Maximum likelihood estimation of the two parameters of a positive negative binomial distribution.
posnegbinomial(zero = "size",
type.fitted = c("mean", "munb", "prob0"),
mds.min = 0.001, nsimEIM = 500, cutoff.prob = 0.999,
eps.trig = 1e-07, max.support = 4000, max.chunk.MB = 30,
lmunb = "loglink", lsize = "loglink", imethod = 1,
imunb = NULL, iprobs.y = NULL,
gprobs.y = ppoints(8), isize = NULL,
gsize.mux = exp(c(-30, -20, -15, -10, -6:3)))An object of class "vglmff"
(see vglmff-class).
The object is used by modelling functions
such as vglm,
rrvglm and vgam.
Link function applied to the munb parameter, which is
the mean Links for more choices.
Parameter link function applied to the dispersion parameter,
called k.
See Links for more choices.
Optional initial value for k, an index parameter.
The value 1/k is known as a dispersion parameter.
If failure to converge occurs try different values (and/or use
imethod).
If necessary this vector is recycled to length equal to the
number of responses.
A value NULL means an initial value for each response
is computed internally using a range of values.
See CommonVGAMffArguments.
Similar to negbinomial.
Similar to negbinomial.
Similar to negbinomial.
See negbinomial.
See CommonVGAMffArguments for details.
This family function is fragile;
at least two cases will lead to numerical problems.
Firstly,
the positive-Poisson model corresponds to k equalling infinity.
If the data is positive-Poisson or close to positive-Poisson,
then the estimated k will diverge to Inf or some
very large value.
Secondly, if the data is clustered about the value 1 because
the munb parameter is close to 0
then numerical problems will also occur.
Users should set trace = TRUE to monitor convergence.
In the situation when both cases hold, the result returned
(which will be untrustworthy) will depend on the initial values.
The negative binomial distribution (NBD) is a strictly unimodal
distribution. Any data set that does not exhibit a mode (in the
middle) makes the estimation problem difficult. The positive
NBD inherits this feature. Set trace = TRUE to monitor
convergence.
See the example below of a data set where posbinomial()
fails; the so-called solution is extremely poor.
This is partly due to a lack of a
unimodal shape because the number of counts decreases only.
This long tail makes it very difficult to estimate the mean
parameter with any certainty. The result too is that the
size parameter is numerically fraught.
This VGAM family function inherits the same warnings as
negbinomial.
And if k is much less than 1 then the estimation may
be slow.
Thomas W. Yee
The positive negative binomial distribution is an ordinary negative binomial distribution but with the probability of a zero response being zero. The other probabilities are scaled to sum to unity.
This family function is based on negbinomial
and most details can be found there. To avoid confusion, the
parameter munb here corresponds to the mean of an ordinary
negative binomial distribution negbinomial. The
mean of posnegbinomial is
The parameters munb and k are not independent in
the positive negative binomial distribution, whereas they are
in the ordinary negative binomial distribution.
This function handles multiple responses, so that a
matrix can be used as the response. The number of columns is
the number of species, say, and setting zero = -2 means
that all species have a k equalling a (different)
intercept only.
Barry, S. C. and Welsh, A. H. (2002). Generalized additive modelling and zero inflated count data. Ecological Modelling, 157, 179--188.
Williamson, E. and Bretherton, M. H. (1964). Tables of the logarithmic series distribution. Annals of Mathematical Statistics, 35, 284--297.
gaitdnbinomial,
pospoisson,
negbinomial,
zanegbinomial,
rnbinom,
CommonVGAMffArguments,
corbet,
logff,
simulate.vlm,
margeff.
if (FALSE) {
pdata <- data.frame(x2 = runif(nn <- 1000))
pdata <- transform(pdata,
y1 = rgaitdnbinom(nn, exp(1), munb.p = exp(0+2*x2), truncate = 0),
y2 = rgaitdnbinom(nn, exp(3), munb.p = exp(1+2*x2), truncate = 0))
fit <- vglm(cbind(y1, y2) ~ x2, posnegbinomial, pdata, trace = TRUE)
coef(fit, matrix = TRUE)
dim(depvar(fit)) # Using dim(fit@y) is not recommended
# Another artificial data example
pdata2 <- data.frame(munb = exp(2), size = exp(3)); nn <- 1000
pdata2 <- transform(pdata2,
y3 = rgaitdnbinom(nn, size, munb.p = munb,
truncate = 0))
with(pdata2, table(y3))
fit <- vglm(y3 ~ 1, posnegbinomial, data = pdata2, trace = TRUE)
coef(fit, matrix = TRUE)
with(pdata2, mean(y3)) # Sample mean
head(with(pdata2, munb/(1-(size/(size+munb))^size)), 1) # Popn mean
head(fitted(fit), 3)
head(predict(fit), 3)
# Example: Corbet (1943) butterfly Malaya data
fit <- vglm(ofreq ~ 1, posnegbinomial, weights = species, corbet)
coef(fit, matrix = TRUE)
Coef(fit)
(khat <- Coef(fit)["size"])
pdf2 <- dgaitdnbinom(with(corbet, ofreq), khat,
munb.p = fitted(fit), truncate = 0)
print(with(corbet,
cbind(ofreq, species, fitted = pdf2*sum(species))), dig = 1)
with(corbet,
matplot(ofreq, cbind(species, fitted = pdf2*sum(species)), las = 1,
xlab = "Observed frequency (of individual butterflies)",
type = "b", ylab = "Number of species", col = c("blue", "orange"),
main = "blue 1s = observe; orange 2s = fitted"))
# Data courtesy of Maxim Gerashchenko causes posbinomial() to fail
pnbd.fail <- data.frame(
y1 = c(1:16, 18:21, 23:28, 33:38, 42, 44, 49:51, 55, 56, 58,
59, 61:63, 66, 73, 76, 94, 107, 112, 124, 190, 191, 244),
ofreq = c(130, 80, 38, 23, 22, 11, 21, 14, 6, 7, 9, 9, 9, 4, 4, 5, 1,
4, 6, 1, 3, 2, 4, 3, 4, 5, 3, 1, 2, 1, 1, 4, 1, 2, 2, 1, 3,
1, 1, 2, 2, 2, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1))
fit.fail <- vglm(y1 ~ 1, weights = ofreq, posnegbinomial,
trace = TRUE, data = pnbd.fail)
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