Zero-Inflated Negative Binomial Distribution Family Function
Fits a zero-inflated negative binomial distribution by full maximum likelihood estimation.
zinegbinomial(zero = "size",
type.fitted = c("mean", "munb", "pobs0", "pstr0",
"onempstr0"),
mds.min = 1e-3, nsimEIM = 500, cutoff.prob = 0.999,
eps.trig = 1e-7, max.support = 4000, max.chunk.MB = 30,
lpstr0 = "logitlink", lmunb = "loglink", lsize = "loglink",
imethod = 1, ipstr0 = NULL, imunb = NULL,
iprobs.y = NULL, isize = NULL,
gprobs.y = (0:9)/10,
gsize.mux = exp(c(-30, -20, -15, -10, -6:3)))
zinegbinomialff(lmunb = "loglink", lsize = "loglink", lonempstr0 = "logitlink",
type.fitted = c("mean", "munb", "pobs0", "pstr0",
"onempstr0"), imunb = NULL, isize = NULL, ionempstr0 =
NULL, zero = c("size", "onempstr0"), imethod = 1,
iprobs.y = NULL, cutoff.prob = 0.999,
eps.trig = 1e-7, max.support = 4000, max.chunk.MB = 30,
gprobs.y = (0:9)/10, gsize.mux = exp((-12:6)/2),
mds.min = 1e-3, nsimEIM = 500)lpstr0, lmunb, lsize |
Link functions for the parameters pstr0,
the mean and k; see |
type.fitted |
See |
ipstr0, isize, imunb |
Optional initial values for pstr0 and k and munb. The default is to compute an initial value internally for both. If a vector then recycling is used. |
lonempstr0, ionempstr0 |
Corresponding arguments for the other parameterization. See details below. |
imethod |
An integer with value |
zero |
Specifies which linear/additive predictors are to be modelled
as intercept-only. They can be such that their absolute values are
either 1 or 2 or 3.
The default is the pstr0 and k parameters
(both for each response).
See |
nsimEIM |
See |
iprobs.y, cutoff.prob, max.support, max.chunk.MB |
See |
mds.min, eps.trig |
See |
gprobs.y, gsize.mux |
These arguments relate to grid searching in the initialization process.
See |
These functions are based on
P(Y=0) = phi + (1- phi) * (k/(k+munb))^k,
and for y=1,2,…,
P(Y=y) = (1- phi) * dnbinom(y, munb, k).
The parameter phi satisfies 0 < phi < 1.
The mean of Y is (1-phi)*munb
(returned as the fitted values).
By default, the three linear/additive predictors
for zinegbinomial()
are (logit(phi),
log(munb), log(k))^T.
See negbinomial, another VGAM family function,
for the formula of the probability density function and other details
of the negative binomial distribution.
Independent multiple responses are handled.
If so then arguments ipstr0 and isize may be vectors
with length equal to the number of responses.
The VGAM family function zinegbinomialff() has a few
changes compared to zinegbinomial().
These are:
(i) the order of the linear/additive predictors is switched so the
NB mean comes first;
(ii) onempstr0 is now 1 minus the probability of a structural 0,
i.e., the probability of the parent (NB) component,
i.e., onempstr0 is 1-pstr0;
(iii) argument zero has a new default so that the onempstr0
is intercept-only by default.
Now zinegbinomialff() is generally recommended over
zinegbinomial().
Both functions implement Fisher scoring and can handle
multiple responses.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
This model can be difficult to fit to data,
and this family function is fragile.
The model is especially difficult to fit reliably when
the estimated k parameter is very large (so the model
approaches a zero-inflated Poisson distribution) or
much less than 1
(and gets more difficult as it approaches 0).
Numerical problems can also occur, e.g., when the probability of
a zero is actually less than, and not more than, the nominal
probability of zero.
Similarly, numerical problems can occur if there is little
or no 0-inflation, or when the sample size is small.
Half-stepping is not uncommon.
Successful convergence is sensitive to the initial values, therefore
if failure to converge occurs, try using combinations of arguments
stepsize (in vglm.control),
imethod,
imunb,
ipstr0,
isize, and/or
zero if there are explanatory variables.
Else try fitting an ordinary negbinomial model
or a zipoisson model.
This VGAM family function can be computationally expensive
and can run slowly;
setting trace = TRUE is useful for monitoring convergence.
If k is large then the use of VGAM family function
zipoisson is probably preferable.
This follows because the Poisson is the limiting distribution of a
negative binomial as k tends to infinity.
The zero-deflated negative binomial distribution
might be fitted by setting lpstr0 = identitylink,
albeit, not entirely reliably. See zipoisson
for information that can be applied here. Else try
the zero-altered negative binomial distribution (see
zanegbinomial).
T. W. Yee
## Not run:
# Example 1
ndata <- data.frame(x2 = runif(nn <- 1000))
ndata <- transform(ndata, pstr0 = logitlink(-0.5 + 1 * x2, inverse = TRUE),
munb = exp( 3 + 1 * x2),
size = exp( 0 + 2 * x2))
ndata <- transform(ndata,
y1 = rzinegbin(nn, mu = munb, size = size, pstr0 = pstr0))
with(ndata, table(y1)["0"] / sum(table(y1)))
nfit <- vglm(y1 ~ x2, zinegbinomial(zero = NULL), data = ndata)
coef(nfit, matrix = TRUE)
summary(nfit)
head(cbind(fitted(nfit), with(ndata, (1 - pstr0) * munb)))
round(vcov(nfit), 3)
# Example 2: RR-ZINB could also be called a COZIVGLM-ZINB-2
ndata <- data.frame(x2 = runif(nn <- 2000))
ndata <- transform(ndata, x3 = runif(nn))
ndata <- transform(ndata, eta1 = 3 + 1 * x2 + 2 * x3)
ndata <- transform(ndata, pstr0 = logitlink(-1.5 + 0.5 * eta1, inverse = TRUE),
munb = exp(eta1),
size = exp(4))
ndata <- transform(ndata,
y1 = rzinegbin(nn, pstr0 = pstr0, mu = munb, size = size))
with(ndata, table(y1)["0"] / sum(table(y1)))
rrzinb <- rrvglm(y1 ~ x2 + x3, zinegbinomial(zero = NULL), data = ndata,
Index.corner = 2, str0 = 3, trace = TRUE)
coef(rrzinb, matrix = TRUE)
Coef(rrzinb)
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