Zero-Inflated Poisson Distribution Family Function
Fits a zero-inflated or zero-deflated Poisson distribution by full maximum likelihood estimation.
zipoisson(lpstr0 = "logitlink", llambda = "loglink", type.fitted =
c("mean", "lambda", "pobs0", "pstr0", "onempstr0"), ipstr0 =
NULL, ilambda = NULL, gpstr0 = NULL, imethod = 1,
ishrinkage = 0.95, probs.y = 0.35, parallel = FALSE, zero = NULL)
zipoissonff(llambda = "loglink", lonempstr0 = "logitlink", 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")lpstr0, llambda |
Link function for the parameter phi
and the usual lambda parameter.
See |
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 |
imethod |
An integer with value |
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 |
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 |
gpstr0, gonempstr0, probs.y |
Details at |
parallel |
Details at |
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,…,
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.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
rrvglm
and vgam.
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.
This family function can be used to estimate the 0-deflated model,
hence pstr0 is not to be interpreted as a probability.
One should set, e.g., lpstr0 = "identitylink".
Likewise, the functions in Zipois
can handle the zero-deflated Poisson distribution too.
Although the iterations
might fall outside the parameter space, the validparams slot
should keep them inside.
A (somewhat) similar alternative for
zero-deflation is to try the zero-altered Poisson model
(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.
T. W. Yee
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.
# Example 1: simulated ZIP data
zdata <- data.frame(x2 = runif(nn <- 1000))
zdata <- transform(zdata, pstr01 = logitlink(-0.5 + 1*x2, inverse = TRUE),
pstr02 = logitlink( 0.5 - 1*x2, inverse = TRUE),
Ps01 = logitlink(-0.5 , inverse = TRUE),
Ps02 = logitlink( 0.5 , inverse = TRUE),
lambda1 = loglink(-0.5 + 2*x2, inverse = TRUE),
lambda2 = loglink( 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 = probitlink, 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 = loglink(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))
head(zdata)
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:
zdata[1, "usepstr0"] # Answer
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)
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