VGAM (version 1.1-14)

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 = "logitlink", lmunb = "loglink", lsize = "loglink",
              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 = "loglink", lsize = "loglink", lonempobs0 = "logitlink",
                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)

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.

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.

See Also

gaitdnbinomial, posnegbinomial, Gaitdnbinom, negbinomial, binomialff, zinegbinomial, zipoisson, spikeplot, dnbinom, CommonVGAMffArguments, simulate.vlm.

Examples

if (FALSE) {
zdata <- data.frame(x2 = runif(nn <- 2000))
zdata <- transform(zdata, pobs0 = logitlink(-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)
head(fitted(fit))
head(predict(fit))
}