# Poisson regression

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## What is Poisson regression ?

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:

• Number of employees in a company on a given year.
• Number of tiles in a certain surface area.
• Count of the viruses in a microscope slide.

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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## Assumptions of Poisson regression

Following are the assumptions of the Poisson regression:

• Y-values are counts. Use Poisson regression only if your response variables are a count of something.
• Counts must be positive integers 0 or greater. As Poisson distribution is a discrete distribution, any count which is a negative integer or a fraction will not work.
• Counts must follow a Poisson distribution. The mean and variance should be the same.
• Explanatory variables are continuous, dichotomous or ordinal.
• Observations must be independent.

## The Poisson regression distribution

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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## The Poisson regression model

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.

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## Interpreting the Poisson regression model:

• exp(α)= effect on the mean μ, when X = 0
• exp(β) = with every unit increase in X, the predictor variable has the multiplicative effect of exp(β) on the mean of Y, that is μ
• If β = 0, then exp(β) = 1, and the expected count is exp(α) and, Y and X are not related.
• If β > 0, then exp(β) > 1, and the expected count is exp(β) times larger than when X = 0
• If β < 0, then exp(β) < 1, and the expected count is exp(β) times smaller than when X = 0

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