the number of defective teeth, thesizeis the total number of teeth for that same
person. The following is the continuation of Example 10.4 on the emergency
service data; but data in Table 10.2 also include information on four covariates.
Example 10.5 The purpose of this study was to examine the data for 44
physicians working for an emergency at a major hospital so as to determine
which of four variables are related to the number of complaints received during
the preceding year. In addition to the number of complaints, served as the
dependend variable, data available consist of the number of visits (which serves
as thesizefor the observation unit, the physician) and four covariates. Table
10.2 presents the complete data set. For each of the 44 physicians there are
two continuous independent variables, the revenue (dollars per hour) and the
workload at the emergency service (hours) and two binary variables, gender
(female/male) and residency traning in emergency services (no/yes).
10.3.1 Simple Regression Analysis
In this section we discuss the basic ideas of simple regression analysis when
only one predictor or independent variable is available for predicting the
response of interest.
Poisson Regression Model In our framework, the dependent variableY is
assumed to follow a Poisson distribution; its valuesyi’s are available fromn
observation units, which is also characterized by an independent variableX. For
the observation unitið 1 anÞ, letsibe the size andxibe the covariate value.
The Poisson regression model assumes that the relationship between the
mean ofYand the covariateXis described by
EðYiÞ¼silðxiÞ
¼siexpðb 0 þb 1 xiÞwherelðxiÞis called theriskof observation unitið 1 anÞ. Under the assump-
tion thatYiis Poisson, the likelihood function is given by
Lðy;bÞ¼Yni¼ 1½silðxiÞyiexp½silðxiÞ
yi!lnL¼Xni¼ 1½yilnsilnyi!þyiðb 0 þb 1 xiÞsiexpðy 0 þb 1 xiÞfrom which estimates forb 0 andb 1 can be obtained by the maximum likelihood
procedure.
Measure of Association Consider the case of a binary covariateX: say, repre-
senting an exposure (1¼exposed, 0¼not exposed). We have:
POISSON REGRESSION MODEL 357