countreg

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", "binomial"),
  zero.dist = c("binomial", "poisson", "negbin", "geometric"),
  link = c("logit", "probit", "cloglog", "cauchit", "log"),
  size = NULL, 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”`.
`size`	size parameter in case the a binomial count model is used (`dist = “binomial”`). By default the maximum count is used.
`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 occurence 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 (occurence of positive counts) either a binomial model can be employed or a censored count distribution. 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`).

References

Cameron AC, Trivedi PK (2013). Regression Analysis of Count Data, 2nd ed. New York: Cambridge University Press.

Cameron AC, Trivedi PK (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 A, Kleiber C, Jackman S (2008). “Regression Models for Count Data in R.” Journal of Statistical Software, 27(8), 1–25. doi:10.18637/jss.v027.i08.

Examples

library("countreg")

## data
data("CrabSatellites", package = "countreg")
cs <- CrabSatellites[, c("satellites", "width", "color")]
cs$color <- as.numeric(cs$color)

## logit-poisson
## "satellites ~ ." is the same as "satellites ~ . | .", i.e.
## "satellites ~ width + color | width + color"
fm_hp1 <- hurdle(satellites ~ ., data = cs)
summary(fm_hp1)


Call:
hurdle(formula = satellites ~ ., data = cs)

Pearson residuals:
    Min      1Q  Median      3Q     Max 
-1.7371 -0.8383 -0.2976  0.6431  4.2699 

Count model coefficients (truncated poisson with log link):
            Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.562699   0.645439   0.872    0.383
width       0.034238   0.022227   1.540    0.123
color       0.007165   0.066627   0.108    0.914
Zero hurdle model coefficients (binomial with logit link):
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -10.0708     2.8065  -3.588 0.000333 ***
width         0.4583     0.1040   4.407 1.05e-05 ***
color        -0.5090     0.2237  -2.276 0.022862 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Number of iterations in BFGS optimization: 13 
Log-likelihood: -362.1 on 6 Df

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


Call:
hurdle(formula = satellites ~ ., data = cs, zero.dist = "geometric")

Pearson residuals:
    Min      1Q  Median      3Q     Max 
-1.7371 -0.8383 -0.2976  0.6431  4.2699 

Count model coefficients (truncated poisson with log link):
            Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.562699   0.645439   0.872    0.383
width       0.034238   0.022227   1.540    0.123
color       0.007165   0.066627   0.108    0.914
Zero hurdle model coefficients (censored geometric with log link):
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -10.0708     2.8065  -3.588 0.000333 ***
width         0.4583     0.1040   4.407 1.05e-05 ***
color        -0.5090     0.2237  -2.276 0.022862 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Number of iterations in BFGS optimization: 13 
Log-likelihood: -362.1 on 6 Df

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

(Intercept)       width       color 
          0           0           0

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


Call:
hurdle(formula = satellites ~ ., data = cs, dist = "negbin")

Pearson residuals:
    Min      1Q  Median      3Q     Max 
-1.3835 -0.7244 -0.2636  0.5557  3.6080 

Count model coefficients (truncated negbin with log link):
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) 0.428567   0.941077   0.455    0.649    
width       0.037845   0.032749   1.156    0.248    
color       0.006929   0.091078   0.076    0.939    
Log(theta)  1.527382   0.352950   4.327 1.51e-05 ***
Zero hurdle model coefficients (binomial with logit link):
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -10.0708     2.8065  -3.588 0.000333 ***
width         0.4583     0.1040   4.407 1.05e-05 ***
color        -0.5090     0.2237  -2.276 0.022862 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Theta: count = 4.6061
Number of iterations in BFGS optimization: 17 
Log-likelihood: -350.4 on 7 Df

## negbin-negbin
## (poorly conditioned zero hurdle, note increased standard errors)
fm_hnb2 <- hurdle(satellites ~ ., data = cs, dist = "negbin", zero = "negbin")
summary(fm_hnb2)


Call:
hurdle(formula = satellites ~ ., data = cs, dist = "negbin", zero.dist = "negbin")

Pearson residuals:
    Min      1Q  Median      3Q     Max 
-1.3984 -0.7163 -0.2724  0.5615  3.6221 

Count model coefficients (truncated negbin with log link):
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) 0.428567   0.941077   0.455    0.649    
width       0.037845   0.032749   1.156    0.248    
color       0.006929   0.091078   0.076    0.939    
Log(theta)  1.527382   0.352950   4.327 1.51e-05 ***
Zero hurdle model coefficients (censored negbin with log link):
            Estimate Std. Error z value Pr(>|z|)
(Intercept)  -8.7334     5.6289  -1.552    0.121
width         0.3872     0.2840   1.363    0.173
color        -0.4097     0.4230  -0.969    0.333
Log(theta)    0.5124     2.5691   0.199    0.842
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Theta: count = 4.6061, zero = 1.6693
Number of iterations in BFGS optimization: 35 
Log-likelihood: -350.3 on 8 Df