Count Outcomes
Count Outcomes
A count needs a clock and a population
“Twelve asthma visits” is incomplete. Twelve visits among how many residents and over how long?
For municipality , let be visits and be person-years at risk. A log-link model is
or equivalently enters as an offset with coefficient fixed at one.
Clearborough records 240 visits across 20,000 person-years before the clean-air zone and 180 across 21,000 after:
| Period | Visits | Person-years | Rate per 1,000 |
|---|---|---|---|
| before | 240 | 20,000 | 12.0 |
| after | 180 | 21,000 | 8.57 |
The crude incidence-rate ratio is , a 28.6% rate reduction. This is descriptive until a counterfactual design handles time trends and composition.
Interpret exponentiated coefficients
In a Poisson model, is a multiplicative change in the conditional mean for a one-unit covariate change. If ,
so the expected rate is 22.1% lower, conditional on the model. The absolute reduction still depends on the baseline rate.
Poisson can be useful without a Poisson variance
Poisson pseudo-maximum likelihood targets a correctly specified conditional mean and can use robust or design-aligned standard errors when variance differs from the mean. Negative binomial models add a particular overdispersion structure; choosing one is not merely “variance exceeds mean, therefore negative binomial.”
| Symptom | Possible cause | Response |
|---|---|---|
| variance exceeds mean | heterogeneity, dependence, wrong mean | inspect design; robust/clustered inference; richer mean |
| many zeros | genuinely no risk, separate participation, rare events | distinguish structural process from chance zeros |
| repeated people | within-person dependence | cluster or model dependence at person level |
| different exposure | time/population at risk varies | offset or explicit rate estimand |
Hurdle and zero-inflated models answer different stories
A hurdle model separates whether any event occurs from the positive count. A zero-inflated model posits a latent group that cannot generate events plus a count process that can also produce zeros.
For student support visits:
- “no need for support” may generate structural zeros;
- “needed support but did not attend” is a behavioural zero;
- “one or more visits” has an intensity process.
Use a two-process model only when that story is meaningful and identified by the data. A better likelihood cannot reveal latent zero types without information.
Fixed effects and short panels
With repeated municipality counts, Poisson fixed-effects estimation can target multiplicative within-unit changes under a conditional-mean assumption. Include time effects for common shocks and align clustering with policy assignment. A clean likelihood does not solve staggered-treatment bias; combine the outcome model with an appropriate DID/event-study design.
Quick check
District A has twice as many incidents as B but four times the population at risk. Which district has the higher rate?