Poisson Regression Analysis: Formula and Example

Q: What is Poisson regression analysis used for?

Modelling count outcomes, such as number of events, visits or absences, as a function of predictors, usually reporting incidence rate ratios.

Q: How do I interpret a Poisson regression coefficient?

A coefficient is the change in the log expected count per one-unit increase in the predictor. Exponentiate it for the IRR; the percent change is (IRR − 1) × 100.

Q: What if my data are overdispersed?

If Pearson χ²/df is well above 1, use quasi-Poisson or negative binomial regression instead; see the comparison linked above.

Q: Is Poisson regression the same as logistic regression?

No. Logistic regression models a binary outcome with a logit link and yields odds ratios; Poisson regression models counts with a log link and yields rate ratios.

Poisson regression analysis models a count outcome (0, 1, 2, …) as a function of predictors by assuming the count follows a Poisson distribution whose mean depends on those predictors through a log link. Researchers use it for events such as clinic visits, infections, publications or days absent. After reading this page you should be able to fit the model, convert coefficients to incidence rate ratios, check overdispersion and report the result.

Definition in one paragraph

Poisson regression is a generalized linear model for count data. In the UCLA OARC description, it models “the log of the expected count as a function of the predictor variables” (UCLA OARC, Poisson Regression | Stata Annotated Output, read 2026-11-07). It is the default choice when the outcome is a non-negative integer count, and it belongs to the same family as generalized linear mixed models, of which it is the fixed-effects special case with a Poisson response. For the wider regression toolkit see regression analysis.

Poisson regression formula and symbols

For observation i with predictors x_i1, …, x_ik:

Y_i ~ Poisson(μ_i), ln(μ_i) = β₀ + β₁x_i1 + … + β_kx_ik

Y_i: the observed count for observation i.
μ_i: the expected count, E(Y_i). In a Poisson distribution the variance also equals μ_i.
ln: the natural logarithm, the log link. It is the default link for the poisson family in R (R documentation for family, read 2026-11-07), and it keeps predicted counts positive.
β₀: the intercept, the log expected count when all predictors are 0.
β_j: the change in the log expected count for a one-unit increase in x_j, other predictors held fixed.

Equivalently, μ_i = exp(β₀ + β₁x_i1 + …). Exponentiating a coefficient gives the incidence rate ratio (IRR) (see incidence rate ratio), exp(β_j). Parameters are estimated by maximum likelihood.

Poisson regression with exposure (offset)

When observations are followed for different lengths of time or cover different population sizes, model a rate instead of a raw count by adding ln(t_i) to the right-hand side with its coefficient fixed at 1: ln(μ_i) = ln(t_i) + β₀ + β₁x_i1 + …. This term is called an offset or exposure. UCLA OARC notes that the plain model assumes the same follow-up time for everyone and that the estimates are biased otherwise, in which case the exposure (offset) option should carry the follow-up length.

Worked example with the arithmetic shown

Illustrative data constructed for this page, not from a real study. Suppose eight patients are observed for the same period, and the outcome is the number of clinic visits. Four patients are in group 0 (reference) and four in group 1 (x = 1).

Group	Counts	Sum	Mean
0 (x = 0)	2, 3, 1, 4	10	2.5
1 (x = 1)	5, 6, 4, 7	22	5.5

With a single binary predictor, the maximum-likelihood fitted mean for each group equals its sample mean, so the coefficients can be done by hand:

β₀ = ln(2.5) = 0.9163 (log expected count in group 0).
β₁ = ln(5.5) − ln(2.5) = ln(2.2) = 0.7885.
IRR = exp(0.7885) = 2.2, which is 5.5 / 2.5.
Standard error of β₁ = √(1/10 + 1/22) = √0.14545 = 0.3814 (the reciprocal of each group’s total count).
Wald z = 0.7885 / 0.3814 = 2.07, two-sided p = 0.039.
95% CI for β₁: 0.7885 ± 1.96 × 0.3814 = 0.041 to 1.536, so the CI for the IRR is exp(0.041) to exp(1.536) = 1.04 to 4.65.

The same fit in statsmodels (GLM, Poisson family) returns β₀ = 0.9163, β₁ = 0.7885, SE = 0.3814 and a likelihood-ratio χ²(1) = 4.61 against the intercept-only model (p = 0.032). The Wald and likelihood-ratio tests are close but not identical, which is normal in small samples.

