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• 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:
Marginal Poisson Regression Model and GEE
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
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
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
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