zapoisson: Zero-Altered Poisson Distribution

Description

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

Usage

zapoisson(lpobs0 = "logitlink", llambda = "loglink", type.fitted =
    c("mean", "lambda", "pobs0", "onempobs0"), imethod = 1,
    ipobs0 = NULL, ilambda = NULL, ishrinkage = 0.95, probs.y = 0.35,
    zero = NULL)
zapoissonff(llambda = "loglink", lonempobs0 = "logitlink", type.fitted =
    c("mean", "lambda", "pobs0", "onempobs0"), imethod = 1,
    ilambda = NULL, ionempobs0 = NULL, ishrinkage = 0.95,
    probs.y = 0.35, zero = "onempobs0")

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 $\mu$ (default) which is given by

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

If type.fitted = "pobs0" then $p_0$ is returned.

Arguments

lpobs0

Link function for the parameter $p_0$, called pobs0 here. See Links for more choices.

llambda

Link function for the usual $\lambda$ parameter. See Links for more choices.

type.fitted

See CommonVGAMffArguments and fittedvlm for information.

lonempobs0

Corresponding argument for the other parameterization. See details below.

imethod, ipobs0, ionempobs0, ilambda, ishrinkage: See CommonVGAMffArguments for information.
probs.y, zero: See CommonVGAMffArguments for information.

Author

T. W. Yee

Details

The response $Y$ is zero with probability $p_0$, else $Y$ 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 for zapoisson() are $(logit(p_0), \log(\lambda))^T$.

The VGAM family function zapoissonff() has a few changes compared to zapoisson(). These are: (i) the order of the linear/additive predictors is switched so the Poisson mean comes first; (ii) argument onempobs0 is now 1 minus the probability of an observed 0, i.e., the probability of the positive Poisson distribution, i.e., onempobs0 is 1-pobs0; (iii) argument zero has a new default so that the onempobs0 is intercept-only by default. Now zapoissonff() is generally recommended over zapoisson(). Both functions implement Fisher scoring and can handle multiple responses.

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.

Yee, T. W. (2014). Reduced-rank vector generalized linear models with two linear predictors. Computational Statistics and Data Analysis, 71, 889--902.

Examples

Run this code

zdata <- data.frame(x2 = runif(nn <- 1000))
zdata <- transform(zdata, pobs0  = logitlink( -1 + 1*x2, inverse = TRUE),
                          lambda = loglink(-0.5 + 2*x2, inverse = TRUE))
zdata <- transform(zdata, y = rgaitdpois(nn, lambda, pobs.mlm = pobs0,
                                        a.mlm = 0))

with(zdata, table(y))
fit <- vglm(y ~ x2, zapoisson, data = zdata, trace = TRUE)
fit <- vglm(y ~ x2, zapoisson, data = zdata, trace = TRUE, crit = "coef")
head(fitted(fit))
head(predict(fit))
head(predict(fit, untransform = TRUE))
coef(fit, matrix = TRUE)
summary(fit)

# 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)
Abdata <- data.frame(yy = with(abdata, rep(y, w)))
fit3 <- vglm(yy ~ 1, zapoisson, data = Abdata, trace = TRUE, crit = "coef")
coef(fit3, matrix = TRUE)
Coef(fit3)  # Estimate lambda (they get 0.6997 with SE 0.1520)
head(fitted(fit3), 1)
with(Abdata, mean(yy))  # Compare this with fitted(fit3)

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