VGAM (version 1.0-4)

# posnegbinomial: Positive Negative Binomial Distribution Family Function

## Description

Maximum likelihood estimation of the two parameters of a positive negative binomial distribution.

## Usage

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 = "loge", lsize = "loge", imethod = 1,
imunb = NULL, iprobs.y = NULL,
gprobs.y = ppoints(8), isize = NULL,
gsize.mux = exp(c(-30, -20, -15, -10, -6:3)))

## Arguments

lmunb

Link function applied to the munb parameter, which is the mean $$\mu_{nb}$$ of an ordinary negative binomial distribution. See Links for more choices.

lsize

Parameter link function applied to the dispersion parameter, called k. See Links for more choices.

isize

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.

nsimEIM, zero, eps.trig

See CommonVGAMffArguments.

mds.min, iprobs.y, cutoff.prob

Similar to negbinomial.

imunb, max.support

Similar to negbinomial.

max.chunk.MB, gsize.mux

Similar to negbinomial.

imethod, gprobs.y

See negbinomial.

type.fitted

See CommonVGAMffArguments for details.

## Value

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

## Warning

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.

## Details

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 $$\mu_{nb} / (1-p(0))$$ where $$p(0) = (k/(k + \mu_{nb}))^k$$ is the probability an ordinary negative binomial distribution has a zero value.

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.

## References

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.

rposnegbin, pospoisson, negbinomial, zanegbinomial, rnbinom, CommonVGAMffArguments, corbet, logff, simulate.vlm.

## Examples

Run this code
# NOT RUN {
pdata <- data.frame(x2 = runif(nn <- 1000))
pdata <- transform(pdata, y1 = rposnegbin(nn, munb = exp(0+2*x2), size = exp(1)),
y2 = rposnegbin(nn, munb = exp(1+2*x2), size = exp(3)))
fit <- vglm(cbind(y1, y2) ~ x2, posnegbinomial, data = 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 = rposnegbin(nn, munb = munb, size = size))
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)  # Population mean

# Example: Corbet (1943) butterfly Malaya data
fit <- vglm(ofreq ~ 1, posnegbinomial, weights = species, data = corbet)
coef(fit, matrix = TRUE)
Coef(fit)
(khat <- Coef(fit)["size"])
pdf2 <- dposnegbin(x = with(corbet, ofreq), mu = fitted(fit), size = khat)
print(with(corbet, cbind(ofreq, species, fitted = pdf2*sum(species))), dig = 1)
# }
# NOT RUN {
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"))
# }
# NOT RUN {
# }
# NOT RUN {
# This 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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