yip88: Zero-Inflated Poisson Distribution (Yip (1988) algorithm)

In VGAMdata: Data Supporting the 'VGAM' Package

Zero-Inflated Poisson Distribution (Yip (1988) algorithm)

Description

Fits a zero-inflated Poisson distribution based on Yip (1988).

Usage

yip88(link = "loglink", n.arg = NULL, imethod = 1)

Arguments

link	Link function for the usual \lambda parameter. See Links for more choices.
n.arg	The total number of observations in the data set. Needed when the response variable has all the zeros deleted from it, so that the number of zeros can be determined.
imethod	Details at CommonVGAMffArguments.

Details

The method implemented here, Yip (1988), maximizes a conditional likelihood. Consequently, the methodology used here deletes the zeros from the data set, and is thus related to the positive Poisson distribution (where P(Y=0) = 0).

The probability function of Y is 0 with probability \phi, and Poisson(\lambda) with probability 1-\phi. Thus

P(Y=0) =\phi + (1-\phi) P(W=0)

where W is Poisson(\lambda). The mean, (1-\phi) \lambda, can be obtained by the extractor function fitted applied to the object.

This family function treats \phi as a scalar. If you want to model both \phi and \lambda as a function of covariates, try zipoisson.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, rrvglm and vgam.

Warning

Under- or over-flow may occur if the data is ill-conditioned. Yip (1988) only considered \phi being a scalar and not modelled as a function of covariates. To get around this limitation, try zipoisson.

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.

Note

The data may be inputted in two ways. The first is when the response is a vector of positive values, with the argument n in yip88 specifying the total number of observations. The second is simply include all the data in the response. In this case, the zeros are trimmed off during the computation, and the x and y slots of the object, if assigned, will reflect this.

The estimate of \phi is placed in the misc slot as @misc$pstr0. However, this estimate is computed only for intercept models, i.e., the formula is of the form y ~ 1.

Author(s)

Thomas W. Yee

References

Yip, P. (1988). Inference about the mean of a Poisson distribution in the presence of a nuisance parameter. The Australian Journal of Statistics, 30, 299–306.

Angers, J-F. and Biswas, A. (2003). A Bayesian analysis of zero-inflated generalized Poisson model. Computational Statistics & Data Analysis, 42, 37–46.

See Also

zipoisson, Zipois, zapoisson, pospoisson, poissonff, dzipois.

Examples

phi <- 0.35; lambda <- 2  # Generate some artificial data
y <- rzipois(n <- 1000, lambda, phi)
table(y)

# Two equivalent ways of fitting the same model
fit1 <- vglm(y ~ 1, yip88(n = length(y)), subset = y > 0)
fit2 <- vglm(y ~ 1, yip88, trace = TRUE, crit = "coef")
(true.mean <- (1-phi) * lambda)
mean(y)
head(fitted(fit1))
fit1@misc$pstr0  # The estimate of phi

# Compare the ZIP with the positive Poisson distribution
pp <- vglm(y ~ 1, pospoisson, subset = y > 0, crit = "c")
coef(pp)
Coef(pp)
coef(fit1) - coef(pp)            # Same
head(fitted(fit1) - fitted(pp))  # Different

# Another example (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, yip88(n = length(yy)), subset = yy > 0)
fit3@misc$pstr0  # phi estimate (they get 0.5154 with SE 0.0707)
coef(fit3, matrix = TRUE)
Coef(fit3)  # Estimate of lambda (they get 0.6997 with SE 0.1520)
head(fitted(fit3))
mean(yy)  # Compare this with fitted(fit3)

VGAMdata documentation built on Sept. 17, 2024, 9:09 a.m.