Fits a positive binomial distribution.
posbinomial(link = "logit", multiple.responses = FALSE, parallel = FALSE,
omit.constant = FALSE, p.small = 1e-4, no.warning = FALSE,
zero = NULL)Details at CommonVGAMffArguments.
Logical.
If TRUE then the constant (lchoose(size, size * yprop)
is omitted from the loglikelihood calculation.
If the model is to be compared using
AIC() or BIC()
(see AICvlm or BICvlm)
to the likes of
posbernoulli.tb etc. then it is important
to set omit.constant = TRUE because all models then
will not have any normalizing constants in the likelihood function.
Hence they become comparable.
This is because the \(M_0\) Otis et al. (1978) model
coincides with posbinomial().
See below for an example.
Also see posbernoulli.t regarding estimating the
population size (N.hat and SE.N.hat) if the
number of trials is the same for all observations.
See posbernoulli.t.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
Under- or over-flow may occur if the data is ill-conditioned.
The positive binomial distribution is the ordinary binomial distribution but with the probability of zero being zero. Thus the other probabilities are scaled up (i.e., divided by \(1-P(Y=0)\)). The fitted values are the ordinary binomial distribution fitted values, i.e., the usual mean.
In the capture--recapture literature this model is called the \(M_0\) if it is an intercept-only model. Otherwise it is called the \(M_h\) when there are covariates. It arises from a sum of a sequence of \(\tau\)-Bernoulli random variates subject to at least one success (capture). Here, each animal has the same probability of capture or recapture, regardless of the \(\tau\) sampling occasions. Independence between animals and between sampling occasions etc. is assumed.
Otis, D. L. et al. (1978) Statistical inference from capture data on closed animal populations, Wildlife Monographs, 62, 3--135.
Patil, G. P. (1962) Maximum likelihood estimation for generalised power series distributions and its application to a truncated binomial distribution. Biometrika, 49, 227--237.
Pearson, K. (1913) A Monograph on Albinism in Man. Drapers Company Research Memoirs.
posbernoulli.b,
posbernoulli.t,
posbernoulli.tb,
binomialff,
AICvlm, BICvlm,
simulate.vlm.
# NOT RUN {
# Number of albinotic children in families with 5 kids (from Patil, 1962) ,,,,
albinos <- data.frame(y = c(rep(1, 25), rep(2, 23), rep(3, 10), 4, 5),
n = rep(5, 60))
fit1 <- vglm(cbind(y, n-y) ~ 1, posbinomial, albinos, trace = TRUE)
summary(fit1)
Coef(fit1) # = MLE of p = 0.3088
head(fitted(fit1))
sqrt(vcov(fit1, untransform = TRUE)) # SE = 0.0322
# Fit a M_0 model (Otis et al. 1978) to the deermice data ,,,,,,,,,,,,,,,,,,,,,,,
M.0 <- vglm(cbind( y1 + y2 + y3 + y4 + y5 + y6,
6 - y1 - y2 - y3 - y4 - y5 - y6) ~ 1, trace = TRUE,
posbinomial(omit.constant = TRUE), data = deermice)
coef(M.0, matrix = TRUE)
Coef(M.0)
constraints(M.0, matrix = TRUE)
summary(M.0)
c( N.hat = M.0@extra$N.hat, # Since tau = 6, i.e., 6 Bernoulli trials per
SE.N.hat = M.0@extra$SE.N.hat) # observation is the same for each observation
# Compare it to the M_b using AIC and BIC
M.b <- vglm(cbind(y1, y2, y3, y4, y5, y6) ~ 1, trace = TRUE,
posbernoulli.b, data = deermice)
sort(c(M.0 = AIC(M.0), M.b = AIC(M.b))) # Okay since omit.constant = TRUE
sort(c(M.0 = BIC(M.0), M.b = BIC(M.b))) # Okay since omit.constant = TRUE
# }
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