The Modeling Limitations of Linear Regression Which Poisson Regression Addresses

Linear regression comes with its own set of challenges/assumptions.

For instance:

• After modeling, the output can be negative for some inputs.

• But this may not make sense at times — predicting the number of goals scored, number of calls received, etc.

• Thus, it is clear that it cannot model count (or discrete) data.

Furthermore, in linear regression:

• Residuals are expected to be normally distributed around the mean.

• Hence, the outcomes on either side of the mean (m-x, m+x) are equally likely.

For instance:

• if the expected number (mean) of calls received is 1...

• ...then, according to linear regression, receiving 3 calls (1+2) is just as likely as receiving -1 (1-2) calls. (This relates to the concept of prediction intervals, which I discussed in one of my previous posts here: Prediction intervals.)

• But in this case, a negative prediction does not make any sense.

Thus, if the above assumptions do not hold, linear regression won’t help.

Instead, what you may need is Poisson regression.

Poisson regression:

• is more suitable if your response (or outcome) is count-based.

• assumes that the response comes from a Poisson distribution.

It is a type of generalized linear model (GLM) that is used to model count data.

It works by estimating a Poisson distribution parameter (λ), which is directly linked to the expected number of events in a given interval.

Contrary to linear regression, in Poisson regression:

• Residuals may follow an asymmetric distribution around the mean (λ).

• Hence, outcomes on either side of the mean (λ-x, λ+x) are NOT equally likely.

For instance:

• if the expected number (mean) of calls received is 1...

• ...then, according to Poisson regression, it is possible to receive 3 (1+2) calls, but it is impossible to receive -1 (1-2) calls.

• This is because its outcome is also non-negative.

The regression fit is mathematically defined as follows:

The effectiveness of Poisson regression is evident from the image below:

The following visual neatly summarizes this post:

While this was just about Poisson regression — one of the many members of the generalized linear models (GLMs) family, here’s a deep dive to learn everything about GLMs: Generalized Linear Models (GLMs): The Supercharged Linear Regression.

👉 Over to you: Can you tell me some limitations or considerations for using Poisson regression?

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