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Lecture 10
Marginal Poisson Regression
Model and GEE
• Examples of count data
– Number of panic attacks occurring during 6-month intervals
after receiving treatment
– Number of infant deaths per month before and after
introduction of a prenatal care program
• The Poisson distribution has been the
most commonly used to model count
data:
Epileptic seizures
• Clinical Trial of 59 epileptics
• For each patient, the number of epileptic
seizures was recorded during a baseline period
of 8 weeks
• Patients were randomized to treatment with the
anti-epileptic drug progabide or placebo
• # of seizures was then recorded in four
consecutive 2-week intervals
• Question: Does progabide reduce the rate of
epileptic seizures?
Poisson Regression Model
Poisson Regression Model
In the progabide example: exp(β) represents
the ratio of average seizure rates, measured
as the number of seizures per 2-week period,
for the treated compared to the controls:
If β<0, then the treatment is effective
relative to the placebo in controlling the
seizure rate
Overdispersed Data
Var(Yij) > E[Yij]
is called the
“overdispersion parameter”
Irregular times
• Suppose that the interval times tij, during
which the events are observed, are not the
same for all subjects. The problem can be
solved by decomposing the marginal
mean E[Yij] as:
Epileptic Seizures
• Clinical Trial of 59 epileptics
– 31 patients received an anti-epileptic drug
progabide
– 28 received placebo
• Patients from the 2 groups are comparable
in terms of age and 8-week baseline
seizure counts
=> High-degree of extra-Poisson variation
Cross-product ratio:
=> Some indication of
treatment effect
Poisson Regression Model and GEE Method
Xi2 allows different
baseline seizure
counts for the
treated and the
control groups
Parameter Interpretation
• exp(β1): ratio of the average seizure rate
after treatment to the average rate before
treatment, for the placebo group
• β3: (parameter of interest) represents the
difference in the log of the post-to-pre
treatment ratio between the progabide
and the placebo groups.
– β3 < 0 corresponds to a greater reduction in
the seizure counts for the progabide group
Results
• If patient 207 is included, then
…suggests very little difference between
treatment and placebo groups in the
change of seizure counts before and after
randomization
• If patient 207 is excluded, then
…suggests modest evidence that
progabide is favored over the placebo
Results (cont’d)
We have completely ignored correlation within
subjects…