Hurdle Models for Count Data Regression
Fit hurdle regression models for count data via maximum likelihood.
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, ...)formula |
symbolic description of the model, see details. |
data, subset, na.action |
arguments controlling formula processing
via |
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 |
control |
a list of control arguments specified via
|
model, y, x |
logicals. If |
... |
arguments passed to |
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.
An object of class "hurdle", i.e., a list with components including
coefficients |
a list with elements |
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 |
control |
the control arguments passed to the |
start |
the starting values for the parameters passed to the |
weights |
the case weights used, |
offset |
a list with elements |
n |
number of observations (with weights > 0), |
df.null |
residual degrees of freedom for the null model (= |
df.residual |
residual degrees of freedom for fitted model, |
terms |
a list with elements |
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 |
dist |
a list with elements |
link |
character string describing the link if a binomial zero hurdle model is used, |
linkinv |
the inverse link function corresponding to |
converged |
logical indicating successful convergence of |
call |
the original function call, |
formula |
the original formula, |
levels |
levels of the categorical regressors, |
contrasts |
a list with elements |
model |
the full model frame (if |
y |
the response count vector (if |
x |
a list with elements |
Achim Zeileis <Achim.Zeileis@R-project.org>
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 http://www.jstatsoft.org/v27/i08/.
## 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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.