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Poisson Regression



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.

 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