Traditional ZIP Models. Excess zeros are said to be present in data when the observed frequency of zeros greatly exceeds the number expected given the distributional assumptions on the data. Data with excess zeros can arise in many applications including industrial [85, 86], ecological [87, 88, 89, 90], horticultural, [91, 92, 93] and medical [94, 95, 96]. Excess zeros are classically described as structural versus random zeros. A structural zero occurs in situations where it is impossible to observe a response, whereas random zeros occur by chance according to the probability distribution describing the observation process [97]. Table 4.1 provides examples of the classical framework for zero inflated models, along with true and false zeros. The most commonly utilized models for data with excess zeros are zero-inflated models and hurdle models [84]. Zero-inflated models are a mixture of a point mass at zero and a standard distribution for count data such as Poisson or negative binomial. A zero-inflated model utilizing a Poisson distribution is known as the zero-inflated Poisson (ZIP) model. Similarly, a hurdle model is a mixture of a point mass at a certain observation(s) and a truncated distribution for the remaining observations. A hurdle model often compared to ZIP model considers a point mass at zero, and a truncated Poisson distribution for the remaining observations. Hurdle models con- sider all zeros together, whereas zero inflated models should be able to partition the zeros into structural and random zeros. Other zero-inflated models account for overdispersion by modeling the count data with the negative binomial or generalized Poisson distributions [100, 98, 96, 84, 101]. A zero-inflated distribution is defined by Pr(Y = 0|p, θ) = p + (1 − p)f (0|θ) Pr(Y = y|p, θ) = (1 − p)f (y|θ), y > 0 (4.1) for some parametric distribution f (y|θ), such as the Poisson distribution, where θy exp(−θ) f (y|θ) = y! . The zero-inflated distribution can be thought of as a joint distribution involving a latent random variable Z that indicates if the zero is a structural zero. In this situ- ation, let p = Pr(Z = 1), which indicates the overall probability that an observation is an structural zero. The joint distribution of Y and Z is Pr(Y = 0, Z = 1|p, θ) = p Pr(Y = y, Z = 0|p, θ) = (1 − p)f (y|θ). Conditionally, Pr(Y = 0|Z = 1) = 1 Pr(Y = y|Z = 0, p, θ) = f (y|θ) Pr(Z = 1|Y = y > 0) = 0 (4.3) T| More interestingly, we can calculate the probability that a zero is a structural zero given that...
