Poisson models for count data often arise when units are observed for different lengths of time. A hospital admission count for a patient followed for 30 days is not directly comparable to a count for a patient followed for 365 days. The natural quantity of interest is the rate — admissions per unit time — not the raw count.
There are three related but distinct specifications, each making a different assumption:
Equivalently, \(\log E[Y] = \beta_0 + \beta_1 X + \log t\), where \(\log t\) is included as a fixed offset (coefficient constrained to 1). This is the standard exposure model: the expected count is proportional to exposure time.
This looks similar, but \(E[Y/t] \ne E[Y]/t\) in a nonlinear (Poisson) model by Jensen’s inequality. In practice the two can give similar point estimates, but the standard errors differ because the variance structure changes when you divide by \(t\).
Here \(\gamma\) is estimated from the data. If \(\hat\gamma \approx 1\), the proportionality assumption is supported. If \(\hat\gamma < 1\), counts grow sub-linearly with exposure — a testable, substantive claim.
25.2 Simulation
library(tidyverse)set.seed(42)n <-800# Simulate a health study: count of adverse events# Covariate: age group (0 = younger, 1 = older)age_old <-rbinom(n, 1, 0.45)# Exposure: days under observation (varies 7–365)exposure <-sample(seq(7, 365, by =7), n, replace =TRUE)# True model: rate = exp(-3 + 1.2 * age_old)# Count ~ Poisson(rate * exposure)true_rate <-exp(-3+1.2* age_old)count <-rpois(n, lambda = true_rate * exposure)df <-data.frame(count, age_old, exposure, log_exposure =log(exposure))summary(df[, c("count", "exposure")])
count exposure
Min. : 0.00 Min. : 7.0
1st Qu.: 7.00 1st Qu.:112.0
Median :13.00 Median :196.0
Mean :18.95 Mean :192.9
3rd Qu.:27.00 3rd Qu.:281.8
Max. :71.00 Max. :364.0
The data has a wide range of exposure times, so ignoring them would badly confound the comparison between age groups.
25.3 Fitting the three models
# Model 1: offset — coefficient on log(exposure) constrained to 1m1 <-glm(count ~ age_old +offset(log(exposure)),family = poisson, data = df)# Model 2: Poisson on the rate Y/t (different response)df$rate <- df$count / df$exposurem2 <-glm(rate ~ age_old,family = poisson, data = df)# Model 3: log(exposure) as a free covariatem3 <-glm(count ~ age_old + log_exposure,family = poisson, data = df)# Collect coefficientsresults <-data.frame(Model =c("1. Offset (constrained γ=1)","2. Rate Y/t as response","3. Unconstrained log(t)"),Intercept =c(coef(m1)[1], coef(m2)[1], coef(m3)[1]),age_old =c(coef(m1)[2], coef(m2)[2], coef(m3)[2]),gamma_log_t =c(1, NA, coef(m3)[3]))knitr::kable(results, digits =3,caption ="Coefficient estimates: true intercept = −3, true age effect = 1.2, true γ = 1")
Reading the table: Model 1 and Model 3 give nearly identical age-group coefficients because the true \(\gamma = 1\) (counts are indeed proportional to exposure time). Model 2 differs slightly in the intercept because the response variable changes. The estimated \(\hat\gamma\) from Model 3 should be close to 1.
25.4 When \(\gamma \ne 1\)
To see why the unconstrained model matters, repeat the simulation with a sub-linear count process (\(\gamma = 0.6\): an initial burst of events that doesn’t grow proportionally with long follow-up):
count2 <-rpois(n, lambda = true_rate * exposure^0.6)df2 <-data.frame(count2, age_old, exposure, log_exposure =log(exposure))m1b <-glm(count2 ~ age_old +offset(log(exposure)),family = poisson, data = df2)m3b <-glm(count2 ~ age_old + log_exposure,family = poisson, data = df2)cat("Offset model age coefficient: ", round(coef(m1b)[2], 3), "\n")
Offset model age coefficient: 1.171
cat("Unconstrained model age coefficient:", round(coef(m3b)[2], 3), "\n")
When \(\gamma \ne 1\), forcing the offset constraint biases the treatment coefficient. The unconstrained model recovers both the true age effect and the true \(\gamma\).
25.5 Stata equivalent
* Model 1: offsetpoissoncount age_old, exposure(exposure)* Model 2: rate as responsegen rate = count / exposurepoisson rate age_old* Model 3: unconstrained log(t)gen log_exposure = log(exposure)poissoncount age_old log_exposure
25.6 Which model to use?
Situation
Recommended model
Counts grow proportionally with time
Model 1 (offset)
Uncertain about proportionality
Model 3 (unconstrained), test \(\hat\gamma = 1\)
Rate is the natural estimand
Model 1 or Model 2; both are valid
Long follow-up with declining event rate
Model 3 or negative binomial with offset
In most health and social-science applications, Model 1 (Poisson with offset) is the default. Model 3 is the diagnostic check: fit it, look at \(\hat\gamma\), and return to Model 1 if the constraint is supported by the data.
