By extending the Poisson conditional mean in this manner, we arrive at the negative binomial regression model. The inclusion of the random error in the conditional mean of the negative binomial regression model is useful, as it allows for the modeling of both observed and selleck screening library unobserved heterogeneity whereas, the Poisson model only accounts for observed heterogeneity. In other words, using the Poisson regression model it was assumed that patients Inhibitors,research,lifescience,medical with the same observation vector would incur the same conditional mean response. The incorporation of the random term in the negative
binomial regression model allows patients with identical observation vectors to experience different conditional mean responses. If we assume (εi) has a mean that of 1 and variance of υ then the conditional
mean of yi is still μi; however, the conditional variance becomes μi(1 +uμi) = μi + uμi2. As u approaches zero binomial regression model converges toward the Poisson model, with a conditional mean that is equal to the conditional Inhibitors,research,lifescience,medical variance, μi [19,21]. For the negative binomial model, the probability that an individual patient incurs yi emergency department visits is dictated by the following density function: P(Yi=yi|xi′v)=Γyi+1vΓ(yi+1)Γ1v1v1v+μi1vμi1v+μiyi Inhibitors,research,lifescience,medical Above, μi represents the mean number of events that is expected for an individual with observation vector xi, u represents the negative binomial dispersion parameter and Γ(·) represents the gamma function. Determination of regression coefficients in negative binomial regression proceeds by maximizing the following log-likelihood function with respect to the unknown parameters: LLNB= ∑i=1nlnΓ(yi+1v)Γ(yi+1)Γ1v-yi+1vln(1+vμi)+yi ln(vμi) The negative binomial regression Inhibitors,research,lifescience,medical model is a useful model for accounting for data in which unobserved heterogeneity or temporal/spatial correlation is present; however, it is
not necessarily an optimal model for dealing with data that contain an excess mass of zeroes at the corner of its empirical distribution. Zero Inflated Poisson (ZIP) regression models were introduced by Lambert Inhibitors,research,lifescience,medical  as a method for modeling the factors influencing the number of defects encountered in a manufacturing application. Greene  introduced the idea of the Zero Inflated Negative Binomial (ZINB) model to handle both excess zeroes and over-dispersion as a result of unobserved heterogeneity which commonly arises in economic MycoClean Mycoplasma Removal Kit problems. Each of the models – ZIP and ZINB – assumes that patients can fall into one of two groups. The first group of patients never experience the outcome (eg. always show zero demand for emergency department services) and the second group of patients show some positive demand which is governed by the Poisson or negative binomial density. A patient falls into group 1 with probability ψi, and a patient falls into group 2 with probability (1 – ψi), where ψi is an estimable parameter from available data.