ns=number of samples, $nx=number of coefficients in capture model, $ny=number of coefficients in survival model, $cov.name=names of coefficients, $ic.name=name of capture history matrix, $mra.version= version number of this package, $R.version=R version used for estimation, $run.date=date the model was estimated.
loglik Maximized CJS likelihood value for the model
deviance Model deviance = -2*loglik. This is relative deviance, see help for F.sat.lik.
aic 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 QAIC (quasi-AIC) = (deviance / vif) + 2(df)
aicc AIC with small sample correction = AIC + (2*df*(df+1)) / (nan - df - 1)
qaicc QAIC with small sample correction = QAIC + (2*df*(df+1))/(nan - df - 1)
vif Variance inflation factor used = estimate of c.hat = chisq.vif / chisq.df
chisq.vif Composite Chi-square statistic from Test 2 and Test 3 used to compute vif, based on pooling rules.
vif.df Degrees of freedom for composite chi-square statistic from Test 2 and Test 3, based on pooling rules.
parameters Vector of all coefficient estimates, NX capture probability coefficients first, then NY survival coefficients. This vector is length NX+NY regardless of estimated DF.
se.param Standard error estimates for all coefficients. Length NX+NY.
se1.param Standard error estimates from 'alternate' method.
capcoef Vector of coefficients in the capture model. Length NX.
se.capcoef Vector of standard errors for coefficients in capture model. Length NX.
surcoef Vector of coefficients in the survival model. Length NY.
se.surcoef Vector of standard errors for coefficients in survival model. Length NY.
covariance Variance-covariance matrix for the estimated model coefficients. Size (NX+NY) X (NX+NY).
p.hat 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.
se.p.hat Matrix of standard errors for estimated capture probabilities. One for each animal each occasion. Size NAN X NS. First column is NA.
s.hat 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.
se.s.hat Matrix of standard errors for estimated survival probabilities. Size NAN X NS. Last column is NA.
df Number of estimable parameters in the model. df is either the estimated number of independent parameters (by default) based on rank of the variance matrix, or NX+NY, or a specified value, depending on the input value of DF parameter.
message A string indicating whether the maximization routine converged.
exit.code Exit code from the maximization routine. Interpretation for exit.code is in message.
cov.code A code indicating the method used to compute the covariance matrix.
cov.meth String indicating method used to compute covariance matrix. Interprets cov.code.
n.hat Vector of Horvitz-Thompson estimates of population size. The Horvitz-Thompson estimator of size is,

N(ij) = sum( h(ij) / p(ij))

Length of n.hat = NS. No estimate for first occasion.

se.n.hat Estimated standard errors for n.hat estimates. Computed using method specified in nhat.v.meth.
n.hat.lower Lower limit of n.hat.conf percent on n.hat. Length NS.
n.hat.upper Upper limit of n.hat.conf percent on n.hat. Length NS.
n.hat.conf Confidence level of intervals on n.hat
nhat.v.meth Code for method used to compute variance of n.hat
num.caught Vector of observed number of animals captured each occasion. Length NS.
fitted 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.
residuals Matrix of Pearson residuals defined as,

(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).

resid.type String describing the type of residuals computed. Currently, only Pearson residuals are returned.

Author(s)

Trent McDonald, WEST-INC, tmcdonald@west-inc.com

References

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.

See Also

print.cjs, residuals.cjs, plot.cjs, F.cjs.covars, F.cjs.gof

Examples



## 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) )
for(j in 1:ncol(dipper.histories)){ assign(paste("x",j,sep=""), xy$x[,,j]) } # Extract 2-D matrices of 0s and 1s
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




[Package mra version 2.2 Index]