Poisson Regression

Poisson Regression for Event Rates and Resource Use

Poisson regression is the basic regression model for counts of health care events per patient or per unit of follow-up, such as hospital admissions, GP consultations or adverse events. It relates the logarithm of the expected count to patient and treatment characteristics, so each exponentiated coefficient is a rate ratio and fitted values can never be negative. In health economics it supplies the event rates and rate ratios that are multiplied by unit costs or turned into transition probabilities in decision models. The model assumes that the variance of a count equals its mean. When that fails, its estimates of the rate stay valid provided the model for the mean is right, but its model-based standard errors do not. It builds on the Poisson distribution, the distribution of a single count, by letting the mean differ between patients.

How a Poisson process becomes a log-linear rate model

The Stata manual sets out the basic assumptions. There is an incidence rate at which events occur, the rate multiplied by exposure gives the expected number of events, more than one event in a very small exposure is unlikely, and non-overlapping exposures are independent. Rodríguez notes that the resulting Poisson count has equal mean and variance, so any factor that changes one changes the other.

A linear model for the mean could predict negative counts. Rodríguez explains that modelling the log of the mean avoids this and gives a generalised linear model with Poisson response and log link, which is the canonical link for the Poisson distribution.

$$y_i \sim \text{Poisson}(\mu_i), \qquad \ln \mu_i = \ln t_i + \beta_0 + \beta_1 x_{1i} + \dots + \beta_k x_{ki}$$

where $y_i$ is the number of events for patient $i$, $\mu_i$ is the expected number, $t_i$ is that patient's follow-up time, $x_{1i}$ to $x_{ki}$ are the covariates and $\beta_0$ to $\beta_k$ are the regression coefficients. Dividing $\mu_i$ by $t_i$ gives the event rate, so $e^{\beta_0}$ is the rate for a patient with every covariate at zero.

Exposure, offsets and grouped trial data

Follow-up differs between patients, so the log of follow-up time enters the linear predictor with its coefficient fixed at one. The Stata manual calls this an exposure variable, and without it every exposure is assumed to equal one.

Rodríguez shows that individual and grouped data give the same likelihood, because a sum of independent Poisson counts is itself Poisson. Fitting the model to each patient's count, or to arm totals with the log of total person-time as the offset, gives exactly the same estimates and standard errors, although the deviances differ.

In evidence synthesis, NICE Decision Support Unit Technical Support Document 2 (TSD 2) uses the same Poisson likelihood and log link for trial arms that report event counts over person-years at risk, assuming a constant hazard in each arm and, for repeated events, independent events. Pooled rate ratios from such a network meta-analysis then feed the economic model.

Estimation by maximum likelihood and checks on fit

The coefficients are found by maximum likelihood estimation. Dropping a constant that does not depend on the coefficients, Rodríguez gives the log-likelihood for $n$ independent counts.

$$\ln L(\beta) = \sum_{i=1}^{n} \left\{ y_i \ln \mu_i - \mu_i \right\}$$

where $\ln L(\beta)$ is the log-likelihood for the coefficient vector $\beta$, $y_i$ is the observed count and $\mu_i$ is the expected count implied by $\beta$. The resulting estimating equations mean that, in a model with a constant, fitted counts add up to the observed total, and for a categorical covariate the fitted total at each level equals the observed total there.

Fit is judged with the deviance and with Pearson's chi-squared statistic, the sum of squared differences between observed and fitted counts divided by the fitted counts. Rodríguez notes that each is approximately chi-squared on $n - p$ degrees of freedom in large samples, where $p$ is the number of parameters. Differences in deviance between nested models give a likelihood ratio test, and single coefficients can be tested with a Wald test.

Reading coefficients as rate ratios and marginal effects

Because of the log link, each coefficient acts on the expected rate by multiplication. Exponentiating it gives an incidence rate ratio, the factor by which the rate changes for a one-unit increase in that covariate with the others held fixed.

$$\mathrm{IRR}_j = e^{\beta_j}, \qquad \mathrm{IRR}_j(\Delta x_j) = e^{\beta_j \Delta x_j}$$

where $\mathrm{IRR}_j$ is the incidence rate ratio for covariate $j$, $\beta_j$ is its coefficient and $\Delta x_j$ is a change in that covariate of any size.

