VGAM (version 1.0-4)

# zanegbinomial: Zero-Altered Negative Binomial Distribution

## Description

Fits a zero-altered negative binomial distribution based on a conditional model involving a binomial distribution and a positive-negative binomial distribution.

## Usage

zanegbinomial(zero = "size", type.fitted = c("mean", "munb", "pobs0"),
mds.min = 1e-3, nsimEIM = 500, cutoff.prob = 0.999,
eps.trig = 1e-7, max.support = 4000, max.chunk.MB = 30,
lpobs0 = "logit", lmunb = "loge", lsize = "loge",
imethod = 1, ipobs0 = NULL,
imunb = NULL, iprobs.y = NULL, gprobs.y = (0:9)/10,
isize = NULL, gsize.mux = exp(c(-30, -20, -15, -10, -6:3)))
zanegbinomialff(lmunb = "loge", lsize = "loge", lonempobs0 = "logit",
type.fitted = c("mean", "munb", "pobs0", "onempobs0"),
isize = NULL, ionempobs0 = NULL, zero = c("size",
"onempobs0"), mds.min = 1e-3, iprobs.y = NULL, gprobs.y = (0:9)/10,
cutoff.prob = 0.999, eps.trig = 1e-7, max.support = 4000,
max.chunk.MB = 30, gsize.mux = exp(c(-30, -20, -15, -10, -6:3)),
imethod = 1, imunb = NULL,
nsimEIM = 500)

## Arguments

lpobs0

Link function for the parameter $$p_0$$, called pobs0 here. See Links for more choices.

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 reciprocal of the dispersion parameter, called k. That is, as k increases, the variance of the response decreases. See Links for more choices.

type.fitted

See CommonVGAMffArguments and fittedvlm for information.

lonempobs0, ionempobs0

Corresponding argument for the other parameterization. See details below.

ipobs0, imunb, isize

Optional initial values for $$p_0$$ and munb and k. If given then it is okay to give one value for each response/species by inputting a vector whose length is the number of columns of the response matrix.

zero

Specifies which of the three linear predictors are modelled as intercept-only. All parameters can be modelled as a function of the explanatory variables by setting zero = NULL (not recommended). A negative value means that the value is recycled, e.g., setting $$-3$$ means all k are intercept-only for zanegbinomial. See CommonVGAMffArguments for more information.

nsimEIM, imethod

See CommonVGAMffArguments.

iprobs.y, gsize.mux, gprobs.y

See negbinomial.

cutoff.prob, eps.trig

See negbinomial.

mds.min, max.support, max.chunk.MB

See negbinomial.

## Value

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

The fitted.values slot of the fitted object, which should be extracted by the generic function fitted, returns the mean $$\mu$$ (default) which is given by $$\mu = (1-p_0) \mu_{nb} / [1 - (k/(k+\mu_{nb}))^k].$$ If type.fitted = "pobs0" then $$p_0$$ is returned.

## Warning

This family function is fragile; it inherits the same difficulties as posnegbinomial. Convergence for this VGAM family function seems to depend quite strongly on providing good initial values.

This VGAM family function is computationally expensive and usually runs slowly; setting trace = TRUE is useful for monitoring convergence.

Inference obtained from summary.vglm and summary.vgam may or may not be correct. In particular, the p-values, standard errors and degrees of freedom may need adjustment. Use simulation on artificial data to check that these are reasonable.

## Details

The response $$Y$$ is zero with probability $$p_0$$, or $$Y$$ has a positive-negative binomial distribution with probability $$1-p_0$$. Thus $$0 < p_0 < 1$$, which is modelled as a function of the covariates. The zero-altered negative binomial distribution differs from the zero-inflated negative binomial distribution in that the former has zeros coming from one source, whereas the latter has zeros coming from the negative binomial distribution too. The zero-inflated negative binomial distribution is implemented in the VGAM package. Some people call the zero-altered negative binomial a hurdle model.

For one response/species, by default, the three linear/additive predictors for zanegbinomial() are $$(logit(p_0), \log(\mu_{nb}), \log(k))^T$$. This vector is recycled for multiple species.

The VGAM family function zanegbinomialff() has a few changes compared to zanegbinomial(). These are: (i) the order of the linear/additive predictors is switched so the negative binomial mean comes first; (ii) argument onempobs0 is now 1 minus the probability of an observed 0, i.e., the probability of the positive negative binomial distribution, i.e., onempobs0 is 1-pobs0; (iii) argument zero has a new default so that the pobs0 is intercept-only by default. Now zanegbinomialff() is generally recommended over zanegbinomial(). Both functions implement Fisher scoring and can handle multiple responses.

## References

Welsh, A. H., Cunningham, R. B., Donnelly, C. F. and Lindenmayer, D. B. (1996) Modelling the abundances of rare species: statistical models for counts with extra zeros. Ecological Modelling, 88, 297--308.

Yee, T. W. (2014) Reduced-rank vector generalized linear models with two linear predictors. Computational Statistics and Data Analysis, 71, 889--902.

dzanegbin, posnegbinomial, negbinomial, binomialff, rposnegbin, zinegbinomial, zipoisson, dnbinom, CommonVGAMffArguments, simulate.vlm.

## Examples

Run this code
# NOT RUN {
zdata <- data.frame(x2 = runif(nn <- 2000))
zdata <- transform(zdata, pobs0 = logit(-1 + 2*x2, inverse = TRUE))
zdata <- transform(zdata,
y1 = rzanegbin(nn, munb = exp(0+2*x2), size = exp(1), pobs0 = pobs0),
y2 = rzanegbin(nn, munb = exp(1+2*x2), size = exp(1), pobs0 = pobs0))
with(zdata, table(y1))
with(zdata, table(y2))

fit <- vglm(cbind(y1, y2) ~ x2, zanegbinomial, data = zdata, trace = TRUE)
coef(fit, matrix = TRUE)