Positive Bernoulli Family Function with Time Effects
Fits a GLM/GAM-like model to multiple Bernoulli responses where each row in the capture history matrix response has at least one success (capture). Sampling occasion effects are accommodated.
posbernoulli.t(link = "logitlink", parallel.t = FALSE ~ 1, iprob = NULL,
p.small = 1e-4, no.warning = FALSE,
type.fitted = c("probs", "onempall0"))link, iprob, parallel.t |
See |
p.small, no.warning |
A small probability value used to give a warning for the
Horvitz–Thompson estimator.
Any estimated probability value less than |
type.fitted |
See |
These models (commonly known as M_t or M_{th} (no prefix h means it is an intercept-only model) in the capture–recapture literature) operate on a capture history matrix response of 0s and 1s (n x tau). Each column is a sampling occasion where animals are potentially captured (e.g., a field trip), and each row is an individual animal. Capture is a 1, else a 0. No removal of animals from the population is made (closed population), e.g., no immigration or emigration. Each row of the response matrix has at least one capture. Once an animal is captured for the first time, it is marked/tagged so that its future capture history can be recorded. Then it is released immediately back into the population to remix. It is released immediately after each recapture too. It is assumed that the animals are independent and that, for a given animal, each sampling occasion is independent. And animals do not lose their marks/tags, and all marks/tags are correctly recorded.
The number of linear/additive predictors is equal to the number of sampling occasions, i.e., M = τ, say. The default link functions are (logit p_(1),…,logit p_(tau))^T where each p_{j} denotes the probability of capture at time point j. The fitted value returned is a matrix of probabilities of the same dimension as the response matrix.
A conditional likelihood is maximized here using Fisher scoring.
Each sampling occasion has a separate probability that
is modelled here. The probabilities can be constrained
to be equal by setting parallel.t = FALSE ~ 0;
then the results are effectively the same as
posbinomial except the binomial constants are
not included in the log-likelihood.
If parallel.t = TRUE ~ 0 then each column should have
at least one 1 and at least one 0.
It is well-known that some species of animals are affected
by capture, e.g., trap-shy or trap-happy. This VGAM
family function does not allow any behavioral effect to be
modelled (posbernoulli.b
and posbernoulli.tb do) because the
denominator of the likelihood function must be free of
behavioral effects.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
Upon fitting the extra slot has a (list) component
called N.hat
which is a point estimate of the population size N
(it is the Horvitz-Thompson (1952) estimator).
And there is a component called SE.N.hat
containing its standard error.
The weights argument of vglm need not be
assigned, and the default is just a matrix of ones.
Fewer numerical problems are likely to occur
for parallel.t = TRUE.
Data-wise, each sampling occasion may need at least one success
(capture) and one failure.
Less stringent conditions in the data are needed when
parallel.t = TRUE.
Ditto when parallelism is applied to the intercept too.
The response matrix is returned unchanged;
i.e., not converted into proportions like posbinomial.
If the response matrix has column names then these are used in the
labelling, else prob1, prob2, etc. are used.
Using AIC() or BIC() to compare
posbernoulli.t,
posbernoulli.b,
posbernoulli.tb
models with a
posbinomial
model requires posbinomial(omit.constant = TRUE)
because one needs to remove the normalizing constant from the
log-likelihood function.
See posbinomial for an example.
Thomas W. Yee.
Huggins, R. M. (1991). Some practical aspects of a conditional likelihood approach to capture experiments. Biometrics, 47, 725–732.
Huggins, R. M. and Hwang, W.-H. (2011). A review of the use of conditional likelihood in capture–recapture experiments. International Statistical Review, 79, 385–400.
Otis, D. L. and Burnham, K. P. and White, G. C. and Anderson, D. R. (1978). Statistical inference from capture data on closed animal populations, Wildlife Monographs, 62, 3–135.
Yee, T. W. and Stoklosa, J. and Huggins, R. M. (2015). The VGAM package for capture–recapture data using the conditional likelihood. Journal of Statistical Software, 65, 1–33. https://www.jstatsoft.org/v65/i05/.
M.t <- vglm(cbind(y1, y2, y3, y4, y5, y6) ~ 1, posbernoulli.t,
data = deermice, trace = TRUE)
coef(M.t, matrix = TRUE)
constraints(M.t, matrix = TRUE)
summary(M.t, presid = FALSE)
M.h.1 <- vglm(Select(deermice, "y") ~ sex + weight, trace = TRUE,
posbernoulli.t(parallel.t = FALSE ~ -1), data = deermice)
coef(M.h.1, matrix = TRUE)
constraints(M.h.1)
summary(M.h.1, presid = FALSE)
head(depvar(M.h.1)) # Response capture history matrix
dim(depvar(M.h.1))
M.th.2 <- vglm(cbind(y1, y2, y3, y4, y5, y6) ~ sex + weight, trace = TRUE,
posbernoulli.t(parallel.t = FALSE), data = deermice)
lrtest(M.h.1, M.th.2) # Test the parallelism assumption wrt sex and weight
coef(M.th.2)
coef(M.th.2, matrix = TRUE)
constraints(M.th.2)
summary(M.th.2, presid = FALSE)
head(model.matrix(M.th.2, type = "vlm"), 21)
M.th.2@extra$N.hat # Estimate of the population size; should be about N
M.th.2@extra$SE.N.hat # SE of the estimate of the population size
# An approximate 95 percent confidence interval:
round(M.th.2@extra$N.hat + c(-1, 1) * 1.96 * M.th.2@extra$SE.N.hat, 1)
# Fit a M_h model, effectively the parallel M_t model, using posbinomial()
deermice <- transform(deermice, ysum = y1 + y2 + y3 + y4 + y5 + y6,
tau = 6)
M.h.3 <- vglm(cbind(ysum, tau - ysum) ~ sex + weight,
posbinomial(omit.constant = TRUE), data = deermice, trace = TRUE)
max(abs(coef(M.h.1) - coef(M.h.3))) # Should be zero
logLik(M.h.3) - logLik(M.h.1) # Difference is due to the binomial constantsPlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.