zeroalt {glmtoolbox}

Zero-Altered Regression Models to deal with Zero-Excess in Count Data

Description

Allows to fit a zero-altered (Poisson or negative binomial) regression model to deal with zero-excess in count data.

Usage

zeroalt(
  formula,
  data,
  offset,
  subset,
  na.action = na.omit(),
  weights,
  family = "poi(log)",
  zero.link = c("logit", "probit", "cloglog", "cauchit", "log"),
  reltol = 1e-13,
  start = list(counts = NULL, zeros = NULL),
  ...
)

Arguments

formula

a Formula expression of the form response ~ x1 + x2 + ...| z1 + z2 + ..., which is a symbolic description of the linear predictors of the models to be fitted to \mu and \pi, respectively. See Formula documentation. If a formula of the form response ~ x1 + x2 + ... is supplied, the same regressors are employed in both components. This is equivalent to response ~ x1 + x2 + ...| x1 + x2 + ....

data

an (optional) data frame in which to look for variables involved in the formula expression, as well as for variables specified in the arguments weights and subset.

offset

this can be used to specify an a priori known component to be included in the linear predictor during fitting. This should be NULL or a numeric vector of length equal to the number of cases.

subset

an (optional) vector specifying a subset of observations to be used in the fitting process.

na.action

a function which indicates what should happen when the data contain NAs. By default na.action is set to na.omit().

weights

an (optional) vector of positive "prior weights" to be used in the fitting process. The length of weights should be the same as the number of observations. As default, weights is set to a vector of 1s.

family

an (optional) character string that allows you to specify the distribution to describe the response variable, as well as the link function to be used in the model for \mu. The following distributions are supported: (zero-altered) negative binomial I ("nb1"), (zero-altered) negative binomial II ("nb2"), (zero-altered) negative binomial ("nbf"), and (zero-altered) poisson ("poi"). Link functions are the same as those available in Poisson models via glm. See family documentation. As default, family is set to be Poisson with log link.

zero.link

an (optional) character string which allows to specify the link function to be used in the model for \pi. Link functions available are the same than those available in binomial models via glm. See family documentation. As default, zero.link is set to "logit".

reltol

an (optional) positive value which represents the relative convergence tolerance for the BFGS method in optim. As default, reltol is set to 1e-13.

start

an (optional) list with two components named "counts" and "zeros", which allows to specify the starting values to be used in the iterative process to obtain the estimates of the parameters in the linear predictors of the models for \mu and \pi, respectively.

...

further arguments passed to or from other methods.

Details

The zero-altered count distributions, also called hurdle models, may be obtained as the mixture between a zero-truncated count distribution and the Bernoulli distribution. Indeed, if Y is a count random variable such that Y|\nu=1 is 0 with probability 1 and Y|\nu=0 ~ ZTP(\mu), where \nu ~ Bernoulli(\pi), then Y is distributed according to the Zero-Altered Poisson distribution, denoted here as ZAP(\mu,\pi).

Similarly, if Y is a count random variable such that Y|\nu=1 is 0 with probability 1 and Y|\nu=0 ~ ZTNB(\mu,\phi,\tau), where \nu ~ Bernoulli(\pi), then Y is distributed according to the Zero-Altered Negative Binomial distribution, denoted here as ZANB(\mu,\phi,\tau,\pi). The Zero-Altered Negative Binomial I (\mu,\phi,\pi) and Zero-Altered Negative Binomial II (\mu,\phi,\pi) distributions are special cases of ZANB when \tau=0 and \tau=-1, respectively.

The "counts" model may be expressed as g(\mu_i)=x_i^{\top}\beta for i=1,\ldots,n, where g(\cdot) is the link function specified at the argument family. Similarly, the "zeros" model may be expressed as h(\pi_i)=z_i^{\top}\gamma for i=1,\ldots,n, where h(\cdot) is the link function specified at the argument zero.link. Parameter estimation is performed using the maximum likelihood method. The parameter vector \gamma is estimated by applying the routine glm.fit, where a binary-response model (1 or "success" if response=0 and 0 or "fail" if response>0) is fitted. Then, the rest of the model parameters are estimated by maximizing the log-likelihood function based on the zero-truncated count distribution through the BFGS method available in the routine optim. The accuracy and speed of the BFGS method are increased because the call to the routine optim is performed using the analytical instead of the numerical derivatives. The variance-covariance matrix estimate is obtained as being minus the inverse of the (analytical) hessian matrix evaluated at the parameter estimates and the observed data. A set of standard extractor functions for fitted model objects is available for objects of class zeroinflation, including methods to the generic functions such as print, summary, model.matrix, estequa, coef, vcov, logLik, fitted, confint, AIC, BIC and predict. In addition, the model fitted to the data may be assessed using functions such as anova.zeroinflation, residuals.zeroinflation, dfbeta.zeroinflation, cooks.distance.zeroinflation and envelope.zeroinflation.

Value

An object of class zeroinflation in which the main results of the model fitted to the data are stored, i.e., a list with components including

coefficients	a list with elements "counts" and "zeros" containing the parameter estimates
	from the respective models,
fitted.values	a list with elements "counts" and "zeros" containing the estimates of \mu_1,\ldots,\mu_n
	and \pi_1,\ldots,\pi_n, respectively,
start	a vector containing the starting values for all parameters in the model,
prior.weights	a vector containing the case weights used,
offset	a list with elements "counts" and "zeros" containing the offset vectors, if any,
	from the respective models,
terms	a list with elements "counts", "zeros" and "full" containing the terms objects for
	the respective models,
loglik	the value of the log-likelihood function avaliated at the parameter estimates and
	the observed data,
estfun	a list with elements "counts" and "zeros" containing the estimating functions
	evaluated at the parameter estimates and the observed data for the respective models,
formula	the formula,
levels	the levels of the categorical regressors,
contrasts	a list with elements "counts" and "zeros" containing the contrasts corresponding
	to levels from the respective models,
converged	a logical indicating successful convergence,
model	the full model frame,
y	the response count vector,
family	a list with elements "counts" and "zeros" containing the family objects used
	in the respective models,
linear.predictors	a list with elements "counts" and "zeros" containing the estimates of
	g(\mu_1),\ldots,g(\mu_n) and h(\pi_1),\ldots,h(\pi_n), respectively,
R	a matrix with the Cholesky decomposition of the inverse of the variance-covariance
	matrix of all parameters in the model,
call	the original function call.

References

Cameron A.C., Trivedi P.K. (1998) Regression Analysis of Count Data. New York: Cambridge University Press.

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

See Also

overglm, zeroinf

Examples

####### Example 1: Roots Produced by the Columnar Apple Cultivar Trajan
data(Trajan)
fit1 <- zeroalt(roots ~ photoperiod, family="nbf(log)", zero.link="logit", data=Trajan)
summary(fit1)

####### Example 2: Self diagnozed ear infections in swimmers
data(swimmers)
fit2 <- zeroalt(infections ~ frequency | location, family="nb1(log)", data=swimmers)
summary(fit2)

####### Example 3: Article production by graduate students in biochemistry PhD programs
bioChemists <- pscl::bioChemists
fit3 <- zeroalt(art ~ fem + kid5 + ment, family="nb1(log)", data = bioChemists)
summary(fit3)