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Modeling Count Data - Chapter 4 Modelling Counts The...

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Chapter 4 Modelling Counts - The Poisson and Negative Binomial Regression In this chapter, we discuss methods that model counts. In a longitudinal setting, these counts typically result from the collapsing repeated binary events on subjects measured over some time period to a single count (e.g., number of episodes of diarrhea, as in the HIV/drinking water study). We start by discussing the most popular distribution for mod- elling counts, the Poisson distribution and Poisson regression (note, in the next chapter, we link Poisson regression directly to survival analysis). The chapter is finished by presenting a slightly bigger model, the negative binomial distribution, which handles some situations where the Poisson model is a poor fit. 4.1 Poisson Distribution The Poisson distribution is often used to model information on counts of various kinds, particularly in situations where there is no natural “denominator”, and thus no upper bound or limit on how large an observed count can be. This is in contrast to the Binomial distribution which focuses on observed proportions. Possible examples of count data where a Poisson model is useful include (i) the number of automobile fatalities in a given region over year intervals, (ii) the number of AIDS cases for a given risk group for a series of monthly intervals, (iii) the number of murders in Chicago by year, (iv) the number of server failures for a web-based company by year, and (v) the number of earthquakes of a certain magnitude in a seismically active region by decade. In each of these examples, there is no reasonable denominator associated with the counts–even in (i) and (iii) where 7
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8 CHAPTER 4. MODELLING COUNTS - THE POISSON AND NEGATIVE BINOMIAL REGRESSION population counts might be appropriate to capture incidence or event proportions, these totals may be difficult to ascertain or define in such a way that members of the population hold at least a similar level of risk of an event. When a Poisson model is appropriate for an outcome Y , the probabilities of observing any specific count, y , are given by the formula: Pr ( Y = y ) = λ y e y y ! , (4.1) where λ is known as the population rate parameter, and y ! = y × ( y 1) ×· · ·× 2 × 1. Such a Poisson random variable Y has expectation E ( Y ) = λ , and variance V ar ( Y ) = λ . The fact that the expectation and variance agree provides a quick check on whether a Poisson model might be appropriate for a sample of observations. In the examples above, the parameter λ describes the (i) rate of automobile fatalities per year, (ii) the AIDS incidence rate per month, etc. The number of pedestrian fatalities due to automobile accidents in Solana County, cal- ifornia was three in 1999. To illustrate the Poisson distribution, suppose that we believe that the annual rate for such fatalaties is two per year and that the distribution is Pois- son. With λ = 2, Figure 4.1 shows the probability density associated with the Poisson distribution. The density indicates that the probability of observing a count of three fa- talities in a specific year is just 2
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