A Poisson regression of hospital readmissions includes age and comorbidity count as predictors. The coefficient for comorbidity count is 0.15. What is the correct interpretation?
AEach additional comorbidity increases readmissions by 0.15 events
BEach additional comorbidity increases the expected readmission count by a factor of exp(0.15) ≈ 1.16, or about 16%
CEach additional comorbidity increases the probability of readmission by 15%
DComorbidity count explains 15% of the variance in readmissions
Poisson regression coefficients are on the log-count (or log-rate) scale. The coefficient of 0.15 means each additional comorbidity increases the log of the expected count by 0.15. Exponentiating gives the rate ratio: exp(0.15) ≈ 1.16, meaning the expected readmission rate increases by about 16% per additional comorbidity. This is a multiplicative effect — the same proportional increase regardless of the baseline count.
Question 2 Multiple Choice
In a Poisson regression of cancer deaths across counties, different counties contribute different population sizes. Why is an offset term for log(population) necessary?
AIt adjusts for the fact that larger counties have more deaths simply because they have more people — the offset converts raw counts to rates
BIt ensures the model produces probabilities between 0 and 1
CIt corrects for overdispersion caused by population heterogeneity
DIt is only needed when population sizes are unknown
A county with 1 million people will have more cancer deaths than a county with 10,000 people regardless of any risk factor — raw counts are not comparable across different exposure times or population sizes. The offset log(population) moves from the right side of the equation, converting the model from log(expected count) = Xβ to log(expected count / population) = Xβ, which is log(rate) = Xβ. The coefficients then represent log rate ratios rather than log count ratios, making comparisons valid.
Question 3 True / False
If the variance of a count outcome substantially exceeds the mean, the Poisson model's standard errors will be too small, but the coefficient estimates themselves remain unbiased.
TTrue
FFalse
Answer: True
Overdispersion (variance > mean) does not bias the Poisson regression point estimates — the coefficients remain consistent. However, the Poisson model assumes variance equals the mean, so it underestimates the true variability when overdispersion is present. This produces standard errors that are too small, confidence intervals that are too narrow, and p-values that are too small — leading to false positives. Solutions include using robust (sandwich) standard errors, quasi-Poisson models, or switching to negative binomial regression which has an additional parameter for overdispersion.
Question 4 Short Answer
Explain why Poisson regression uses a log link rather than modeling counts directly as a linear function of predictors.
Think about your answer, then reveal below.
Model answer: Counts are non-negative integers, but a linear function of predictors can produce negative values, which are meaningless as counts. The log link maps the positive real line to all reals, ensuring that the expected count exp(Xβ) is always positive regardless of predictor values. The log link also provides a natural multiplicative interpretation: effects are proportional changes in the rate rather than additive changes, which matches how most biological exposures affect disease rates.
The log link is the canonical link for the Poisson distribution in the generalized linear model framework. It also connects directly to epidemiological thinking — rate ratios (multiplicative comparisons) are the standard measure of association for incidence data, and the log link produces exactly these quantities as exponentiated coefficients.