zapoisson: Zero-Altered Poisson Distribution

In VGAM: Vector Generalized Linear and Additive Models

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")

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.

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.

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.

Note

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 p_0 to be modelled as functions of the covariates.

This family function effectively combines pospoisson and binomialff into one family function. This family function can handle multiple responses, e.g., more than one species.

It is recommended that Gaitdpois be used, e.g., rgaitdpois(nn, lambda, pobs.mlm = pobs0, a.mlm = 0) instead of rzapois(nn, lambda, pobs0 = pobs0).

Author(s)

T. W. Yee

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.

See Also

Gaitdpois, rzapois, zipoisson, gaitdpoisson, pospoisson, posnegbinomial, spikeplot, binomialff, CommonVGAMffArguments, simulate.vlm.

Examples

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)

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.