# Alternatives to the Poisson and Binomial Regression Models under the presence of Overdispersion. Allows to fit regression models based on the negative binomial, beta-binomial, and random-clumped binomial. distributions, which are alternatives to the Poisson and binomial regression models under the presence of overdispersion. ```r overglm( formula, offset, family = "nb1(log)", weights, data, subset, na.action = na.omit(), reltol = 1e-13, start = NULL, ... ) ``` ## Arguments - `formula`: a `formula` expression of the form `response ~ x1 + x2 + ...`, which is a symbolic description of the linear predictor of the model to be fitted to the data. - `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. - `family`: A character string that allows you to specify the distribution describing the response variable. In addition, it allows you to specify the link function to be used in the model for $\mu$. The following distributions are supported: negative binomial I ("nb1"), negative binomial II ("nb2"), negative binomial ("nbf"), zero-truncated negative binomial I ("ztnb1"), zero-truncated negative binomial II ("ztnb2"), zero-truncated negative binomial ("ztnbf"), zero-truncated poisson ("ztpoi"), beta-binomial ("bb") and random-clumped binomial ("rcb"). Link functions available for these models are the same as those available for Poisson and binomial models via glm . See family documentation. - `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. - `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`. - `subset`: an (optional) vector specifying a subset of individuals 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()`. - `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) vector of starting values for the parameters in the linear predictor. - `...`: further arguments passed to or from other methods. ## Returns an object of class **overglm** in which the main results of the model fitted to the data are stored, i.e., a list with components including ||| |:--|:--| |`coefficients`|a vector containing the parameter estimates,| ||| |`fitted.values`|a vector containing the estimates of $\mu_1,\ldots,\mu_n$ ,| ||| |`start`|a vector containing the starting values used,| ||| |`prior.weights`|a vector containing the case weights used,| ||| |`offset`|a vector containing the offset used,| ||| |`terms`|an object containing the terms objects,| ||| |`loglik`|the value of the log-likelihood function avaliated at the parameter estimates,| ||| |`estfun`|a vector containing the estimating functions evaluated at the parameter estimates| ||and the observed data,| ||| |`formula`|the formula,| ||| |`levels`|the levels of the categorical regressors,| ||| |`contrasts`|an object containing the contrasts corresponding to levels,| ||| |`converged`|a logical indicating successful convergence,| ||| |`model`|the full model frame,| ||| |`y`|the response count vector,| ||| |`family`|an object containing the family object used,| ||| |`linear.predictors`|a vector containing the estimates of $g(\mu_1),\ldots,g(\mu_n)$ ,| ||| |`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.| ## Details 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 . ## Examples ```r ### Example 1: Ability of retinyl acetate to prevent mammary cancer in rats data(mammary) fit1 <- overglm(tumors ~ group, family="nb1(identity)", data=mammary) summary(fit1) ### Example 2: Self diagnozed ear infections in swimmers data(swimmers) fit2 <- overglm(infections ~ frequency + location, family="nb1(log)", data=swimmers) summary(fit2) ### Example 3: Urinary tract infections in HIV-infected men data(uti) fit3 <- overglm(episodes ~ cd4 + offset(log(time)), family="nb1(log)", data = uti) summary(fit3) ### Example 4: Article production by graduate students in biochemistry PhD programs bioChemists <- pscl::bioChemists fit4 <- overglm(art ~ fem + kid5 + ment, family="nb1(log)", data = bioChemists) summary(fit4) ### Example 5: Agents to stimulate cellular differentiation data(cellular) fit5 <- overglm(cbind(cells,200-cells) ~ tnf + ifn, family="bb(logit)", data=cellular) summary(fit5) ### Example 6: Teratogenic effects of phenytoin and trichloropropene oxide data(ossification) model6 <- cbind(fetuses,litter-fetuses) ~ pht + tcpo fit6 <- overglm(model6, family="rcb(cloglog)", data=ossification) summary(fit6) ### Example 7: Germination of orobanche seeds data(orobanche) model7 <- cbind(germinated,seeds-germinated) ~ specie + extract fit7 <- overglm(model7, family="rcb(cloglog)", data=orobanche) summary(fit7) ``` ## References Crowder M. (1978) Beta-binomial anova for proportions, **Journal of the Royal Statistical Society Series C (Applied Statistics)** 27, 34-37. Lawless J.F. (1987) Negative binomial and mixed poisson regression, **The Canadian Journal of Statistics** 15, 209-225. Morel J.G., Neerchal N.K. (1997) Clustered binary logistic regression in teratology data using a finite mixture distribution, **Statistics in Medicine** 16, 2843-2853. Morel J.G., Nagaraj N.K. (2012) **Overdispersion Models in SAS**. SAS Institute Inc., Cary, North Carolina, USA. ## See Also zeroalt , zeroinf

overglm function