Poisson Regression Basic Concepts

Introduction

Poisson regression is similar to multinomial logistic regression in that the dependent variable can take only non-negative integer values. With this regression method, the dependent variable takes values 0, 1, …, r for some known value of r, while with Poisson regression there is no predetermined r value, i.e. any count value is possible.

In Poisson regression, we assume that we have k independent variables x1, x2, …, xk. We further assume that we have a random sample of size n, whose ith element is xi1, xi2, …, xik. Corresponding to each k-tuple in the sample is an element yi which is taken from a population that has a Poisson distribution with mean μi. Thus, for each i, the probability of yi events occurring is

Poisson pdf

Since the mean and variance of a Poisson distribution are equal, for each i we also have

Poisson mean equals varianceRegression Model

We could now study the linear regression equations

Linear regression equation

and use an ordinary least squares approach, but this would not assure that the values of μi are non-negative. Instead, we use

Poisson regression equation

where exp(x) is the link function. This is equivalent to

Alternative Poisson regression form

We can assume that x1 = 1 (thus, xi1 = 1 for all i) so that we can have a constant term as for other regression models.

For any values of x2, x3, …, xk, we can use the model to predict the risk μ of a rare event occurring in a specified unit of time or space as follows

Poisson regression general form

where b1, b2, …, bk are estimates of the regression coefficients and x1 = 1.

Log-Likelihood Function

The log-likelihood function is

Log-likelihood Poisson regression

where LL is considered to be a function of the bi (which are used to define the μi) and the xij and yi are considered to be fixed.

Predictions

For each observation, we can also explicitly specify the exposure time or space ti, in which case

Poisson regression with frequency

Thus, for any values of x2, x3, …, xk, we can use the model to predict the number of occurrences μ of a rare event in time (or space) t by

Poisson regression predictive form

References

Hintze, J. L. (2007) Poisson regression. NCSS
https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/Poisson_Regression.pdf

Nussbaum, E. M., Elsadat, S., Khago, A. H. (2007) Best practices in evaluating count data, Chapter 21: Poisson regression.
http://www.academia.edu/438746

Penn State (2017) Poisson regression. STAT 504: Analysis of discrete data.
https://online.stat.psu.edu/stat504/lesson/9

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