umber of events occurring within a given time interval. The Poisson distribution models the log-odds as a linear function of the observed covariates. This gives the generalized linear model with Poisson response and ling log.
If the number of occurrence has a variable Y which has a poisson distribution with parameter μ and it takes integer values of y = 0, 1, 3, … then the probability distribution is given by
P(Y = y) = (μ^y e^(-λ))/y! ; λ > 0 3.1
where λ is the shape parameter which indicates the average number of events in the given time interval.
The poisson distribution has mean and the variance that can be shown as
E(Y ) = var(Y ) = μ
If it is true that the mean is equal to the variance, then any factor that affects one will also affect the other. The Poisson distribution can only be applied under the following assumptions;
1. the event is something that can be counted in whole numbers;
2. occurrences are independent, so that one occurrence neither diminishes nor increases the chance of another
log(μ) = β_0 + β_1 x 3.3
Where; x denotes the vector of explanatory variables and β the vector of regression parameters.
However, this model was not did not fit the data for the study since the mean is not equal to the variance even though it is a count data. This was due to the excess zeros in the data which were not sampling error but outcome. A Zero-Inflated-Poisson was proposed.

Zero-Inflated-Poisson (zip)

The data 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