Fit zero-inflated regression models for count data via maximum likelihood.
zeroinfl(formula, data, subset, na.action, weights, offset,
dist = c("poisson", "negbin", "geometric"),
link = c("logit", "probit", "cloglog", "cauchit", "log"),
control = zeroinfl.control(...),
model = TRUE, y = TRUE, x = FALSE, ...)An object of class "zeroinfl", i.e., a list with components including
a list with elements "count" and "zero"
containing the coefficients from the respective models,
a vector of raw residuals (observed - fitted),
a vector of fitted means,
a list with the output from the optim call for
minimizing the negative log-likelihood,
the control arguments passed to the optim call,
the starting values for the parameters passed to the optim call,
the case weights used,
a list with elements "count" and "zero"
containing the offset vectors (if any) from the respective models,
number of observations (with weights > 0),
residual degrees of freedom for the null model (= n - 2),
residual degrees of freedom for fitted model,
a list with elements "count", "zero" and
"full" containing the terms objects for the respective models,
estimate of the additional \(\theta\) parameter of the negative binomial model (if a negative binomial regression is used),
standard error for \(\log(\theta)\),
log-likelihood of the fitted model,
covariance matrix of all coefficients in the model (derived from the
Hessian of the optim output),
character string describing the count distribution used,
character string describing the link of the zero-inflation model,
the inverse link function corresponding to link,
logical indicating successful convergence of optim,
the original function call,
the original formula,
levels of the categorical regressors,
a list with elements "count" and "zero"
containing the contrasts corresponding to levels from the
respective models,
the full model frame (if model = TRUE),
the response count vector (if y = TRUE),
a list with elements "count" and "zero"
containing the model matrices from the respective models
(if x = TRUE),
symbolic description of the model, see details.
arguments controlling formula processing
via model.frame.
optional numeric vector of weights.
optional numeric vector with an a priori known component to be included in the linear predictor of the count model. See below for more information on offsets.
character specification of count model family (a log link is always used).
character specification of link function in the binary zero-inflation model (a binomial family is always used).
a list of control arguments specified via
zeroinfl.control.
logicals. If TRUE the corresponding components
of the fit (model frame, response, model matrix) are returned.
arguments passed to zeroinfl.control in the
default setup.
Achim Zeileis <Achim.Zeileis@R-project.org>
Zero-inflated count models are two-component mixture models combining a point mass at zero with a proper count distribution. Thus, there are two sources of zeros: zeros may come from both the point mass and from the count component. Usually the count model is a Poisson or negative binomial regression (with log link). The geometric distribution is a special case of the negative binomial with size parameter equal to 1. For modeling the unobserved state (zero vs. count), a binary model is used that captures the probability of zero inflation. in the simplest case only with an intercept but potentially containing regressors. For this zero-inflation model, a binomial model with different links can be used, typically logit or probit.
The formula can be used to specify both components of the model:
If a formula of type y ~ x1 + x2 is supplied, then the same
regressors are employed in both components. This is equivalent to
y ~ x1 + x2 | x1 + x2. Of course, a different set of regressors
could be specified for the count and zero-inflation component, e.g.,
y ~ x1 + x2 | z1 + z2 + z3 giving the count data model y ~ x1 + x2
conditional on (|) the zero-inflation model y ~ z1 + z2 + z3.
A simple inflation model where all zero counts have the same
probability of belonging to the zero component can by specified by the formula
y ~ x1 + x2 | 1.
Offsets can be specified in both components of the model pertaining to count and
zero-inflation model: y ~ x1 + offset(x2) | z1 + z2 + offset(z3), where
x2 is used as an offset (i.e., with coefficient fixed to 1) in the
count component and z3 analogously in the zero-inflation component. By the rule
stated above y ~ x1 + offset(x2) is expanded to
y ~ x1 + offset(x2) | x1 + offset(x2). Instead of using the
offset() wrapper within the formula, the offset argument
can also be employed which sets an offset only for the count model. Thus,
formula = y ~ x1 and offset = x2 is equivalent to
formula = y ~ x1 + offset(x2) | x1.
All parameters are estimated by maximum likelihood using optim,
with control options set in zeroinfl.control.
Starting values can be supplied, estimated by the EM (expectation maximization)
algorithm, or by glm.fit (the default). Standard errors
are derived numerically using the Hessian matrix returned by optim.
See zeroinfl.control for details.
The returned fitted model object is of class "zeroinfl" and is similar
to fitted "glm" objects. For elements such as "coefficients" or
"terms" a list is returned with elements for the zero and count component,
respectively. For details see below.
A set of standard extractor functions for fitted model objects is available for
objects of class "zeroinfl", including methods to the generic functions
print, summary, coef,
vcov, logLik, residuals,
predict, fitted, terms,
model.matrix. See predict.zeroinfl for more details
on all methods.
Cameron, A. Colin and Pravin K. Trevedi. 1998. Regression Analysis of Count Data. New York: Cambridge University Press.
Cameron, A. Colin and Pravin K. Trivedi. 2005. Microeconometrics: Methods and Applications. Cambridge: Cambridge University Press.
Lambert, Diane. 1992. “Zero-Inflated Poisson Regression, with an Application to Defects in Manufacturing.” Technometrics. 34(1):1-14
Zeileis, Achim, Christian Kleiber and Simon Jackman 2008. “Regression Models for Count Data in R.” Journal of Statistical Software, 27(8). URL http://www.jstatsoft.org/v27/i08/.
## data
data("bioChemists", package = "pscl")
## without inflation
## ("art ~ ." is "art ~ fem + mar + kid5 + phd + ment")
fm_pois <- glm(art ~ ., data = bioChemists, family = poisson)
fm_qpois <- glm(art ~ ., data = bioChemists, family = quasipoisson)
fm_nb <- MASS::glm.nb(art ~ ., data = bioChemists)
## with simple inflation (no regressors for zero component)
fm_zip <- zeroinfl(art ~ . | 1, data = bioChemists)
fm_zinb <- zeroinfl(art ~ . | 1, data = bioChemists, dist = "negbin")
## inflation with regressors
## ("art ~ . | ." is "art ~ fem + mar + kid5 + phd + ment | fem + mar + kid5 + phd + ment")
fm_zip2 <- zeroinfl(art ~ . | ., data = bioChemists)
fm_zinb2 <- zeroinfl(art ~ . | ., data = bioChemists, dist = "negbin")
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