hurdle: Hurdle Models for Count Data Regression

In pscl: Political Science Computational Laboratory

Hurdle Models for Count Data Regression

Description

Fit hurdle regression models for count data via maximum likelihood.

Usage

hurdle(formula, data, subset, na.action, weights, offset,
  dist = c("poisson", "negbin", "geometric"),
  zero.dist = c("binomial", "poisson", "negbin", "geometric"),
  link = c("logit", "probit", "cloglog", "cauchit", "log"),
  control = hurdle.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.
weights	optional numeric vector of weights.
offset	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.
dist	character specification of count model family.
zero.dist	character specification of the zero hurdle model family.
link	character specification of link function in the binomial zero hurdle (only used if zero.dist = "binomial".
control	a list of control arguments specified via hurdle.control.
model, y, x	logicals. If TRUE the corresponding components of the fit (model frame, response, model matrix) are returned.
...	arguments passed to hurdle.control in the default setup.

Details

Hurdle count models are two-component models with a truncated count component for positive counts and a hurdle component that models the zero counts. Thus, unlike zero-inflation models, there are not two sources of zeros: the count model is only employed if the hurdle for modeling the occurrence of zeros is exceeded. The count model is typically a truncated 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 hurdle, either a binomial model can be employed or a censored count distribution. The outcome of the hurdle component of the model is the occurrence of a non-zero (positive) count. Thus, for most models, positive coefficients in the hurdle component indicate that an increase in the regressor increases the probability of a non-zero count. Binomial logit and censored geometric models as the hurdle part both lead to the same likelihood function and thus to the same coefficient estimates. A censored negative binomial model for the zero hurdle is only identified if there is at least one non-constant regressor with (true) coefficient different from zero (and if all coefficients are close to zero the model can be poorly conditioned).

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 zero hurdle component, e.g., y ~ x1 + x2 | z1 + z2 + z3 giving the count data model y ~ x1 + x2 conditional on (|) the zero hurdle model y ~ z1 + z2 + z3.

Offsets can be specified in both parts of the model pertaining to count and zero hurdle 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 part and z3 analogously in the zero hurdle part. 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 hurdle.control. Starting values can be supplied, otherwise they are estimated by glm.fit (the default). By default, the two components of the model are estimated separately using two optim calls. Standard errors are derived numerically using the Hessian matrix returned by optim. See hurdle.control for details.

The returned fitted model object is of class "hurdle" 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 components, respectively. For details see below.

A set of standard extractor functions for fitted model objects is available for objects of class "hurdle", including methods to the generic functions print, summary, coef, vcov, logLik, residuals, predict, fitted, terms, model.matrix. See predict.hurdle for more details on all methods.

Value

An object of class "hurdle", 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 (of lists) with the output(s) from the optim call(s) for minimizing the negative log-likelihood(s),
control	the control arguments passed to the optim call,
start	the starting values for the parameters passed to the optim call(s),
weights	the case weights used,
offset	a list with elements "count" and "zero" containing the offset vectors (if any) from the respective models,
n	number of observations (with weights > 0),
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(s) (if negative binomial component is used),
SE.logtheta	standard error(s) 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(s)),
dist	a list with elements "count" and "zero" with character strings describing the respective distributions used,
link	character string describing the link if a binomial zero hurdle model is used,
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).

Author(s)

Achim Zeileis <[e-mail]>

References

Cameron, A. Colin and Pravin K. Trivedi. 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.

Mullahy, J. 1986. Specification and Testing of Some Modified Count Data Models. Journal of Econometrics. 33:341–365.

Zeileis, Achim, Christian Kleiber and Simon Jackman 2008. “Regression Models for Count Data in R.” Journal of Statistical Software, 27(8). URL https://www.jstatsoft.org/v27/i08/.

See Also

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

Examples

## data
data("bioChemists", package = "pscl")

## logit-poisson
## "art ~ ." is the same as "art ~ . | .", i.e.
## "art ~ fem + mar + kid5 + phd + ment | fem + mar + kid5 + phd + ment"
fm_hp1 <- hurdle(art ~ ., data = bioChemists)
summary(fm_hp1)

## geometric-poisson
fm_hp2 <- hurdle(art ~ ., data = bioChemists, zero = "geometric")
summary(fm_hp2)

## logit and geometric model are equivalent
coef(fm_hp1, model = "zero") - coef(fm_hp2, model = "zero")

## logit-negbin
fm_hnb1 <- hurdle(art ~ ., data = bioChemists, dist = "negbin")
summary(fm_hnb1)

## negbin-negbin
## (poorly conditioned zero hurdle, note the standard errors)
fm_hnb2 <- hurdle(art ~ ., data = bioChemists, dist = "negbin", zero = "negbin")
summary(fm_hnb2)

pscl documentation built on May 29, 2024, 9:09 a.m.