zapoisson {VGAM} | R Documentation |
Fits a zero-altered Poisson distribution based on a conditional model involving a binomial distribution and a positive-Poisson distribution.
zapoisson(lp0 = "logit", llambda = "loge", ep0=list(), elambda=list(), zero=NULL)
lp0 |
Link function for the parameter p0, called p0 here.
See Links for more choices.
|
llambda |
Link function for the usual lambda parameter.
See Links for more choices.
|
ep0, elambda |
Extra argument for the respective links.
See earg in Links for general information.
|
zero |
Integer valued vector, usually assigned -1 or 1 if used
at all. Specifies which of the two linear/additive predictors are
modelled as an intercept only.
By default, both linear/additive predictors are modelled using
the explanatory variables.
If zero=1 then the p0 parameter
(after lp0 is applied) is modelled as a single unknown
number that is estimated. It is modelled as a function of the
explanatory variables by zero=NULL . A negative value
means that the value is recycled, so setting -1 means all p0
are intercept-only (for multivariate responses).
|
The response Y is zero with probability p0, or Y has a positive-Poisson(lambda) distribution with probability 1-p0. Thus 0 < p0 < 1, which is modelled as a function of the covariates. The zero-altered Poisson distribution differs from the zero-inflated Poisson distribution in that the former has zeros coming from one source, whereas the latter has zeros coming from the Poisson distribution too. Some people call the zero-altered Poisson a hurdle model.
For one response/species, by default, the two linear/additive predictors are (logit(p0), log(lambda))^T.
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) * lambda / [1 - exp(-lambda)].
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.
There are subtle differences between this family function and
zipoisson
and yip88
.
In particular, zipoisson
is a
mixture model whereas zapoisson
and yip88
are conditional models.
Note this family function allows p0 to be modelled
as functions of the covariates. It can be thought of an extension
of yip88
, which is also a conditional model but its
phi parameter is a scalar only.
This family function effectively combines pospoisson
and binomialff
into one family function.
This family function can handle a multivariate response, e.g., more than one species.
T. W. Yee
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.
Angers, J-F. and Biswas, A. (2003) A Bayesian analysis of zero-inflated generalized Poisson model. Computational Statistics & Data Analysis, 42, 37–46.
Documentation accompanying the VGAM package at http://www.stat.auckland.ac.nz/~yee contains further information and examples.
zipoisson
,
yip88
,
pospoisson
,
posnegbinomial
,
binomialff
,
rpospois
.
x = runif(n <- 1000) p0 = logit(-1 + 1*x, inverse=TRUE) lambda = loge(-0.3 + 2*x, inverse=TRUE) y = ifelse(runif(n) < p0, 0, rpospois(n, lambda)) table(y) fit = vglm(y ~ x, zapoisson, trace=TRUE) fit = vglm(y ~ x, zapoisson, trace=TRUE, crit="c") fitted(fit)[1:5] predict(fit)[1:5,] predict(fit, untransform=TRUE)[1:5,] coef(fit, matrix=TRUE) # Another example ------------------------------ # Data from Angers and Biswas (2003) y = 0:7; w = c(182, 41, 12, 2, 2, 0, 0, 1) y = y[w>0] w = w[w>0] yy = rep(y,w) fit3 = vglm(yy ~ 1, zapoisson, trace=TRUE, crit="c") coef(fit3, matrix=TRUE) Coef(fit3) # Estimate of lambda (they get 0.6997 with standard error 0.1520) fitted(fit3)[1:5] mean(yy) # compare this with fitted(fit3)