- If the observation unit is exposed,
lnliðexposedÞ¼b 0 þb 1whereas- If the observation unit is not exposed,
lnliðnot exposedÞ¼b 0or, in other words,
liðexposedÞ
liðnot exposedÞ¼eb^1This quantity is called therelative risk associated with the exposure.
Similarly, we have for a continuous covariateXand any valuexofX,
lnliðX¼xÞ¼b 0 þb 1 x
lnliðX¼xþ 1 Þ¼b 0 þb 1 ðxþ 1 Þso that
liðX¼xþ 1 Þ
liðX¼xÞ¼eb^1representing the relative risk associated with a 1-unit increase in the value ofX.
The basic rationale for using the termsriskandrelative riskis the approxi-
mation of the binomial distribution by the Poisson distribution. Recall from
Section 10.2 that when n!y,p!0 whiley¼np remains constant, the
binomial distributionBðn;pÞcan be approximated by the Poisson distribution
PðyÞ. The numbernis the size of the observation unit; so the ratio between the
mean and the size represents thep[orlðxÞin the new model], the probability or
riskand the ratio of risks is the risks ratio orrelative risk.
Example 10.6 Refer to the emergency service data in Example 10.5 (Table
10.2) and suppose that we want to investigate the relationship between the
number of complaints (adjusted for number of visits) and residency training. It
may be perceived that by having training in the specialty a physician would
perform better and therefore would be less likely to provoke complaints. An
application of the simple Poisson regression analysis yields the results shown in
Table 10.3.
The result indicates that the common perception is almost true, that the
relationship between the number of complaints and no residency training in
emergency service is marginally significantðp¼ 0 : 0779 Þ; the relative risk asso-
POISSON REGRESSION MODEL 359