pscl (version 0.73)

zeroinfl: Zero-inflated Count Data Regression

Description

Fit zero-inflated regression models for count data via maximum likelihood.

Usage

zeroinfl(formula, data, subset, na.action,
  dist = c("poisson", "negbin", "geometric"),
  link = c("logit", "probit", "cloglog", "cauchit", "log"),
  control = zeroinfl.control(...),
  model = TRUE, y = TRUE, x = FALSE, ...)

Arguments

formula

symbolic description of the model, see details.

data, subset, na.action

arguments controlling formula processing via model.frame.

dist

character specification of count model family (a log link is always used).

link

character specification of link function in the binary zero-inflation model (a binomial family is always used).

control

a list of control arguments specified via zeroinfl.control.

model, y, x

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.

Value

• An object of class "zeroinfl", i.e., a list with components including
• coefficients
  a list with elements "count" and "zero" containing the coefficients from the respective models,
• residuals
  a vector of raw residuals (observed - fitted),
• fitted.values
  a vector of fitted means,
• optim
  a list with the output from the optim call for minimizing the negative log-likelihood,
• control
  the control arguments passed to the optim call,
• start
  the starting values for the parameters passed to the optim call,
• n
  number of observations,
• df.null
  residual degrees of freedom for the null model (= n - 2),
• df.residual
  residual degrees of freedom for fitted model,
• terms
  a list with elements "count", "zero" and "full" containing the terms objects for the respective models,
• theta
  estimate of the additional $\theta$ parameter of the negative binomial model (if a negative binomial regression is used),
• SE.logtheta
  standard error for $\log(\theta)$,
• loglik
  log-likelihood of the fitted model,
• vcov
  covariance matrix of all coefficients in the model (derived from the Hessian of the optim output),
• dist
  character string describing the count distribution used,
• link
  character string describing the link of the zero-inflation model,
• linkinv
  the inverse link function corresponding to link,
• converged
  logical indicating successful convergence of optim,
• call
  the original function call,
• formula
  the original formula,
• levels
  levels of the categorical regressors,
• contrasts
  a list with elements "count" and "zero" containing the contrasts corresponding to levels from the respective models,
• model
  the full model frame (if model = TRUE),
• y
  the response count vector (if y = TRUE),
• x
  a list with elements "count" and "zero" containing the model matrices from the respective models (if x = TRUE),

Details

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: 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 mainly describes the count data model, i.e., y ~ x1 + x2 specifies a count data regression where all zero counts have the same probability of belonging to the zero component. This is equivalent to the model y ~ x1 + x2 | 1, making it more explicit that the zero-inflation model only has an intercept. Additionally, further regressors can be added to the zero-inflation model so that not all zeros have the same probability for belonging to the point mass component or to the count component. A typical formula is y ~ x1 + x2 | z1 + z2. The regressors in the zero and the count component can be overlapping (or identical). 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). The latter corresponds to the first iteration of the EM algorithm and initializes the unobserved state as y > 0, i.e., all zeros are in the perfect component and only the non-zero counts in the count component. 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.

References

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.V34(1):1-14

See Also

zeroinfl.control, glm, glm.fit, glm.nb, hurdle

Examples

## from Long (1997)
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 <- glm.nb(art ~ ., data = bioChemists)

## with simple inflation
## (no regressors for 0 component)
fm_zip <- zeroinfl(art ~ ., data = bioChemists)
fm_zinb <- zeroinfl(art ~ ., data = bioChemists, dist = "negbin", EM = TRUE)

## inflation with regressors (choose starting values by EM)
## ("art ~ . | ." is "art ~ fem + mar + kid5 + phd + ment | fem + mar + kid5 + phd + ment")
fm_zip2 <- zeroinfl(art ~ . | ., data = bioChemists, EM = TRUE)
fm_zinb2 <- zeroinfl(art ~ . | ., data = bioChemists, dist = "negbin", EM = TRUE)