STA303 Blog 4

Hello, today's topic is the introduction of the Poisson Regression.

When we are predicting a response using one or more explanatory variables, although these examples have response variables that are counts per some unit of time or space, the Poisson Regression Model is our best choice.

As the Poisson random variables are often used to model counts, their minimum value is zero and, in theory, the maximum is unbounded. Hence, they take value from 0 to positive infinity.

We’d like to model our main parameter λ, the average number of occurrences per unit of time or space, as a function of one or more covariates. ( For example, in the first question above, λi represents the average number of motorcycle deaths in a year for state i, and we hope to show that state-to-state variability in λi can be explained by state helmet laws.) As a result, we model the Poisson parameter λi as a linear function of an explanatory variable.

Hence, the model will look like this:

This model exactly looks like a linear regression model; however, the equal variance assumption in linear regression inference is violated because as the mean rate for a Poisson variable increases, the variance also increases.

The solution for these problems is to model log(λi) instead of λi as a function of the covariates. The log(λi) takes on values from negative infinity to positive infinity (−∞,∞). We can also take into account the increase in the variance with an increasing mean using this approach.

Hence, the Poisson Regression Model looks like this:

where the observed values Yi∼ Poisson with λ=λi for a given xi.

Reference:

Legler, P. (2021). Chapter 4 Poisson Regression | Beyond Multiple Linear Regression. Bookdown.org. Retrieved 30 March 2021, from https://bookdown.org/roback/bookdown-BeyondMLR/ch-poissonreg.html.