Positive Bernoulli Family Function with Time and Behavioural 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 and behavioural effects are accommodated.
posbernoulli.tb(link = "logitlink", parallel.t = FALSE ~ 1,
parallel.b = FALSE ~ 0, drop.b = FALSE ~ 1,
type.fitted = c("likelihood.cond", "mean.uncond"),
imethod = 1, iprob = NULL,
p.small = 1e-4, no.warning = FALSE,
ridge.constant = 0.0001, ridge.power = -4)link, imethod, iprob |
See |
parallel.t, parallel.b, drop.b |
A logical, or formula with a logical as the response.
See Suppose the model is intercept-only.
Setting The default model has a different intercept for each sampling occasion, a time-parallelism assumption for all other covariates, and a dummy variable representing a single behavioural effect (also in the intercept). The most flexible model is to set
|
type.fitted |
Character, one of the choices for the type of fitted value returned.
The default is the first one.
Partial matching is okay.
For |
ridge.constant, ridge.power |
Determines the ridge parameters at each IRLS iteration.
They are the constant and power (exponent) for the ridge adjustment
for the working weight matrices (the capture probability block
matrix, hence the first τ diagonal values).
At iteration a of the IRLS algorithm
a positive value is added to the first tau
diagonal elements of the working weight matrices to make
them positive-definite. This adjustment is the
mean of the diagonal elements of |
p.small, no.warning |
See |
This model
(commonly known as M_{tb}/M_{tbh} in the capture–recapture literature)
operates on a response matrix of 0s and 1s (n x tau).
See posbernoulli.t
for information that is in common.
It allows time and behavioural effects to be modelled.
Evidently,
the expected information matrix (EIM) seems not
of full rank (especially in early iterations), so
ridge.constant and ridge.power are used to
try fix up the problem.
The default link functions
are
(logit p_{c1},…,logit p_{c,tau},logit p_{r2},…,logit p_{r,tau})^T
where the subscript c denotes capture,
the subscript r denotes recapture,
and it is not possible to recapture the animal at sampling occasion 1.
Thus M=2*tau-1.
The parameters are currently prefixed by pcapture and precapture
for the capture and recapture probabilities.
This VGAM family function may be further modified in the future.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
It is a good idea to apply the parallelism assumption to each sampling
occasion except possibly with respect to the intercepts.
Also, a simple behavioural effect such as being modelled using the intercept
is recommended; if the behavioural effect is not parallel and/or
allowed to apply to other covariates
then there will probably be too many parameters, and hence,
numerical problems. See M_tbh.1 below.
It is a good idea to monitor convergence.
Simpler models such as the M_0/M_h models
are best fitted with posbernoulli.t or
posbernoulli.b or
posbinomial.
Thomas W. Yee.
See posbernoulli.t.
posbernoulli.b (including N.hat),
posbernoulli.t,
posbinomial,
Select,
fill1,
Huggins89table1,
Huggins89.t1,
deermice,
prinia.
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# Example 1: simulated data
nTimePts <- 5 # (aka tau == # of sampling occasions)
nnn <- 1000 # Number of animals
pdata <- rposbern(n = nnn, nTimePts = nTimePts, pvars = 2)
dim(pdata); head(pdata)
M_tbh.1 <- vglm(cbind(y1, y2, y3, y4, y5) ~ x2,
posbernoulli.tb, data = pdata, trace = TRUE)
coef(M_tbh.1) # First element is the behavioural effect
coef(M_tbh.1, matrix = TRUE)
constraints(M_tbh.1, matrix = TRUE)
summary(M_tbh.1, presid = FALSE) # Standard errors are approximate
head(fitted(M_tbh.1))
head(model.matrix(M_tbh.1, type = "vlm"), 21)
dim(depvar(M_tbh.1))
M_tbh.2 <- vglm(cbind(y1, y2, y3, y4, y5) ~ x2,
posbernoulli.tb(parallel.t = FALSE ~ 0),
data = pdata, trace = TRUE)
coef(M_tbh.2) # First element is the behavioural effect
coef(M_tbh.2, matrix = TRUE)
constraints(M_tbh.2, matrix = TRUE)
summary(M_tbh.2, presid = FALSE) # Standard errors are approximate
head(fitted(M_tbh.2))
head(model.matrix(M_tbh.2, type = "vlm"), 21)
dim(depvar(M_tbh.2))
# Example 2: deermice subset data
fit1 <- vglm(cbind(y1, y2, y3, y4, y5, y6) ~ sex + weight,
posbernoulli.t, data = deermice, trace = TRUE)
coef(fit1)
coef(fit1, matrix = TRUE)
constraints(fit1, matrix = TRUE)
summary(fit1, presid = FALSE) # Standard errors are approximate
# fit1 is the same as Fit1 (a M_{th} model):
Fit1 <- vglm(cbind(y1, y2, y3, y4, y5, y6) ~ sex + weight,
posbernoulli.tb(drop.b = TRUE ~ sex + weight,
parallel.t = TRUE), # No parallelism for the intercept
data = deermice, trace = TRUE)
constraints(Fit1)
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