Example: Modeling Count Data

We illustrate models for discrete data using the horseshoe crab dataset introduced in Sec. 1.5.1. The response variable for the n = 173 mating female crabs is y = number of " satellites " — male crabs that group around the female and may fertilize her eggs. Explanatory variables are the female crab's color, spine condition, weight, and carapace width. To illustrate the Poisson, negative binomial, ZIP, and ZINB distributions introduced in this chapter, we first investigate the marginal distribution of satellite counts. From Sec. 1.5.1, the mean of 2.919 and variance of 9.912 suggest overdispersion relative to the Poisson.The histogram (Figure 7.2) shows a strong mode at 0 but slightly elevated frequencies for satellite counts of 3 through 6 before decreasing substantially. Because the distribution may not be unimodal, the negative binomial may not fit as well as a zero-inflated distribution. Figure 7.2. Histogram for sample distribution of y = number of horseshoe crab satellites.