Poisson regression interpretation

Read each exponentiated coefficient as a multiplicative change in the expected count (or rate). The percent change is (IRR − 1) × 100.

Quantity	Reading	Example
β_j > 0, IRR > 1	Expected count rises as x_j rises	IRR 2.2 above: (2.2 − 1) × 100 = 120% higher expected count in group 1
β_j = 0, IRR = 1	No association with the expected count	A 95% CI that contains 1 is consistent with no effect
β_j < 0, IRR < 1	Expected count falls as x_j rises	IRR 0.9965 per unit: about 0.35% lower per one-unit increase
Continuous predictor	IRR applies per one-unit increase and compounds over larger changes	IRR^k for a k-unit increase

UCLA OARC’s annotated example (316 students, outcome days absent) reports an IRR of 1.4932 for female students and describes the rate as “1.493 times greater”; it also reports a McFadden pseudo R² of 0.0536 and advises caution when interpreting pseudo R². The odds-ratio analogue for binary outcomes is covered under odds ratio; an IRR compares rates, not odds.

Assumptions, overdispersion and common mistakes

Count outcome: non-negative integers. Do not use it for continuous or percentage outcomes.
Mean equals variance (equidispersion): the Poisson family fixes the dispersion at 1. UCLA OARC states the model assumes the outcome is not over-dispersed and does not have an excessive number of zeros, and names negative binomial and zero-inflated models as alternatives.
Independent observations: repeated measures or clustered counts need random effects or robust errors (see GLMM).
Log-linear form: predictors act multiplicatively on the expected count.
Same exposure, or an offset: forgetting the offset when follow-up differs is a frequent mistake.

Checking overdispersion

Compute the Pearson χ² statistic divided by the residual degrees of freedom. Values near 1 are consistent with the Poisson assumption; values well above 1 indicate overdispersion and standard errors that are too small. In the worked example, Pearson χ² = 2.91 on 6 residual degrees of freedom gives 0.48, so there is no sign of overdispersion (if anything, slight underdispersion in this tiny sample). The R documentation notes that quasipoisson does not fix the dispersion at one and so can model overdispersion. For the negative binomial option see negative binomial regression for overdispersion, and for software syntax Poisson and negative binomial regression in Stata. Choosing between quasi-Poisson and negative binomial is the subject of quasi-Poisson vs negative binomial regression.

How to report Poisson regression results (APA example)

Report the coefficient or IRR, its confidence interval, the test statistic and p, the overdispersion check and the model fit. Pattern, using the worked example:

A Poisson regression showed that group 1 had a higher expected number of clinic visits than group 0, IRR = 2.20, 95% CI [1.04, 4.65], Wald z = 2.07, p = .039. The Pearson χ²/df ratio was 0.48, indicating no evidence of overdispersion.

State the software, the link function and whether an offset was used.

Poisson regression in R, Python and Excel

R

y <- c(2,3,1,4,5,6,4,7)
x <- c(0,0,0,0,1,1,1,1)
fit <- glm(y ~ x, family = poisson(link = "log"))
summary(fit)
exp(coef(fit))                       # IRRs
exp(confint.default(fit))            # Wald CIs
sum(residuals(fit, "pearson")^2) / fit$df.residual   # dispersion
# with exposure: glm(y ~ x + offset(log(t)), family = poisson)

Python (statsmodels)

import numpy as np, statsmodels.api as sm
y = np.array([2,3,1,4,5,6,4,7]); x = np.array([0]*4 + [1]*4)
fit = sm.GLM(y, sm.add_constant(x), family=sm.families.Poisson()).fit()
print(fit.summary())
print(np.exp(fit.params))               # IRRs
print(fit.pearson_chi2 / fit.df_resid)  # dispersion
# with exposure: sm.GLM(y, X, family=..., offset=np.log(t))

Excel

For a single binary predictor you can reproduce the worked example with worksheet functions: =LN(SUM(group1)/SUM(group0)) gives β₁, =EXP(that) gives the IRR, and =SQRT(1/SUM(group0)+1/SUM(group1)) gives its standard error. Models with several predictors need R, Python or another statistics package.

Frequently asked questions

What is Poisson regression analysis used for?