Distinguishing between the two sources of zeroes is not possible, as it is a form of discrete unobserved heterogeneity [21]. The probability density function for the ZIP model is given below: P(Yi=yi|xi′ψi)=ψi+(1-ψi)e-μiyi=0(1-ψi)e-μiμiyiyi!yi>0 Similarly, for zero-inflated Cabozantinib negative binomial model, the probability density function is given by: P(Yi=yi|xi′ψi′v)=ψi+(1-ψi)1(1+vμi)1∕vyi=0(1-ψi)Γyi+1vΓ(yi+1)Γ1v(vμi)yi(1+vμi)yi+1vyi>0 For both the ZIP and ZINB models

the probability Inhibitors,research,lifescience,medical of an excess zero, ψi, the is modeled using logistic regression (although, any binary regression framework will suffice). As a result, the probability of an excess zero is given by: ψi=11+eηi=11+eziγ In other words, the probability of an excess zero is a function of some observed linear predictor, ηi, which itself is formed from a set of predictor variables, zi, multiplied by their associated logistic regression coefficients, ε(nb. the Inhibitors,research,lifescience,medical set zi, in the logistic of model need not equal the set of variables, xi, in the Poisson or negative binomial component regression models). For the ZIP model the conditional mean and variance are: E(yi|xi′zi)=μi-μiψiVar(yi|xi′zi)=μi(1-ψi)(1+μiψi) Inhibitors,research,lifescience,medical For the ZINB model, the conditional mean

is the same Inhibitors,research,lifescience,medical as for the ZIP model; however, the conditional variance differs. The equations for both the conditional mean and variance of the ZINB model are given below: E(yi|xi′zi)=μi-μiψiVar(yi|xi′zi)=μi(1-ψi)(1+μi(ψi+v)) Considering ψi as the probability of excess zeroes, it can be observed that as ψi tends toward

zero then the probability densities, as well as the conditional mean and variances of the ZIP and ZINB models converge toward the corresponding formulas for the Poisson and negative binomial models, respectively [18,19,21]. Determination of regression coefficients for the ZIP Inhibitors,research,lifescience,medical and ZINB models once again occurs by maximization of the log-likelihood functions, which are given below. LLZIP=∑i=1n[I(yi=0)ln[(ψi+(1−ψi)exp(−μi)]+I(yi≥1)[ln(1−ψi)+yiln(μi)−μi−ln(yi!)]] LLZINB=∑i=1n[I(yi=0)ln(ψi+(1−ψi)1(1+vμi)1v)+I(yi≥1)[ln(1−ψi)+ln[Γ(yi+1v)]−ln[Γ(yi+1)]−ln[Γ(1v)]+yiln(vμi)−(yi+1/v)ln(1+vμi)] Here I(·) is an indicator function. found One issue with the application of zero-inflated modeling strategies for emergency department demand is that interpretively some of the zeroes in ZIP/ZINB models are considered to be structural; whereas, others are assumed to arise as a result of a sampling process. Conceptually, it is hard to imagine even the healthiest individuals in the Ontario population not being “at risk” for an emergency department visit and hence representing a structural zero.