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Types of Poisson Regression
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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

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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

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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

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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 ?,
  which happens to be both its mean and variance.
  While convenient to remember, its not often
  realistic.
• A distribution of counts will usually have a
  variance thats 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
  doesnt fit well.

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• One approach that addresses this issue is
  Negative Binomial Regression.
• We go for Negative Binomial Regression when
  Variance gt 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)µµ2/k

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Zero Inflated Regression
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