Introductory Biostatistics

obtain the expected frequencies; for example,

E 2 ¼ð 44 Þð 0 : 198 Þ

¼ 8 : 71

To avoid having any expected frequencies less than 1, we combine the last five
groups together, resulting in eight groups available for testing goodness of fit
with

O 8 ¼ 2 þ 2 þ 1 þ 1 þ 1

¼ 7

E 8 ¼ 1 : 45 þ 0 : 62 þ 0 : 22 þ 0 : 09 þ 0 : 04

¼ 2 : 42

The result is

X^2 ¼

ð 1 
 1 : 59 Þ^2

1 : 19

þ

ð 12 
 5 : 19 Þ^2

5 : 19

þþ

ð 7 
 2 : 42 Þ^2

2 : 42

¼ 28 : 24

with 8
2 ¼6 degrees of freedom, indicating a significant deviation from the
Poisson distributionðp< 0 : 005 Þ. A simple inspection of Table 10.1 reveals an
obvious overdispersion.

10.3 POISSON REGRESSION MODEL

As mentioned previously, the Poisson model is often used when the random
variableXis supposed to represent the number of occurrences of some random
event in an interval of time or space, or some volume of matter, and numerous
applications in health sciences have been documented. In some of these appli-
cations, one may be interested to see if the Poisson-distributed dependent
variable can be predicted from or explained by other variables. The other vari-
ables are called predictors, explanatory variables, or independent variables. For
example, we may be interested in the number of defective teeth per person as a
function of gender and age of a child, brand of toothpaste, and whether or not
the family has dental insurance. In this and other examples, the dependent
variableYis assumed to follow a Poisson distribution with meany.
The Poisson regression model expresses this mean as a function of certain
independent variablesX 1 ;X 2 ;...;Xk, in addition to thesizeof the observation
unit from which one obtained the count of interest. For example, ifYis the
number of virus in a solution, thesizeis the volume of the solution; or ifYis

356 METHODS FOR COUNT DATA