Source Code
---title: "Poisson model with exposure"date: "2024-04-20"---## Poisson with exposure or Poisson ratePoisson models for count data often arise when units are observed for differentlengths of time. A hospital admission count for a patient followed for 30 daysis not directly comparable to a count for a patient followed for 365 days. Thenatural quantity of interest is the *rate* — admissions per unit time — notthe raw count.There are three related but distinct specifications, each making a differentassumption:**Model 1 — Poisson with offset (rate model):**$$\log\!\bigl(E[Y]/t\bigr) = \beta_0 + \beta_1 X$$Equivalently, $\log E[Y] = \beta_0 + \beta_1 X + \log t$, where $\log t$ isincluded as a *fixed* offset (coefficient constrained to 1). This is thestandard exposure model: the expected count is proportional to exposure time.**Model 2 — Poisson on the ratio $Y/t$:**$$\log\!\bigl(E[Y/t]\bigr) = \beta_0 + \beta_1 X$$This looks similar, but $E[Y/t] \ne E[Y]/t$ in a nonlinear (Poisson) modelby Jensen's inequality. In practice the two can give similar point estimates,but the standard errors differ because the variance structure changes when youdivide by $t$.**Model 3 — Unconstrained log-exposure:**$$\log E[Y] = \beta_0 + \beta_1 X + \gamma \log t$$Here $\gamma$ is estimated from the data. If $\hat\gamma \approx 1$, theproportionality assumption is supported. If $\hat\gamma < 1$, counts growsub-linearly with exposure — a testable, substantive claim.## Simulation```{r}#| label: sim-data#| cache: true#| warning: false#| message: falselibrary(tidyverse)set.seed(42)n <-800# Simulate a health study: count of adverse events# Covariate: age group (0 = younger, 1 = older)age_old <-rbinom(n, 1, 0.45)# Exposure: days under observation (varies 7–365)exposure <-sample(seq(7, 365, by =7), n, replace =TRUE)# True model: rate = exp(-3 + 1.2 * age_old)# Count ~ Poisson(rate * exposure)true_rate <-exp(-3+1.2* age_old)count <-rpois(n, lambda = true_rate * exposure)df <-data.frame(count, age_old, exposure, log_exposure =log(exposure))summary(df[, c("count", "exposure")])```The data has a wide range of exposure times, so ignoring them would badlyconfound the comparison between age groups.## Fitting the three models```{r}#| label: fit-models#| cache: true#| warning: false# Model 1: offset — coefficient on log(exposure) constrained to 1m1 <-glm(count ~ age_old +offset(log(exposure)),family = poisson, data = df)# Model 2: Poisson on the rate Y/t (different response)df$rate <- df$count / df$exposurem2 <-glm(rate ~ age_old,family = poisson, data = df)# Model 3: log(exposure) as a free covariatem3 <-glm(count ~ age_old + log_exposure,family = poisson, data = df)# Collect coefficientsresults <-data.frame(Model =c("1. Offset (constrained γ=1)","2. Rate Y/t as response","3. Unconstrained log(t)"),Intercept =c(coef(m1)[1], coef(m2)[1], coef(m3)[1]),age_old =c(coef(m1)[2], coef(m2)[2], coef(m3)[2]),gamma_log_t =c(1, NA, coef(m3)[3]))knitr::kable(results, digits =3,caption ="Coefficient estimates: true intercept = −3, true age effect = 1.2, true γ = 1")```**Reading the table:** Model 1 and Model 3 give nearly identical age-groupcoefficients because the true $\gamma = 1$ (counts are indeed proportional toexposure time). Model 2 differs slightly in the intercept because the responsevariable changes. The estimated $\hat\gamma$ from Model 3 should be close to 1.## When $\gamma \ne 1$To see why the unconstrained model matters, repeat the simulation with asub-linear count process ($\gamma = 0.6$: an initial burst of events thatdoesn't grow proportionally with long follow-up):```{r}#| label: nonlinear-exposure#| cache: true#| warning: falsecount2 <-rpois(n, lambda = true_rate * exposure^0.6)df2 <-data.frame(count2, age_old, exposure, log_exposure =log(exposure))m1b <-glm(count2 ~ age_old +offset(log(exposure)),family = poisson, data = df2)m3b <-glm(count2 ~ age_old + log_exposure,family = poisson, data = df2)cat("Offset model age coefficient: ", round(coef(m1b)[2], 3), "\n")cat("Unconstrained model age coefficient:", round(coef(m3b)[2], 3), "\n")cat("Estimated gamma (log-exposure): ", round(coef(m3b)[3], 3)," (true = 0.6)\n")```When $\gamma \ne 1$, forcing the offset constraint biases the treatmentcoefficient. The unconstrained model recovers both the true age effect andthe true $\gamma$.## Stata equivalent```stata* Model 1: offsetpoisson count age_old, exposure(exposure)* Model 2: rate as responsegen rate = count / exposurepoisson rate age_old* Model 3: unconstrained log(t)gen log_exposure = log(exposure)poisson count age_old log_exposure```## Which model to use?| Situation | Recommended model ||---|---|| Counts grow proportionally with time | Model 1 (offset) || Uncertain about proportionality | Model 3 (unconstrained), test $\hat\gamma = 1$ || Rate is the natural estimand | Model 1 or Model 2; both are valid || Long follow-up with declining event rate | Model 3 or negative binomial with offset |In most health and social-science applications, Model 1 (Poisson with offset)is the default. Model 3 is the diagnostic check: fit it, look at $\hat\gamma$,and return to Model 1 if the constraint is supported by the data.