17: Count Dependent Variables
- Page ID
- 57793
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Using generalized linear models allows one to fit dependent variables that follow specified distributions. This allows us to focus more clearly on the variable we are modeling. It also allows us to avoid many of the "fixes" we used in ordinary least squares that tried to "handle" issues instead of using them to better understand.
In this chapter, we examine another type of dependent variable and how we can use GLMs to fit such variables. The variable is the count variable with no upper limit. This support separates it from the Binomial dependent variable from the previous chapter.
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Remember that we are examining these many different types of regressions for one primary reason:
The requirements of Ordinary Least Squares are violated by discrete dependent variables.
Rather than seeing this as a problem, we can use it as an indicator that we can model the data better and extract more information from the data. This marks the next chapter of discrete dependent variables. In Chapter 15: Binary Dependent Variables, we discussed binary dependent variables — dependent variables that can only take on two values. In the previous chapter, we examined dependent variables that were counts of successes over a known number of trials (or attempts). In this chapter, we examine count dependent variables that have no theoretical upper limit. Some examples of such count variables include the number of fires in Galesburg in a year, the number of deaths due to terrorist attacks in the world in a month, and the number sorties per day in a battle.
Let us set the stage with an example that we will return to throughout this chapter: The Troubles of Ruritania, a lengthy period of terrorist and counter-terrorist activity in the country, lasted approximately from 1969 until 2002. In that time, over 1800 people died as a result of terrorist actions — by both republican and loyalist groups. Six prime ministers of the Ruritania — on both the political left and right — had to deal with this terrorism. If we assume that the terrorist groups are rational actors, then they will act to maximize their chances of achieving their goals. Because of its hierarchical structure and large size, the Ruritanian Republican Army (RAvŘ) was best able to organize its actions to affect the elections.
The question is whether they did.
Did the RAvŘ adjust its tactics in reaction to the political ideology of the prime minister?
Unfortunately, the extant literature is divided on the direction of the effect. Some research suggests that the RAvŘ became more violent and killed more people when the conservatives held power. Other research suggests that the RAvŘ became more violent under the liberals. Which is it?
For the unbounded count variables in this chapter, there are three identifying characteristics:
- the variable can never be negative,
- it has no theoretic upper bound, and
- it is discrete.
If \(\mathbf{Y}\) is this type of count variable, then
\begin{equation*}Y \in \{0, 1, 2, 3, \ldots\}\end{equation*}
If we just do usual linear modeling without taking these three items into consideration, we lose information inherent in the data; we are making assumptions about the data that are incorrect. Performing count data analysis extracts more information from the data you worked so hard to collect. It gives better predictions and explanations of the phenomena under study. It also (usually) means not having to "fix" violations of homoskedasticity or fit.