zipoisson {VGAM}R Documentation

Zero-Inflated Poisson Distribution Family Function

Description

Fits a zero-inflated Poisson distribution by full maximum likelihood estimation.

Usage

zipoissonff(llambda = "loge", lprobp = "logit",
            elambda = list(), eprobp = list(),
            ilambda = NULL, iprobp = NULL, imethod = 1,
            shrinkage.init = 0.8, zero = -2)
zipoisson(lphi = "logit", llambda = "loge",
          ephi = list(), elambda = list(),
          iphi = NULL,   ilambda = NULL, imethod = 1,
          shrinkage.init = 0.8, zero = NULL)

Arguments

lphi, llambda, ephi, elambda

Link function and extra argument for the parameter phi and the usual lambda parameter. See Links for more choices, and earg in Links for general information. See CommonVGAMffArguments for more information.

iphi, ilambda

Optional initial values for phi, whose values must lie between 0 and 1. Optional initial values for lambda, whose values must be positive. The defaults are to compute an initial value internally for each. If a vector then recycling is used.

lprobp, eprobp, iprobp

Corresponding arguments for the other parameterization. See details below.

imethod

An integer with value 1 or 2 which specifies the initialization method for lambda. If failure to converge occurs try another value and/or else specify a value for shrinkage.init and/or else specify a value for iphi. See CommonVGAMffArguments for more information.

shrinkage.init

How much shrinkage is used when initializing lambda. The value must be between 0 and 1 inclusive, and a value of 0 means the individual response values are used, and a value of 1 means the median or mean is used. This argument is used in conjunction with imethod. See CommonVGAMffArguments for more information.

zero

An integer specifying which linear/additive predictor is modelled as intercepts only. If given, the value must be either 1 or 2, and the default is none of them. Setting zero = 1 makes phi a single parameter. See CommonVGAMffArguments for more information.

Details

This model is a mixture of a Poisson distribution and the value 0; it has value 0 with probability phi else is Poisson(lambda) distributed. Thus there are two sources for zero values, and phi is the probability of a structural zero. The model for zipoisson() can be written

P(Y = 0) = phi + (1-phi) * exp(-lambda),

and for y=1,2,…,

P(Y = y) = (1-phi) * exp(-lambda) * lambda^y / y!.

Here, the parameter phi satisfies 0 < phi < 1. The mean of Y is (1-phi)*lambda and these are returned as the fitted values. The variance of Y is (1-phi)*lambda*(1 + phi lambda). By default, the two linear/additive predictors are (logit(phi), log(lambda))^T. This function implements Fisher scoring.

The VGAM family function zipoissonff() has a few changes compared to zipoisson(). These are: (i) the order of the linear/additive predictors is switched so the Poisson mean comes first; (ii) probp is now the probability of the Poisson component, i.e., probp is 1-phi; (iii) it can handle multiple responses; (iv) argument zero has a new default so that the probp is an intercept-only by default. Now zipoissonff() is generally recommended over zipoisson(), and definitely recommended over yip88.

Value

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

Warning

Numerical problems can occur, e.g., when the probability of zero is actually less than, not more than, the nominal probability of zero. For example, in the Angers and Biswas (2003) data below, replacing 182 by 1 results in nonconvergence. Half-stepping is not uncommon. If failure to converge occurs, try using combinations of imethod, shrinkage.init, iphi, and/or zipoisson(zero = 1) if there are explanatory variables. The default for zipoissonff() is to model the structural zero probability as an intercept-only.

Note

For intercept-models, the misc slot has a component called p0 which is the estimate of P(Y = 0). Note that P(Y = 0) is not the parameter phi. This family function currently cannot handle a multivariate response.

The zero-deflated Poisson distribution cannot be handled with this family function. It can be handled with the zero-altered Poisson distribution; see zapoisson.

The use of this VGAM family function with rrvglm can result in a so-called COZIGAM or COZIGLM. That is, a reduced-rank zero-inflated Poisson model (RR-ZIP) is a constrained zero-inflated generalized linear model. See COZIGAM. A RR-ZINB model can also be fitted easily; see zinegbinomial. Jargon-wise, a COZIGLM might be better described as a COZIVGLM-ZIP.

Author(s)

T. W. Yee

References

Thas, O. and Rayner, J. C. W. (2005) Smooth tests for the zero-inflated Poisson distribution. Biometrics, 61, 808–815.

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

Cameron, A. C. and Trivedi, P. K. (1998) Regression Analysis of Count Data. Cambridge University Press: Cambridge.

Yee, T. W. (2010) Two-parameter reduced-rank vector generalized linear models. In preparation.

See Also

zapoisson, Zipois, yip88, rrvglm, zipebcom, rpois.

Examples

# Example 1: simulated ZIP data
zdata <- data.frame(x2 = runif(nn <- 2000))
zdata <- transform(zdata, phi1    = logit(-0.5 + 1*x2, inverse = TRUE),
                          phi2    = logit( 0.5 - 1*x2, inverse = TRUE),
                          Phi1    = logit(-0.5       , inverse = TRUE),
                          Phi2    = logit( 0.5       , inverse = TRUE),
                          lambda1 =  loge( 0.5 + 2*x2, inverse = TRUE),
                          lambda2 =  loge( 0.5 + 2*x2, inverse = TRUE))
zdata <- transform(zdata, y1 = rzipois(nn, lambda1, Phi1),
                          y2 = rzipois(nn, lambda2, Phi2))

with(zdata, table(y1)) # Eyeball the data
with(zdata, table(y2))
fit1 <- vglm(y1 ~ x2, zipoisson(zero = 1), zdata, crit = "coef")
fit2 <- vglm(y2 ~ x2, zipoisson(zero = 1), zdata, crit = "coef")
coef(fit1, matrix = TRUE)  # These should agree with the above values
coef(fit2, matrix = TRUE)  # These should agree with the above values

# Fit all two simultaneously, using a different parameterization:
fit12 <- vglm(cbind(y1, y2) ~ x2, zipoissonff, zdata, crit = "coef")
coef(fit12, matrix = TRUE)  # These should agree with the above values


# Example 2: McKendrick (1926). Data from 223 Indian village households
cholera <- data.frame(ncases = 0:4, # Number of cholera cases,
                      wfreq = c(168, 32, 16, 6, 1)) # Frequencies
fit <- vglm(ncases ~ 1, zipoisson, wei = wfreq, cholera, trace = TRUE)
coef(fit, matrix = TRUE)
with(cholera, cbind(actual = wfreq,
                    fitted = round(dzipois(ncases, lambda = Coef(fit)[2],
                                           phi = Coef(fit)[1]) *
                                   sum(wfreq), dig = 2)))

# Example 3: 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)
fit <- vglm(y ~ 1, zipoisson(lphi = probit, iphi = 0.3),
            abdata, weight = w, trace = TRUE)
fit@misc$prob0  # Estimate of P(Y = 0)
coef(fit, matrix = TRUE)
Coef(fit)  # Estimate of phi and lambda
fitted(fit)
with(abdata, weighted.mean(y, w)) # Compare this with fitted(fit)
summary(fit)


# Example 4: This RR-ZIP is known as a COZIGAM or COZIVGLM-ZIP
rrzip <- rrvglm(Alopacce ~ bs(WaterCon), zipoissonff(zero = NULL),
                hspider, trace = TRUE)
coef(rrzip, matrix = TRUE)
Coef(rrzip)
summary(rrzip)
## Not run: plotvgam(rrzip, lcol = "blue")

[Package VGAM version 0.8-4 Index]