zanegbinomial {VGAM} | R Documentation

## 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(lp0="logit", lmunb = "loge", lk = "loge",
                  ep0=list(), emunb =list(), ek = list(), ipnb0 = NULL,
                  ik = NULL, zero = -3, cutoff = 0.995, method.init=1,
                  shrinkage.init=0.95)

### Arguments

`lp0` | Link function for the parameter p0, called `p0` here. See `Links` for more choices.
`lmunb` | Link function applied to the `munb` parameter, which is the mean munb of an ordinary negative binomial distribution. See `Links` for more choices.
`lk` | 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.
`ep0, emunb, ek` | List. Extra argument for the respective links. See `earg` in `Links` for general information.
`ipnb0` | Optional initial values for the probability p0. If given, they must be in (0,1), and it is ok to give one value each for each response/species.
`ik` | Optional initial values for `k`. If given, they must be positive, and give one value for each response/species.
`zero` | Integer valued vector, usually assigned -3 or 3 if used at all. Specifies which of the three linear predictors are modelled as an intercept only. By default, the `k` parameter (after `lk` is applied) for each response is modelled as a single unknown number that is estimated. It can be modelled as a function of the explanatory variables by setting `zero=NULL`. A negative value means that the value is recycled, so setting -3 means all `k` are intercept only.
`cutoff` | A numeric which is close to 1 but never exactly 1. Used to specify how many terms of the infinite series are actually used. The sum of the probabilites are added until they reach this value or more. It is like specifying `p` in an imaginary function `qnegbin(p)`.
`method.init, shrinkage.init` | See `negbinomial`.

### Details

The response Y is zero with probability p0, or Y has a positive-negative binomial distribution with probability 1-p0. Thus 0 < p0 < 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 are (logit(p0), log(munb), log(k))^T. This vector is recycled for multiple species.

### 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 which is given by

mu = (1-p0) * munb / [1 - (k/(k+munb))^k].

### Warning

Convergence for this VGAM family function seems to depend quite strongly on providing good initial values.

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.

### Note

Note this family function allows p0 to be modelled as functions of the covariates. It is a conditional model, not a mixture model.

This family function effectively combines `posnegbinomial` and `binomialff` into one family function.

This family function can handle a multivariate response, e.g., more than one species.

### Author(s)

T. W. Yee

### 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.

### See Also

`dzanegbin`, `posnegbinomial`, `negbinomial`, `binomialff`, `rposnegbin`, `zinegbinomial`, `zipoisson`, `dnbinom`.

### Examples

    x = runif(n <- 2000)
    p0 = logit(-1 + 2*x, inverse=TRUE)
    y1 = rposnegbin(n, munb=exp(0+2*x), size=exp(1))   # With covariates
    y2 = rposnegbin(n, munb=exp(1+2*x), size=exp(1))   # With covariates
    y1 = ifelse(runif(n) < p0, 0, y1)
    y2 = ifelse(runif(n) < p0, 0, y2)
    table(y1)
    table(y2)

    fit = vglm(cbind(y1,y2) ~ x, zanegbinomial, trace=TRUE)
    coef(fit, matrix=TRUE)
    head(fitted(fit))
    head(predict(fit))

[Package VGAM version 0.7-9 Index]