closedp.t and closedp.0 fit various loglinear models for closed populations in
capture-recapture experiments. For back compatibility, closedp.t is also named closedp.
closedp.t fits more models than closedp.0 but for data set with more than 20 capture occasions,
the function might fail. However, closedp.0 works with fairly large data sets (see Details).closedp(X, dfreq=FALSE, neg=TRUE)
closedp.t(X, dfreq=FALSE, neg=TRUE)
closedp.0(X, dfreq=FALSE, dtype=c("hist","nbcap"), t=NULL, t0=NULL,
neg=TRUE)
## S3 method for class 'closedp':
print(x, \dots)
## S3 method for class 'closedp':
boxplot(x, main="Boxplots of Pearson Residuals", \dots)
## S3 method for class 'closedp':
plot(x, main="Residual plots for some heterogeneity models", \dots)Rcapture-package
for a description of the accepted formats).FALSE, which means that X has one row per unit.
If TRUE, it indicates that the matrix X contains frequencies in its last column."hist" or "nbcap", to specify the type of data.
"hist", the default, means that X contains complete observed capture histories.
"nbcadtype="nbcap". A numeric specifying the total number of
capture occasions in the experiment. For closedp.0, the value t=Inf
is accepted. It indicates that captures occur it0 times.
By default, if t is not equal to Inf, t0=t. When t=Inf, the default value
closedp function, to print or to plot.print.default, boxplot.default and plot.default).X.closedp.0 only: the value of the argument t0 used in the computations.glm.fit function generates
one or more warnings when fitting a model, a copy of these warnings are
stored in glm.warn$Mname where Mname is a short name to identify
the model. A NULL list element means that glm.fit did not
produce any warnings for the considered model.NULL, likelihood ratio tests cannot be
conducted to test whether a particular heterogeneous model is adequate by
comparing it's deviance to the deviance of Chao's lower bound model
(see Details).dfreq argument given in the function call.closedp.t fits models M0, Mt, Mh Chao (LB), Mh Poisson2, Mh Darroch, Mh Gamma3.5,
Mth Chao (LB), Mth Poisson2, Mth Darroch, Mth Gamma3.5, Mb and Mbh. closedp.0 fits
only models M0, Mh Chao (LB), Mh Poisson2, Mh Darroch and Mh Gamma3.5. However,
closedp.0 can be used with larger data sets than closedp.t.
This is explained by the fact that closedp.t fits models using the frequencies of
the observable capture histories (vector of size $2^t-1$), whereas closedp.0
uses the numbers of units captured i times, for $i=1,\ldots,t$ (vector of size $t$).
Multinomial profile confidence intervals for the abundance are constructed by closedpCI.t
and closedpCI.0.
To calculate bias corrected abundance estimates, use the closedp.bc function.
CHAO'S LOWER BOUND MODEL
Chao's (or LB) models estimate a lower bound for the abundance, both with a time
effect (Mth Chao) and without one (Mh Chao). The estimate obtained under Mh Chao is Chao's (1987)
moment estimator. Rivest and Baillargeon (2007) exhibit a loglinear model underlying this
estimator and provide a generalization to Mth. For these two models, a small deviance means
that there is an heterogeneity in capture probabilities; it does not mean that the lower
bound estimate is unbiased. To test whether a certain model for heterogeneity is adequate,
one can conduct a likelihood ratio test by subtracting the deviance of Chao's model
to the deviance of the heterogeneous model under study. If this heterogeneous model
includes a time effect, it must be compared to model Mth Chao. If it does not include a time
effect, it must be compared to model Mh Chao.
Under the null hypothesis of equivalence between the two models, the difference of deviances
follows a chi-square distribution with degrees of freedom equal to the difference between
the models' degrees of freedom.
Chao's lower bound models contain $t-2$ parameters, called
eta parameters, for the heterogeneity. These parameters should theoretically be greater
or equal to zero (see Rivest and Baillargeon (2007)). When the argument neg is set
to TRUE (the default), negative eta parameters are set to zero (to do so, columns are
removed from the design matrix of the model). Consequently, heterogeneous models are no longer
particular cases of Chao's model. Therefore, likelihood ratio tests cannot be conducted
anymore to test whether a chosen heterogeneous model is adequate. Also, the degrees
of freedom of Chao's model increase when eta parameters are set to zero.
OTHER MODELS FOR HETEROGENEITY
Other models for heterogeneity are defined as follows :
closedpCI.t and
closedpCI.0 functions.
Darroch's models for Mh and Mth are considered by Darroch et al. (1993) and Agresti (1994).
Poisson and Gamma models are discussed in Rivest and Baillargeon (2007). Poisson models
typically yield smaller corrections for heterogeneity than Darroch's model since the capture
probabilities are bounded from below under these models. On the other hand, Gamma models
can lead to very large estimators of abundance. We suggest considering this estimator only in
experiments where very small capture probabilities are likely.
PLOT METHODS AND FUNCTIONS
The boxplot.closedp function produces boxplots of the Pearson residuals of the fitted loglinear models that converged.
The plot.closedp function produces scatterplots of the Pearson residuals in terms of $f_i$
(number of units captured i times) for the heterogeneous models Mh Poisson2, Mh Darroch and Mh Gamma3.5 if they converged.closedpCI.t, closedpCI.0, closedp.bc, closedp.Mtb, uifit.data(hare)
hare.closedp<-closedp.t(hare)
hare.closedp
boxplot(hare.closedp)
data(mvole)
period3<-mvole[,11:15]
closedp.t(period3)
data(BBS2001)
BBS.closedp<-closedp.0(BBS2001,dfreq=TRUE,dtype="nbcap",t=50,t0=20)
BBS.closedp
plot(BBS.closedp)
### Seber (1982) p.107
# When there is 2 capture occasions, the heterogeneity models cannot be fitted
X <- matrix(c(1,1,167,1,0,781,0,1,254),byrow=TRUE,ncol=3)
closedp.t(X,dfreq=TRUE)
### Example of captures in continuous time
### Illegal immigrants data
data(ill)
closedp.0(ill, dtype="nbcap", dfreq=TRUE, t=Inf)Run the code above in your browser using DataLab