Rates and rate ratios (Poisson)

Incidence rate ratios (IRR) with Poisson regression and person-time - and when to choose it over Cox

Published

July 2, 2026

Warning

Under development. This page few examples. More detail to come.

A rate is the number of events divided by the time people were at risk (person-time). An incidence rate ratio (IRR) compares the rate in two groups: if the rate is twice as high among the exposed, the IRR is 2. It is one of the most common effect measures (measures of effect size) in Danish register papers, precisely because registers follow people over time and count up events.

In Descriptive tables you computed a crude rate per group. This page goes one step further: a Poisson regression gives an IRR that is adjusted for confounding (age, sex, other covariates), the same way logistic regression gives an adjusted odds ratio.

Note

The code examples use generic path and variable names. Adapt them to your project. The packages (survival, gtsummary, MASS) must be installed in your R environment at Statistics Denmark.

When do you choose Poisson?

Three models can be used on a “did the event happen?” outcome. The difference is whether time enters, and how:

  • Logistic regression (Regression) only looks at yes/no and ignores time. Gives an odds ratio. Fits when everyone is followed equally long, or when time is irrelevant.
  • Cox regression (Time-to-event) looks at the time until the first event per person and gives a hazard ratio. It models the rate without assuming a particular shape over time.
  • Poisson regression (this page) looks at the number of events per person-time and gives a rate ratio (IRR). It is strong when you want to work with rates across age groups or calendar periods, or when you have counted-up events and person-time rather than one time-to-event per person.
Tip

Cox or Poisson? They often answer almost the same question, and HR and IRR are close when the event is rare. Choose Cox if you think in individual follow-up time and time-to-first-event. Choose Poisson if you think in rates per person-year split by age/calendar time (classic in register studies of e.g. cancer or mortality), or if your data are already counted up (events and person-years per group).

Starting point

You need two things per person, exactly as for the crude rate: the number of events (event, here 0 or 1) and the person-time (followup_years, follow-up time in years). Both are built in Phase 12.

library(dplyr)                         # %>% (pipe)
library(gtsummary)                     # tbl_regression() for tidy output

df <- readRDS("path/to/analysis.rds")  # analysis-ready dataset, one row per person

Poisson regression: rate ratio (IRR)

The model is a glm with family = poisson. The new part, compared with logistic regression, is the offset: it tells the model how long each person was at risk, so it models the rate (events per time) and not just the number of events.

model <- glm(event ~ exposure + age + sex +          # outcome (event count) explained by exposure + covariates
               offset(log(followup_years)),          # offset = person-time: turns counts into a RATE
             data = df, family = poisson)             # family = poisson -> rate model

model %>%                                # send the model on to a table
  tbl_regression(exponentiate = TRUE)    # exponentiate = TRUE -> rate ratios (IRR)
  • family = poisson makes it a rate model.
  • offset(log(followup_years)) is the person-time. It must be logged (log()), because the Poisson model works on the log scale. Without the offset you model the number of events, not the rate - and that is almost never what you want.
  • exponentiate = TRUE shows rate ratios (IRR) instead of log-rate coefficients.

An IRR reads exactly like an OR or HR: 1 = no difference, above 1 = higher rate, below 1 = lower. See Reading your result.

Note

Here event is 0/1 (one row per person), and the offset is the person’s total follow-up time. That gives a valid IRR adjusted for the covariates you have at baseline (at study start/index). If you want the rate to change with age or calendar year during follow-up, the person-time must be split - see Rates that vary over time below.

Assumptions

The Poisson model rests on a few assumptions. If they fail, the confidence intervals in particular become misleading:

  • Constant rate within each “band”. The model assumes the rate is the same throughout the person-time a group contributes. If the rate changes strongly with e.g. age, you satisfy the assumption by splitting the person-time into age groups (see below), so the rate only has to be constant within each narrow band.
  • Events are Poisson-distributed (mean = variance). If the variance is larger than the mean (overdispersion), the standard errors become too small and the confidence intervals artificially narrow. This is common in register data - see the box below.
  • Independent observations. One row (or one cluster of person-time) per person. If the same person appears several times (e.g. matched with replacement or recurrent events), use cluster-robust standard errors as in Regression.
  • Log-linearity. Continuous variables are related to the log rate in a straight line. Check as for logistic regression, e.g. with splines.
Check for overdispersion (and switch to negative binomial if needed)

Overdispersion means the events vary more than a pure Poisson distribution allows. The consequence is confidence intervals that are too narrow. Two common fixes:

library(MASS)                            # glm.nb(): negative binomial regression

# Negative binomial: like Poisson, but with an extra parameter that allows overdispersion
glm.nb(event ~ exposure + age + sex + offset(log(followup_years)),
       data = df) %>%
  tbl_regression(exponentiate = TRUE)    # -> IRR with confidence intervals that account for overdispersion

Alternatively you can keep the Poisson model and use family = quasipoisson, which scales the standard errors up without changing the distribution. Both give wider, more honest confidence intervals than a pure Poisson when there is overdispersion.

Rates that vary over time

The model above assumes one rate throughout the person’s follow-up. But in a register study the person ages along the way, and the calendar year changes - and the rate of e.g. cancer or death changes strongly with both. If you want the rate to vary with attained age or calendar period, the follow-up time must be split into intervals: one row per person per interval, so each row sits in one age band and one calendar year. You then count up events and person-time and fit the same Poisson model with age and period bands as covariates.

This is a standard move in register epidemiology, but it needs a bit more machinery than this page covers. Use survival::survSplit() or Epi::splitLexis() (Lexis objects), or the popEpi package, which is built for exactly rates, person-time and standardisation on register data. See Read more.

Note

SMR and SIR. If you want to compare your population’s rate with a reference population (e.g. the rates for all of Denmark from Statistics Denmark), you use indirect standardisation: the standardized mortality/incidence ratio (SMR/SIR) is observed events divided by those you would expect from the reference rates. Epi and popEpi have ready-made functions. It is a descriptive standardisation, not an adjusted regression, so choose according to your question.

Note

Remember: anything leaving Statistics Denmark must pass output control - no small cells, only aggregated results. See Phase 14 - Export and sending home.

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