Lecture 10

• 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

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 DataVar(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…