Types of Poisson Regression

Offset Regression  A variant of Poisson Regression  Count data often have an exposure variable, which indicates the number of times the event could have happened  This variable should be incorporated into a Poisson model with the use of the offset option

Offset Regression  If all the students have same exposure to math (program), the number of awards are comparable  But if there is variation in the exposure, it could affect the count  A count of 5 awards out of 5 years is much bigger than a count of 1 out of 3  Rate of awards is count/exposure  In a model for awards count, the exposure is moved to the right side  Then if the algorithm of count is logged & also the exposure, the final model contains ln(exposure) as term that is added to the regression equation  This logged variable, ln(exposure) or a similarity constructed variable is called the offset variable

Offset Poisson Regression  A data frame with 63 observations on the following 4 variables. (lung.cancer)  years.smok a factor giving the number of years smoking  cigarettes a factor giving cigarette consumption  Time man-years at risk  y number of deaths

Negative Binomial Regression  One potential drawback of Poisson regression is that it may not accurately describe the variability of the counts  A Poisson distribution is parameterized by λ, which happens to be both its mean and variance. While convenient to remember, it’s not often realistic.  A distribution of counts will usually have a variance that’s not equal to its mean. When we see this happen with data that we assume (or hope) is Poisson distributed, we say we have under- or over dispersion, depending on if the variance is smaller or larger than the mean.  Performing Poisson regression on count data that exhibits this behavior results in a model that doesn’t fit well.

 One approach that addresses this issue is Negative Binomial Regression.  We go for Negative Binomial Regression when Variance > Mean (over dispersion)  The negative binomial distribution, like the Poisson distribution, describes the probabilities of the occurrence of whole numbers greater than or equal to 0.  The variance of a negative binomial distribution is a function of its mean and has an additional parameter, k, called the dispersion parameter. var(Y)=μ+μ2/k

Zero Inflated Regression