In public health data analysis, disease counts, proportions, and rates are often used as outcome variables. These discrete outcomes differ from the continuous variables typically associated with linear regression. When analyzing count data, especially when the variable of interest is an integer that can take any value in the set of natural numbers, the Poisson Model frequently comes into play, depending on the specific context of the analysis.
The Poisson Model
The Poisson distribution is a member of the exponential family of distributions and uses the natural logarithm as its canonical link function. This makes it well-suited for modeling count data where the mean and variance of the response variable are equal.
For instance, consider counts \(Y_1,Y_2,…,Y_n\), which are independently and identically distributed as Poisson random variables with a mean \(E(Y_i)\). The Poisson regression approach models the expected value \(E(Y_i)\) as a function of covariates:
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