13 Research methods, statistics, critical review and evidence-based practice13.10 Correlation and regression
13.10.18 Poisson regression
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Written by Resident Doctor Committee · Published
New in the 2027 syllabus
The College added this topic for exams from January 2027.
What it is
Poisson regression models counts: the number of times something happens, such as hospital admissions. Counts are whole numbers and are never negative. The model assumes they follow a Poisson distribution.
It usually uses a log link, so it predicts the log of the expected count. This keeps the fitted mean counts positive. Once you exponentiate a coefficient, its effect is multiplicative: it multiplies the expected count rather than adding to it.
Counts and rates
People are often followed up for different lengths of time. To allow for this, include the log of each person's exposure time as an offset. The model can then compare event rates. With an offset, twice the follow-up implies twice the expected events, as long as the rate and other predictors are unchanged.
Quantity | What it means |
|---|---|
Count outcome | The number of events during observation |
Offset | The known log exposure time, with its coefficient fixed at 1 |
Exponentiated coefficient | A count ratio, or an incidence rate ratio when exposure is included |
An incidence rate ratio describes event rates, not the probability of having at least one event.
Worked example: predicting a rate
Assume a baseline rate of 0.4 events per person-year and an estimated rate ratio of 1.5 for a specified comparison.
Input | Value |
|---|---|
Baseline event rate | 0.4 per person-year |
Rate ratio | 1.5 |
Observation time | 2 years |
Comparison rate = 0.4 x 1.5 = 0.6 events per person-year.
Expected count over two years = 0.6 x 2 = 1.2 events.
The expected count can be a fraction, even though every observed event count is a whole number.
Assumptions and alternatives
Equal mean and variance. A Poisson distribution assumes the mean equals the variance, given the predictors. The unadjusted sample mean and variance need not match exactly.
Overdispersion means the counts vary more, given the predictors, than the model allows. The usual standard errors may then be too small.
Negative binomial models can allow for the extra variation. Other approaches adjust the inference instead, without treating the Poisson variance assumption as exact.
Excess zeros may need a separate explanation. But many zero counts alone do not prove that you need a zero-inflated model.
Repeated events and clustering need attention because of dependence. Repeated admissions by the same person are not automatically independent.