Positive Binomial Distribution Family Function
Fits a positive binomial distribution.
posbinomial(link = "logitlink", multiple.responses = FALSE, parallel = FALSE,
omit.constant = FALSE, p.small = 1e-4, no.warning = FALSE,
zero = NULL)link, multiple.responses, parallel, zero |
Details at |
omit.constant |
Logical.
If |
p.small, no.warning |
See |
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 τ-Bernoulli random variates subject to at least one success (capture). Here, each animal has the same probability of capture or recapture, regardless of the τ sampling occasions. Independence between animals and between sampling occasions etc. is assumed.
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 input for this family function is the same as
binomialff.
If multiple.responses = TRUE then each column of the matrix response
should be a count (the number of successes), and the
weights argument should be a matrix of the same dimension
as the response containing the number of trials.
If multiple.responses = FALSE then the response input should be the same
as binomialff.
Yet to be done: a quasi.posbinomial() which estimates a
dispersion parameter.
Thomas W. Yee
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.
# 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 = TRUEPlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.