# Poission regression

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Chapter 4

Poisson Models for Count Data
In this chapter we study log-linear models for count data under the assumption of a Poisson error structure. These models have many applications, not only to the analysis of counts of events, but also in the context of models for contingency tables and the analysis of survival data.

4.1

Introduction to Poisson Regression

As usual, we start byintroducing an example that will serve to illustrative regression models for count data. We then introduce the Poisson distribution and discuss the rationale for modeling the logarithm of the mean as a linear function of observed covariates. The result is a generalized linear model with Poisson response and link log.

4.1.1

The Children Ever Born Data

Table 4.1, adapted from Little (1978),comes from the Fiji Fertility Survey and is typical of the sort of table published in the reports of the World Fertility Survey. The table shows data on the number of children ever born to married women of the Indian race classiﬁed by duration since their ﬁrst marriage (grouped in six categories), type of place of residence (Suva, other urban and rural), and educational level (classiﬁed in fourcategories: none, lower primary, upper primary, and secondary or higher). Each cell in the table shows the mean, the variance and the number of observations. In our analysis of these data we will treat the number of children ever
G. Rodr´ ıguez. Revised September, 2007

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CHAPTER 4. POISSON MODELS FOR COUNT DATA

Table 4.1: Number of Children Ever Born to Women of Indian Race By MaritalDuration, Type of Place of Residence and Educational Level
(Each cell shows the mean, variance and sample size) Marr. Dur. 0–4 Suva LP UP 1.14 0.90 0.73 0.67 21 42 2.67 2.04 0.99 1.87 30 24 3.67 2.90 2.31 1.57 27 20 4.94 3.15 1.46 0.81 31 13 5.06 3.92 4.64 4.08 18 12 6.74 5.38 11.66 4.27 27 8 Urban LP UP 0.85 1.05 1.59 0.73 27 39 2.65 2.68 1.51 0.97 37 44 3.33 3.62 2.99 1.96 43 29 5.36 4.60 2.973.83 42 20 5.88 5.00 4.44 4.33 25 13 7.51 7.54 10.53 12.60 45 13 Rural LP UP 0.96 0.97 0.81 0.80 102 107 2.71 2.47 1.36 1.30 117 81 4.14 3.94 3.31 3.28 132 50 5.59 4.50 3.23 3.29 86 30 6.34 5.74 5.72 5.20 68 23 7.81 5.80 7.57 7.07 59 10

5–9

10–14

15–19

20–24

25–29

N 0.50 1.14 8 3.10 1.66 10 4.08 1.72 12 4.21 2.03 14 5.62 4.15 21 6.60 12.40 47

S+ 0.73 0.48 51 1.73 0.68 22 2.001.82 12 2.75 0.92 4 2.60 4.30 5 2.00 – 1

N 1.17 1.06 12 4.54 3.44 13 4.17 2.97 18 4.70 7.40 23 5.36 7.19 22 6.52 11.45 46

S+ 0.69 0.54 51 2.29 0.81 21 3.33 1.52 15 3.80 0.70 5 5.33 0.33 3 – – –

N 0.97 0.88 62 2.44 1.93 70 4.14 3.52 88 5.06 4.91 114 6.46 8.20 117 7.48 11.34 195

S+ 0.74 0.59 47 2.24 1.19 21 3.33 2.50 9 2.00 – 1 2.50 0.50 2 – – –

born to each woman as the response, andher marriage duration, type of place of residence and level of education as three discrete predictors or factors.

4.1.2

The Poisson Distribution

A random variable Y is said to have a Poisson distribution with parameter µ if it takes integer values y = 0, 1, 2, . . . with probability Pr{Y = y} = e−µ µy y! (4.1)

for µ > 0. The mean and variance of this distribution can be shown to be E(Y )= var(Y ) = µ. Since the mean is equal to the variance, any factor that aﬀects one will also aﬀect the other. Thus, the usual assumption of homoscedasticity would not be appropriate for Poisson data.

4.1. INTRODUCTION TO POISSON REGRESSION

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The classic text on probability theory by Feller (1957) includes a number of examples of observations ﬁtting the Poisson distribution, includingdata on the number of ﬂying-bomb hits in the south of London during World War II. The city was divided into 576 small areas of one-quarter square kilometers each, and the number of areas hit exactly k times was counted. There were a total of 537 hits, so the average number of hits per area was 0.9323. The observed frequencies in Table 4.2 are remarkably close to a Poisson distribution with mean µ...