Question

1. This assignment focuses on the architecture of the Poisson, Negative Binomial, Zero-Inflated Poisson and Zero-Inflated Negative Binomial regression models.

A. In which context Poisson regression can be employed, please provide some examples?

B. What is the main difference between the Poisson regression model and the Negative Binomial regression model? Please explain.

C. Why is it important to use Zero-Inflated models such as the Zero-Inflation Poisson regression model or the Zero-Inflated Negative Binomial regression model in the first place?

D. What is the specific contribution of the Vuong test into the count outcomes regression analysis? Be specific.

Answer 1

A. In which context Poisson regression can be employed, please provide some examples?

-Poisson regression – Poisson regression is often used for modeling count data. Poisson regression has a number of extensions useful for count models. Poisson Regression models are best used for modeling events where the outcomes are counts.

Example

For instance of events such as the arrival of a customer in particular shop. The events must be independent in the sense that the arrival of one customer will not make another more or less likely, but the probability per unit time of events is understood to be related to covariates such as time of day. i.e more number of customer will arrive in evening time , and may be less customer in afternnon time . For such case Poisson Regression models is appropriate , i.e to predict number of arrival of customers .

B. What is the main difference between the Poisson regression model and the Negative Binomial regression model? Please explain.

-Negative binomial regression is for modeling count variables, usually for over-dispersed count outcome variables.

Suppose collage wants to know attendance behavior of students at selected say two different collage. Predictors of the number of days of absence include the differenct type of program in which the student is enrolled and a standardized test in some subject

So in these example respone variable of interest is days absent, and the average numbers of days absent by program type and seems to suggest that program type is a good candidate for predicting the number of days absent, our outcome variable, because the mean value of the outcome that is number of days absent appears to vary by different type of programe .

So the main difference between the Poisson regression model and the Negative Binomial regression model, as because Negative Binomial regression model loosens the highly restrictive assumption that the variance is equal to the mean made by the Poisson model.

C. Why is it important to use Zero-Inflated models such as the Zero-Inflation Poisson regression model or the Zero-Inflated Negative Binomial regression model in the first place?

-Sometimes when analyzing a response variable that is a count variable, the number of zeroes may seem excessive.

Zero-inflated regression model – Zero-inflated models attempt to account for excess zeros. In other words, two kinds of zeros are thought to exist in the data, "true zeros" and "excess zeros". Zero-inflated models estimate two equations simultaneously, one for the count model and one for the excess zeros

For for example we taken for Negative Binomial regression model , where we wish to model number of days absent by a student .

Now , a student might be absent zero days during the school year if he never gets sick and never skips school.

So number of absent days will be zero , and hence count will be zero ( let it denote by "excess zero".). Another student might be absent zero days during the school year because her parents may be techer of that collage and may insist them go to school every day, regardless of illness or desire to skip school. So this is again we get count to be zero ( let it denote by "true zero").

But note that in both zero count reason for absent is different.

So in such situation , negative binomial model would not distinguish between these two processes i.e , but a zero-inflated model allows for and accommodates this complication.

So this is important of Zero-Inflated models .

D. What is the specific contribution of the Vuong test into the count outcomes regression analysis? Be specific.

- The Vuong test comparing a Poisson and a zero-inflated Poisson model . Vuong test compares the goodness of fit of the Poisson model with the zero-inflated Poisson , ZIP model .

The statistic tests the null hypothesis that the two models are equally close to the true data generating process, against the alternative that one model is closer.

If the two models fit the data equally well, then their likelihood functions would be nearly identical. Otherwise, differences between the likelihood functions provide an indication of which model fits the data better. The Vuong test is predicated upon this principle.

Often the Poisson model is treated as the null, and the alternative is the second model such as the ZIP model. The null is rejected only if the Vuong test is significant in favor of the second model. In other words, when the Vuong test is not significant or significant but in favor of the Poisson model, we do not reject the null.

So , specific contribution of the Vuong test into the count outcomes regression analysis is that to determine whether estimating a zero-inflation ZIP model component is appropriate or whether a single-equation count model like Poisson model should be used .