VGAM (version 0.8-1)

zapoisson: Zero-Altered Poisson Distribution

Description

Fits a zero-altered Poisson distribution based on a conditional model involving a binomial distribution and a positive-Poisson distribution.

Usage

zapoisson(lp0 = "logit", llambda = "loge", ep0=list(),
          elambda=list(), zero=NULL)

Arguments

lp0

Link function for the parameter $p_0$, 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.

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 $$\mu = (1-p_0) \lambda / [1 - \exp(-\lambda)].$$

Warning

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.

Details

The response $Y$ is zero with probability $p_0$, or $Y$ has a positive-Poisson($\lambda)$ distribution with probability $1-p_0$. Thus $0 < p_0 < 1$, 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 $(logit(p_0), \log(\lambda))^T$.

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.

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.

See Also

rzapois, zipoisson, pospoisson, posnegbinomial, binomialff, rpospois.

Examples

zapdata = data.frame(x = runif(nn <- 1000))
zapdata = transform(zapdata, p0 = logit(-1 + 1*x, inverse=TRUE),
                             lambda = loge(-0.5 + 2*x, inverse=TRUE))
zapdata = transform(zapdata, y = rzapois(nn, lambda, p0=p0))

with(zapdata, table(y))
fit = vglm(y ~ x, zapoisson, zapdata, trace=TRUE)
fit = vglm(y ~ x, zapoisson, zapdata, trace=TRUE, crit="c")
head(fitted(fit))
head(predict(fit))
head(predict(fit, untransform=TRUE))
coef(fit, matrix=TRUE)


# Another example ------------------------------
# Data from Angers and Biswas (2003)
abdata = data.frame(y = 0:7, w = c(182, 41, 12, 2, 2, 0, 0, 1))
abdata = subset(abdata, w>0)
yy = with(abdata, 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)
head(fitted(fit3), 1)
mean(yy) # compare this with fitted(fit3)