Statistical Methods in Medical Research 4ed

Example 14.2
Bishop (2000) followed up 207 patients admitted to hospital following injury and recorded
functional outcome after 3 months using a modified Glasgow Outcome Score (GOS). This
score had five ordered categoriesÐfull recovery, mild disability, moderate disability,
severe disability, and dead or vegetative state.
The relationship between functional outcome and a number of variables relating to the
patient and the injury was analysed using the cumulative logits model (14.9) of
polytomous regression. The final model included seven bs indicating the relationship
between outcome and seven variables, which included age, whether the patient was
transferred from a peripheral hospital, three variables representing injury severity, two
interaction terms, and four as representing the splits between the five categories of GOS.
The model fitted well and was assessed in terms of ability to predict GOS for each
patient. For 88 patients there was exact agreement between observed and predicted GOS
scores compared with 52 4 expected by chance if the model had no predicting ability, and
there were three patients who differed by three or four categories on the GOS scale
compared with 23 3 expected. As discussed in §13.3, this is likely to be overoptimistic as
far as the ability of the model to predict the categories of future patients is concerned.
14.4 Poisson regression
Poisson distribution
The expectation of a Poisson variable is positive and so limited to the range 0 to
1. A link function is required to transform this to the unlimited range 1 to 1.
The usual transformation is the logarithmic transformation
leading to the log-linear model
Example 14.3
g…m† ˆln m,
14.4 Poisson regression 499
ln m ˆ b 0 ‡ b 1x 1 ‡ b 2x 2 ‡ ...‡ bpxp: …14:12†
Table 14.2 shows the number of cerebrovascular accidents experienced during a certain
period by 41 men, each of whom had recovered from a previous cerebrovascular accident
and was hypertensive. Sixteen of these men received treatment with hypotensive drugs and
25 formed a control group without such treatment. The data are shown in the form of a
frequency distribution, as the number of accidents takes only the values 0, 1, 2 and 3. This
was not a controlled trial with random allocation, but it was nevertheless useful to enquire
whether the difference in the mean numbers of accidents for the two groups was significant,
and since the age distributions of the two groups were markedly different it was
thought that an allowance for age might be important.
The data consist of 41 men, classified by three age groups and two treatment groups,
and the variable to be analysed is the number of cerebrovascular accidents, which takes
integral values. If the number of accidents is taken as having a Poisson distribution with