ata that has excess of zero counts is model by zip regression model. Theory suggests that the excess zeros are generated by a separate process from the count values and that the excess zeros are modeled independently. The zip model has two parts, the first part use Poisson to mode the count model and the second use logit model to predict excess zeros. Zero-inflated models estimate two equations simultaneously, one for the count model and one for the excess zeros.
Pr(yi = 0) = π + (1- π)e^(-μ) 3.4
Pr(Yi = yi) = (1 – π) (μ^(y_i ) e^(-μ))/y_i , y > 0 3.5
Where yi is the outcome variable with any non-zero value, μ is the expected Poisson count for the ith individual and is the probability of the extra zeros. The zip regression model has mean to be (1- π)μ and the variance is μ(1- π) (1+ μπ). This model fit best if the data is not over dispersed with the mean larger than the variance.

Negative Binomial Regression Model (NB)
The negative binomial regression model is a parametric model that is more dispersed than the Poisson which can handle the over dispersed situation in the data. Given y to be the respondent variable of the number of death occurrence in a year and that y ∼ Poisson (μ), whereas μ is a random variable with a gamma distribution. Now if
y/μ ~ Poisson (μ) and μ ~ Gamma(α,β),
Where the gamma distribution has mean αβ and variance αβ2, with probability density
P(μ)= 1/(β^α Γ(μ)) μ^(α-1) exp⁡(-μ/β); μ>0 3.6
Then the negative binomial with unconditional distribution of y is
P(y) = (Γ (α+y))/(Γ (α)y !) (β/(1+ β))^y (1/(1+ β))^α, y = 0, 1, 2, … 3.7
This distribution has mean
E(y) = E[E(y / μ)] = E(μ) = αβ
and variance Var(y) = E[Var(y / μ)] + Var[E(y / μ)]
= Var (μ) + E (μ) = αβ+ αβ^2
Expressing the negative binomial distribution in terms of the parameters μ = αβ and k = 1/α, that the E(y) = μ and Var (y) = μ + kμ^2 (function is quadratic)
Therefore the distribution of y is given by
P(y) = (Γ (k^(-1)+y))/(Γ (k^(-1) ) y !) ((k μ)/(1+ β))^y (1/(1+ k μ))^□(1/k), 3.8
Note that the negative binomial distribution approache Poisson (μ) as k → 0.
To model the negative binomial, let yi ~ Negative (μ_i,k) with the log link, so that