Zero-Inflated Poisson Distribution (Yip (1988) algorithm)
Fits a zero-inflated Poisson distribution based on Yip (1988).
yip88(link = "loglink", n.arg = NULL, imethod = 1)
link |
Link function for the usual lambda parameter.
See |
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 |
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.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
rrvglm and vgam.
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.
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$pstr0. However, this estimate is computed only for intercept
models, i.e., the formula is of the form y ~ 1.
Thomas W. Yee
Yip, P. (1988). Inference about the mean of a Poisson distribution in the presence of a nuisance parameter. The Australian Journal of Statistics, 30, 299–306.
Angers, J-F. and Biswas, A. (2003). A Bayesian analysis of zero-inflated generalized Poisson model. Computational Statistics & Data Analysis, 42, 37–46.
phi <- 0.35; lambda <- 2 # Generate some artificial data y <- rzipois(n <- 1000, lambda, phi) table(y) # Two equivalent ways of fitting the same model fit1 <- vglm(y ~ 1, yip88(n = length(y)), subset = y > 0) fit2 <- vglm(y ~ 1, yip88, trace = TRUE, crit = "coef") (true.mean <- (1-phi) * lambda) mean(y) head(fitted(fit1)) fit1@misc$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 # Estimate of phi (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)
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