Regression models for count data

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Are you wondering how to create a regression model for count data? Well then you are in the right place! In this article we tell you everything you need to know to select the best regression model for your count data.

First we discuss what count data is and when you should use a regression model for count data. After that, we talk about different regression models for count data such as Poisson regression, zero-inflated Poisson regression, negative binomial regression, and zero-inflated negative binomial regression. Finally, we explain how to choose which model is most appropriate for your count data.

An introduction to count data

Before we explain the differences between different regression models for count data, we will talk about what count data is and when you should use a regression model for count data.

What is count data?

The first thing we will talk about is what count data is. Count data is any data such that the ultimate number that is recorded is a count of the number of events or occurrences of something. In general, count data should be strictly non-negative and should take the form of whole numbers or integers. This means that count data is discrete rather than continuous.

Here are a few examples of variables that qualify as count data.

  • The number of sunny days in a year
  • The number of doctor visits a person makes in a year
  • The number of points a player scores in a soccer game

When to use a regression model for count data?

There are some cases where you will have count variables in your analysis but you will not necessarily need to use a regression model that is made for count data. So when do you use a regression model that is made for count data?

You should use a regression model for count data specifically when your outcome variable represents a count. There is no need to use a regression model for count data if you are including features in your model that are counts, so long as your outcome variable is not a count.

Considerations for working for count data

Now we will talk about a few common concerns you should look out for when you are working with count data. These concerns will help to inform what type of regression model you should use when you are working with count data.

Overdispersion in count data

First we will talk about overdispersion in count data. If your data is overdispersed, it simply means that the variance of your data is higher than the mean of your data. If the distribution of your count data has a very long tail, the data may be overdispersed.

It is important to look out for overdispersion in count data because the most common distribution that is used to model count data, the Poisson distribution, makes the assumption that the mean and variance of the distribution are the same. If this assumption does not hold true for your count data, you should consider using a different distribution to model your count data.

Two charts with examples of count data. One has overdispersion and one does not.

Zero inflation in count data

The next concern we will talk about is zero-inflation. Zero-inflation simply means that there are more zero values than expected in the count data. Count data that is zero-inflated is generally easy to recognize. All you need to do is plot your data and look for an overwhelmingly large spike at zero.

Two charts with examples of count data. One has zero-inflation and one does not.

Common regression models for count data

Now that we have gone over some concerns that you should look for in count data, we will go over some of the most common regression models for count data. We will also talk about the different situations where you should use one model over the other.

A diagram with suggestions for what regression models for count data to use if your data is overdispersed or zero inflated.

Poisson regression

We will start by talking about Poisson regression. This is because Poisson regression is the most simple regression model for count data. The Poisson regression model has the fewest number of parameters that need to be estimated because it makes the assumption that the mean of the distribution is the same as the variance of the distribution.

Since the Poisson regression model is the most simple regression model for count data, it is a great baseline model to start out with before you move on to more complex models. If you do try out some more complex regression models for count data, you should compare the performance of the other models to the Poisson regression model to ensure that the added complexity improves the performance of the model.

Since the Poisson regression model is the most simple and least flexible count regression model, it works best in situations where there is no zero-inflation and no overdispersion.

Zero-inflated Poisson regression

The next model we will talk about is the zero-inflated Poisson model. This model is similar to the Poisson model, however it accommodates situations where there are an excessive number of zeros in the distribution.

The zero-inflated Poisson model accommodates the additional zeros by treating that data as if it comes from two different distributions – one that is made up entirely of zeros and another more standard Poisson distribution. An additional parameter is introduced in this model that represents whether an observation belongs to the traditional poisson distribution or the distribution of zeros.

Since the zero-inflated Poisson model is still based on the Poisson distribution, zero-inflated Poisson regression works best in cases where the data has zero-inflation but no overdispersion.

Negative binomial regression

So what do you do if you believe that the variance of your distribution is greater than the mean of your distribution? This is the situation where you would want to use a negative binomial regression model. The negative binomial distribution is similar to the Poisson distribution in that it is intended to be used for count data.

The main difference between the negative binomial model and the Poisson model is that the negative binomial model introduces another parameter that allows the mean and variance of the distribution to differ. That means that negative binomial regression is great for cases where your data is overdispersed but not zero-inflated.

Zero-inflated negative binomial regression

The final regression model for count data that we will talk about is the zero-inflated negative binomial model. The zero-inflated negative binomial model works in the same way that the zero-inflated poisson model works. That is to say, that the model makes the assumption that some of the data comes from a traditional negative binomial distribution and the other data comes from a distribution made up of all zeros.

The zero-inflated negative binomial model is a good model to turn to in cases where you have zero-inflation or overdispersion.

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