The negative binomial distribution can be obtained as mixture of the Poisson and Gamma distributions. If
\(Y | \lambda\) ~ Poisson\((\lambda)\), where E\((Y | \lambda)=\) Var\((Y | \lambda)=\lambda\), and
\(\lambda\) ~ Gamma\((\theta,\nu)\), in which E\((\lambda)=\theta\) and Var\((\lambda)=\nu\theta^2\), then
\(Y\) is distributed according to the negative binomial distribution. As follows, some special cases are described:
(1) If \(\theta=\mu\) and \(\nu=\phi\) then \(Y\) ~ Negative Binomial I,
E\((Y)=\mu\) and Var\((Y)=\mu(1 + \phi\mu)\).
(2) If \(\theta=\mu\) and \(\nu=\phi/\mu\) then \(Y\) ~ Negative Binomial II,
E\((Y)=\mu\) and Var\((Y)=\mu(1 +\phi)\).
(3) If \(\theta=\mu\) and \(\nu=\phi\mu^\tau\) then \(Y\) ~ Negative Binomial,
E\((Y)=\mu\) and Var\((Y)=\mu(1 +\phi\mu^{\tau+1})\).
Therefore, the regression models based on the negative binomial and
zero-truncated negative binomial distributions are alternatives,
under overdispersion, to those based on the Poisson and
zero-truncated Poisson distributions, respectively.
The beta-binomial distribution can be obtained as a mixture of the binomial and beta distributions. If
\(mY | \pi\) ~ Binomial\((m,\pi)\), where E\((Y | \pi)=\pi\) and Var\((Y | \pi)=m^{-1}\pi(1-\pi)\),
and \(\pi\) ~ Beta\((\mu,\phi)\), in which E\((\pi)=\mu\) and Var\((\pi)=(\phi+1)^{-1}\mu(1-\mu)\),
with \(\phi>0\), then \(mY\) ~ Beta-Binomial\((m,\mu,\phi)\), so that E\((Y)=\mu\) and
Var\((Y)=m^{-1}\mu(1-\mu)[1 + (\phi+1)^{-1}(m-1)]\). Therefore, the regression model based on the
beta-binomial distribution is an alternative, under overdispersion, to the binomial regression model.
The random-clumped binomial distribution can be obtained as a mixture of the binomial and Bernoulli distributions. If
\(mY | \pi\) ~ Binomial\((m,\pi)\), where E\((Y | \pi)=\pi\) and Var\((Y | \pi)=m^{-1}\pi(1-\pi)\),
whereas \(\pi=(1-\phi)\mu + \phi\) with probability \(\mu\), and \(\pi=(1-\phi)\mu\) with probability \(1-\mu\),
in which E\((\pi)=\mu\) and Var\((\pi)=\phi^{2}\mu(1-\mu)\), with \(\phi \in (0,1)\), then \(mY\) ~ Random-clumped
Binomial\((m,\mu,\phi)\), so that E\((Y)=\mu\) and Var\((Y)=m^{-1}\mu(1-\mu)[1 + \phi^{2}(m-1)]\). Therefore,
the regression model based on the random-clumped binomial distribution is an alternative, under
overdispersion, to the binomial regression model.
In all cases, even where the response variable is described by a
zero-truncated distribution, the fitted model describes the way in
which \(\mu\) is dependent on some covariates. Parameter estimation
is performed using the maximum likelihood method. The model
parameters are estimated by maximizing the log-likelihood function
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 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.overglm,
residuals.overglm, dfbeta.overglm, cooks.distance.overglm, localInfluence.overglm,
gvif.overglm and envelope.overglm. The variable selection may be accomplished using the routine
stepCriterion.overglm.