Introduction to Categorical Data Analysis

3.3 GENERALIZED LINEAR MODELS FOR COUNT DATA 75
Chapter 7 presents Poisson GLMs for counts in contingency tables that cross
classify categorical response variables. This section introduces Poisson GLMs for
modeling counts for a single discrete response variable.
3.3.1 Poisson Regression
The Poisson distribution has a positive mean. GLMs for the Poisson mean can use
the identity link, but it is more common to model the log of the mean. Like the linear
predictor α + βx, the log of the mean can take any real-number value. A Poisson
loglinear model is a GLM that assumes a Poisson distribution for Y and uses the log
link function.
For a single explanatory variable x, the Poisson loglinear model has form
The mean satisfies the exponential relationship
log μ = α + βx (3.5)
μ = exp(α + βx) = e α (e β ) x
(3.6)
A one-unit increase in x has a multiplicative impact of e β on μ: The mean of Y
at x + 1 equals the mean of Y at x multiplied by e β .Ifβ = 0, then e β = e 0 = 1 and
the multiplicative factor is 1. Then, the mean of Y does not change as x changes. If
β>0, then e β > 1, and the mean of Y increases as x increases. If β30.25), and calculated the sample mean number of satellites in each
category. Figure 3.5 plots these sample means against the sample mean width for the
female crabs in each category.