VGAM (version 1.0-4)

# zipoisson: Zero-Inflated Poisson Distribution Family Function

## Description

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

## Usage

zipoisson(lpstr0 = "logit", llambda = "loge", type.fitted =
c("mean", "lambda", "pobs0", "pstr0", "onempstr0"), ipstr0 =
NULL, ilambda = NULL, gpstr0 = NULL, imethod = 1,
ishrinkage = 0.95, probs.y = 0.35, zero = NULL)
zipoissonff(llambda = "loge", lonempstr0 = "logit", type.fitted =
c("mean", "lambda", "pobs0", "pstr0", "onempstr0"),
ilambda = NULL, ionempstr0 = NULL, gonempstr0 = NULL,
imethod = 1, ishrinkage = 0.95, probs.y = 0.35, zero =
"onempstr0")

## Arguments

lpstr0, llambda

Link function for the parameter $$\phi$$ and the usual $$\lambda$$ parameter. See Links for more choices; see CommonVGAMffArguments for more information. For the zero-deflated model see below.

ipstr0, 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.

lonempstr0, ionempstr0

Corresponding arguments for the other parameterization. See details below.

type.fitted

Character. The type of fitted value to be returned. The first choice (the expected value) is the default. The estimated probability of an observed 0 is an alternative, else the estimated probability of a structural 0, or one minus the estimated probability of a structural 0. See CommonVGAMffArguments and fittedvlm for more information.

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 ishrinkage and/or else specify a value for ipstr0. See CommonVGAMffArguments for more information.

ishrinkage

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

Specifies which linear/additive predictors are to be modelled as intercept-only. If given, the value can 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.

gpstr0, gonempstr0, probs.y

Details at CommonVGAMffArguments.

## 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, ishrinkage, ipstr0, 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.

## Details

These models are 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 $$y=1,2,\ldots$$, $$P(Y = y) = (1-\phi) \exp(-\lambda) \lambda^y / y!.$$ Here, the parameter $$\phi$$ satisfies $$0 < \phi < 1$$. The mean of $$Y$$ is $$(1-\phi) \lambda$$ and these are returned as the fitted values, by default. The variance of $$Y$$ is $$(1-\phi) \lambda (1 + \phi \lambda)$$. By default, the two linear/additive predictors of zipoisson() are $$(logit(\phi), \log(\lambda))^T$$.

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) onempstr0 is now 1 minus the probability of a structural 0, i.e., the probability of the parent (Poisson) component, i.e., onempstr0 is 1-pstr0; (iii) argument zero has a new default so that the onempstr0 is intercept-only by default. Now zipoissonff() is generally recommended over zipoisson() (and definitely recommended over yip88). Both functions implement Fisher scoring and can handle multiple responses.

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

zapoisson, Zipois, yip88, rrvglm, zipebcom, rpois, simulate.vlm, hdeff.vglm.

## Examples

Run this code
# NOT RUN {
# Example 1: simulated ZIP data
zdata <- data.frame(x2 = runif(nn <- 1000))
zdata <- transform(zdata, pstr01  = logit(-0.5 + 1*x2, inverse = TRUE),
pstr02  = logit( 0.5 - 1*x2, inverse = TRUE),
Ps01    = logit(-0.5       , inverse = TRUE),
Ps02    = 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, lambda = lambda1, pstr0 = Ps01),
y2 = rzipois(nn, lambda = lambda2, pstr0 = Ps02))

with(zdata, table(y1))  # Eyeball the data
with(zdata, table(y2))
fit1 <- vglm(y1 ~ x2, zipoisson(zero = 1), data = zdata, crit = "coef")
fit2 <- vglm(y2 ~ x2, zipoisson(zero = 1), data = 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, data = zdata, crit = "coef")
coef(fit12, matrix = TRUE)  # These should agree with the above values

# For the first observation compute the probability that y1 is
# due to a structural zero.
(fitted(fit1, type = "pstr0") / fitted(fit1, type = "pobs0"))[1]

# 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],
pstr0 = Coef(fit)[1]) *
sum(wfreq), digits = 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(lpstr0 = probit, ipstr0 = 0.8),
data = abdata, weight = w, trace = TRUE)
fitted(fit, type = "pobs0")  # Estimate of P(Y = 0)
coef(fit, matrix = TRUE)
Coef(fit)  # Estimate of pstr0 and lambda
fitted(fit)
with(abdata, weighted.mean(y, w))  # Compare this with fitted(fit)
summary(fit)

# Example 4: zero-deflated model for intercept-only data
zdata <- transform(zdata, lambda3 = loge(0.0, inverse = TRUE))
zdata <- transform(zdata, deflat.limit = -1 / expm1(lambda3))  # Boundary
# The 'pstr0' parameter is negative and in parameter space:
zdata <- transform(zdata, usepstr0 = deflat.limit / 2)  # Not too near the boundary
zdata <- transform(zdata, y3 = rzipois(nn, lambda3, pstr0 = usepstr0))
with(zdata, table(y3))  # A lot of deflation
fit3 <- vglm(y3 ~ 1, zipoisson(zero = -1, lpstr0 = "identitylink"),
data = zdata, trace = TRUE, crit = "coef")
coef(fit3, matrix = TRUE)
# Check how accurate it was:
coef(fit3)[1]         # Estimate
Coef(fit3)
vcov(fit3)  # Is positive-definite

# Example 5: This RR-ZIP is known as a COZIGAM or COZIVGLM-ZIP
set.seed(123)
rrzip <- rrvglm(Alopacce ~ sm.bs(WaterCon, df = 3), zipoisson(zero = NULL),
data = hspider, trace = TRUE, Index.corner = 2)
coef(rrzip, matrix = TRUE)
Coef(rrzip)
summary(rrzip)
# }
# NOT RUN {
plotvgam(rrzip, lcol = "blue")
# }


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