Introductory Biostatistics

Poisson Regression Model with Several Covariates Suppose that we want to
consider k covariates, X 1 ;X 2 ;...;Xk, simultaneously. The simple Poisson
regression model of Section 10.3.1 can easily be generalized and expressed as

EðYiÞ¼silðxji’sÞ

¼siexp b 0 þ

Xk

j¼ 1

bjxji

!

whereYis the Poisson-distributed dependent variable andlðxji’sÞis theriskof
observation unitið 1 anÞ.
Under the assumption thatYiis Poisson, the likelihood function is given by

Lðy;bÞ¼

Yn

i¼ 1

½silðxiފyiexp½
silðxji’sފ

yi!

¼

Xn

i¼ 1

yilnsi
lnyi!þyi b 0 þ

Xk

j¼ 1

bjxji

!

siexp y 0 þ

Xk

j¼ 1

bjxji

"#!

from which estimates forb 0 ;b 1 ;...;bkcan be obtained by the maximum like-
lihood procedure.
Also similar to the simple regression case, expðbiÞrepresents:

1. The relative risk associated with an exposure ifXiis binary (exposed
   Xi¼1 versus unexposedXi¼0), or


2. The relative risk due to a 1-unit increase ifXiis continuous (Xi¼xþ 1
   versusXi¼x).

Afterbb^iiand its standard error have been obtained, a 95% confidence inter-
val for the relative risk above is given by

exp½bb^iG 1 :96SEðbb^iފ

These results are necessary in the e¤ort to identify important risk factors for the
Poisson outcome, the count. Of course, before such analyses are done, the
problem and the data have to be examined carefully. If some of the variables
are highly correlated, one or fewer of the correlated factors are likely to be as
good predictors as all of them; information from other similar studies also has
to be incorporated so as to drop some of these correlated explanatory variables.
The use of products such asX 1 X 2 and higher power terms such asX 12 may be
necessary and can improve the goodness of fit. It is important to note that we
are assuming a(log)linear regression modelin which, for example, the relative
risk due to a 1-unit increase in the value of a continuousXi(Xi¼xþ1 versus
Xi¼x) is independent ofx. Therefore, if thislinearityseems to be violated, the

POISSON REGRESSION MODEL 361