Fits a zero-altered negative binomial distribution based on a conditional model involving a binomial distribution and a positive-negative binomial distribution.
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)Link function for the parameter \(p_0\), called pobs0 here.
See Links for more choices.
Link function applied to the munb parameter, which is the mean
\(\mu_{nb}\) of an ordinary negative binomial distribution.
See Links for more choices.
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.
See CommonVGAMffArguments
and fittedvlm for information.
Corresponding argument for the other parameterization. See details below.
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.
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.
See negbinomial.
See negbinomial.
See negbinomial.
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.
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.
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.
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.
# NOT RUN {
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))
# }
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