VGAM (version 1.1-8)

posbinomial: Positive Binomial Distribution Family Function

Description

Fits a positive binomial distribution.

Usage

posbinomial(link = "logitlink", multiple.responses = FALSE,
    parallel = FALSE, omit.constant = FALSE, p.small = 1e-4,
    no.warning = FALSE, zero = NULL)

Value

An object of class "vglmff"

(see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Arguments

link, multiple.responses, parallel, zero

Details at CommonVGAMffArguments.

omit.constant

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.

p.small, no.warning

See posbernoulli.t.

Author

Thomas W. Yee

Warning

Under- or over-flow may occur if the data is ill-conditioned.

Details

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.

References

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.

See Also

posbernoulli.b, posbernoulli.t, posbernoulli.tb, binomialff, AICvlm, BICvlm, simulate.vlm.

Examples

# 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$N.hat,     # As tau = 6, i.e., 6 Bernoulli trials
  SE.N.hat = M.0@extra$SE.N.hat)  # per obsn is the same for each obsn

# 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)))  # Ok since omit.constant=TRUE
sort(c(M.0 = BIC(M.0), M.b = BIC(M.b)))  # Ok since omit.constant=TRUE