The Poisson model is used for count data, that is; where each data point \(y_i\) can equal 0,1,2,….N. Poisson models are commonly used for assessing disease and/or mortality counts particularly for rare diseases.
Model Setup
We assume that \(y_i\) follows a Poisson distribution: \[
y_i \sim Poisson(\theta_i) = \frac{e^{-\theta_i}\theta_i^{y_i}}{y_i !}
\]
where \(\theta_i>0\) is the expected count for observation \(i\).
An important characteristic of the Poisson distribution is that the mean and variance are equal:
\[
E(y_i ) = Var(y_i) = \theta_i
\] - This is notable because it is a characteristic that determines whether Poisson is an appropriate model assumption.
As with linear and logistic regression, the variation in \(y\) can be explained with linear predictors \(X\).
\[
\theta_i = exp(X_i \beta)
\]
Here the link is the log() link. log() has domain \((0, \infty)\) and range \((-\infty, \infty)\) and is strictly increasing.
which is interpreted as the incidence rate ratio (IRR) — the multiplicative change in the expected count for a one-unit increase in \(( X_{ij} )\), holding other variables constant.
Fitting a Poisson Regression in R
We are going to use the preterm birth data from Georgia to illustrate our approach. For this analysis, the data have been aggregated by 5-year maternal age groups and stratified by infant sex and maternal tobacco use. This aggregation allows us to model counts of preterm births (events) relative to the number of total live births (exposure) using a Poisson regression.
library(lme4)
Loading required package: Matrix
library(lmerTest)
Attaching package: 'lmerTest'
The following object is masked from 'package:lme4':
lmer
The following object is masked from 'package:stats':
step
library(dplyr)
Attaching package: 'dplyr'
The following objects are masked from 'package:stats':
filter, lag
The following objects are masked from 'package:base':
intersect, setdiff, setequal, union
`summarise()` has grouped output by 'age_group', 'male'. You can override using
the `.groups` argument.
head(agg_data)
# A tibble: 6 × 5
# Groups: age_group, male [3]
age_group male tobacco events total
<fct> <fct> <int> <int> <int>
1 16-19 F 0 315 3184
2 16-19 F 1 31 245
3 16-19 M 0 325 3297
4 16-19 M 1 26 276
5 20-24 F 0 696 8257
6 20-24 F 1 61 663
Here we can see the number of preterm births (events) for each age-group–sex–tobacco combination, as well as the total number of live births (total) in that subgroup.
To fit a Poisson regression, we model the counts of events with a log link function, using the total number of births as an offset:
Call:
glm(formula = events ~ age_group + male + tobacco, family = poisson,
data = agg_data, offset = log(total))
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -2.37223 0.04040 -58.713 < 2e-16 ***
age_group20-24 -0.11935 0.04533 -2.633 0.00847 **
age_group25-29 -0.24692 0.04539 -5.441 5.31e-08 ***
age_group30-34 -0.21900 0.04570 -4.792 1.65e-06 ***
age_group35-39 -0.04205 0.05075 -0.829 0.40736
age_group40-44 0.12036 0.08087 1.488 0.13668
maleM 0.06634 0.02474 2.682 0.00733 **
tobacco 0.34917 0.05029 6.943 3.83e-12 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 148.530 on 23 degrees of freedom
Residual deviance: 29.892 on 16 degrees of freedom
AIC: 202.82
Number of Fisher Scoring iterations: 4
Model Interpretation:
The intercept (\(\hat{\beta}_0 = -2.37\)) represents the log rate of preterm birth for the reference group: mothers aged 16–19, with female infants and no tobacco use. Converting to a rate: \(e^{-2.37} \approx 0.093\), meaning about 9 preterm births per 100 live births in this reference group.
Age effects: Compared to mothers aged 16–19: Ages 20–24 have a 11% lower rate of preterm birth: \(e^{-0.119} \approx 0.89\) (p = 0.008). Ages 25–29 have a 22% lower rate: \(e^{-0.247} \approx 0.78\) (p < 0.001). Ages 30–34 have a 20% lower rate: \(e^{-0.219} \approx 0.80\) (p < 0.001). No significant difference is observed for ages 35–39 or 40–44 (p > 0.05).
Infant sex: Male infants have a 6.9% higher preterm birth rate compared to female infants: \(e^{0.066} \approx 1.07\) (p = 0.007).
Tobacco use: Mothers who smoke have a 42% higher preterm birth rate compared to non-smokers: \(e^{0.349} \approx 1.42\) (p < 0.001).
Goodness-of-fit: The residual deviance (29.89 on 16 df) suggests an adequate fit, and the model explains a substantial portion of variation compared to the null model (null deviance = 148.53).
What is an Offset?
In the code above you saw a parameter in the glm \(\text{offset = log(total)}\). This is called an offset in Poisson regression.
When modeling rates rather than raw counts, it is critical to account for the population-at-risk or the exposure time for each observation. In the context of preterm birth data, the total number of live births within each age–sex–tobacco subgroup serves as the population-at-risk. Without this adjustment, groups with larger populations would naturally have higher counts simply due to their size, not because they have a higher risk of preterm births.
Why Use an Offset?
The Poisson regression models the expected count of events \(y_i\) as: \[
y_i \sim \text{Poisson}(\theta_i)
\] where \(\theta_i\) is the expected count for the \(i\)th subgroup.
To model rates, we can express: \[
\text{Rate}_i = \frac{\theta_i}{N_i}
\] where \(N_i\) is the population size (e.g., total births in the subgroup).
Taking logs: \[
\log(\theta_i) = \log(N_i) + \log(\text{Rate}_i).
\] Here, \(\log(N_i)\) is treated as a known constant rather than a parameter to estimate.
This motivates the use of an offset term: \[
\log(\theta_i) = \log(N_i) + X_i \beta,
\] which can be written in R as:
Differences between the Binomial and Poisson models
The Poisson model is similar to the binomial model for count data but is applied in slightly different situations.
If each data point \(y_i\) can be interpreted as the number of “successes” out of \(n_i\) trials then it is standard to use the binomial/logistic model.
If each data point \(y_i\) does not have a natural limit- it is not based on a number of independent trials- then it is standard to use the Poisson/log repression model.
Summary
Section
Description
What is the model?
Poisson Regression: Models count data (\(y_i = 0,1,2,...\)) assuming \(y_i \sim \text{Poisson}(\theta_i)\), where \(E(y_i) = \theta_i\) and \(Var(y_i) = \theta_i\). The mean count is linked to predictors by \(\log(\theta_i) = X_i \beta\).
Link Function
Uses the log link, \(g(\theta_i) = \log(\theta_i)\), mapping \((0, \infty) \rightarrow (-\infty, \infty)\). Coefficients represent changes in the log of the expected count.
Interpretation
\(e^{\beta_j}\) is the incidence rate ratio (IRR): the multiplicative change in the expected count for a one-unit increase in \(X_j\), holding other variables constant.
Offsets
Adjusts for population-at-risk or exposure time. The model becomes \(\log(\theta_i) = \log(N_i) + X_i \beta\), where \(\log(N_i)\) is an offset term (not estimated).
Model Assumptions
- Counts are independent. - Mean equals variance (no overdispersion). - Correct link function (log).
Key Differences
- Use Binomial/Logistic regression when counts are bounded by a number of trials (\(n_i\)). - Use Poisson regression when counts have no natural upper limit and represent rates or events over a fixed exposure.
Example Findings
Tobacco use increases preterm birth rate by 42% (\(IRR \approx 1.42\)), while mothers aged 25–29 have a 22% lower rate (\(IRR \approx 0.78\)) compared to ages 16–19.
