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Poisson regression is a tool that helps you model the response variables that are observed counts. It determines which explanatory variables have statistically significant effect on the response variables. In other words, we can say that the response variable is a dependent variable and the explanatory variable is an independent variable.
If we talk about the graphical point of view, the response variables are the Y-values and the explanatory variables are the X-values. Hence, Poisson regression helps you identify which y-values influence which x-values.
As we said Poisson regression works with the count of the responsive variables, the examples of the same can be:
As normal distribution is considered as a suitable assumption for the Poisson distribution, it is bets to use Poisson regression when the count of the responsive variable is a small integer. Poisson regression only uses numeric and continuous data.
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Following are the assumptions of the Poisson regression:
Poisson distribution basically models the probability of y events happening within a certain timeframe. It assumes that the occurrences are unaffected by the timings of the previous y occurrences.
Poisson distribution is represented by a formula:
µ is the mean incidence rate of a rare event per unit of exposure. Exposure is nothing but time, space, distance, population size, etc. as exposure is represented based on time, and we use the t symbol to represent exposure. When no given values, the exposure value is taken as 1.
The parameter µ can be interpreted as the risk of a new occurrence of the event during a specified exposure period, t. The probability of y events is then given by
One of the properties of Poisson distribution is that its mean and variance are always equal.
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In Poisson regression, we suppose that the Poisson incidence rate µ is determined by a set of k regressor variables (the X’s). The expression binding these quantities:
Often, X1 corresponds to 1 and β1 is called the intercept. The regression coefficients β1, β2, β3 . . . βk are unknown parameters and are estimated from a set of data.
Using this notation, the fundamental Poisson regression model for an observation “i” is written as:
For a given set of values of the regressor variables, the outcome follows the Poisson distribution.