Cameron explains that a coefficient of 0.05 means a one-unit change in the covariate goes with about a 5% change in the expected count. The effect in absolute numbers of events is the expected count times the coefficient, so it varies between patients, and the average marginal effect is the mean of these individual effects. For costing, the number of events avoided depends on the baseline rate as well as the rate ratio.

When the variance exceeds the mean

Rodríguez notes that count data often show overdispersion, a variance larger than the mean, which is expected in health care because patients with the same recorded characteristics still differ. Cameron's summary states that the Poisson estimator stays consistent even when the data are not Poisson distributed, provided the mean is correctly specified. The usual standard errors, however, are wrong unless the variance equals the mean, and with marked overdispersion the uncorrected t statistics can be much too large.

Two corrections keep the Poisson estimates and repair the standard errors. One is a sandwich estimator, with cluster-robust standard errors where observations are grouped, for example repeated periods for the same patient. The other is the overdispersed Poisson model that Rodríguez describes, in which the variance is a constant multiple of the mean.

$$\hat{\phi} = \frac{\chi^2_P}{n - p}, \qquad \mathrm{SE}_{\text{corrected}} = \sqrt{\hat{\phi}}\ \mathrm{SE}_{\text{Poisson}}$$

where $\hat{\phi}$ is the estimated dispersion factor, $\chi^2_P$ is Pearson's chi-squared statistic, $n$ is the number of observations, $p$ is the number of parameters and SE is a standard error. Rodríguez cautions that this treats all lack of fit as pure error, so it needs confidence that the model for the mean is right.

Negative binomial regression models the extra variation directly. Cameron describes its efficiency gain over Poisson as small in practice, but recommends it when the aim is to predict the probability of each count as well as the mean.

Worked example with admissions in a two-arm trial

All numbers in this example are illustrative. A trial of a community nursing service for older adults, with 400 patients per arm, records unplanned admissions. Usual care has 182 admissions over 520 person-years and the service has 135 over 540 person-years. A Poisson model with a treatment indicator and a log follow-up offset is fitted.

1. Estimate the rates. With one binary covariate the fitted totals equal the observed totals, so the model reproduces the crude rates. The usual-care rate is 182 / 520 = 0.35 admissions per person-year and the service rate is 135 / 540 = 0.25.

2. Form the rate ratio. The rate ratio is 0.25 / 0.35 = 0.714, so the treatment coefficient is ln 0.714 = minus 0.336, a 28.6% lower admission rate.

3. Model-based standard error. For two groups the variance of the log rate ratio is the sum of the reciprocals of the event counts. The reciprocals 0.00549 and 0.00741 sum to 0.01290, so the standard error is 0.1136. The 95% interval is minus 0.336 plus or minus 1.96 × 0.1136 = 0.223, or minus 0.559 to minus 0.114, a rate ratio of 0.57 to 0.89.

4. Correct for overdispersion. Suppose the patient-level fit gives a Pearson statistic of 1,430 on 798 degrees of freedom (800 patients less 2 parameters). The dispersion factor is 1,430 / 798 = 1.792, its square root is 1.339, and the corrected standard error is 0.1136 × 1.339 = 0.1521. The interval becomes minus 0.336 plus or minus 1.96 × 0.1521 = 0.298, or minus 0.634 to minus 0.038, a rate ratio of 0.53 to 0.96. The point estimate is unchanged and the interval is about a third wider.

5. Turn rates into costs. At an illustrative GBP 3,200 per admission, expected admission costs are 0.35 × 3,200 = GBP 1,120 per patient-year under usual care and 0.25 × 3,200 = GBP 800 with the service, a saving of GBP 320.

6. Turn rates into cycle probabilities. A Markov cohort model with annual cycles needs a transition probability, the chance of at least one admission in a year. Under a constant rate, TSD 2 gives this as one minus $e^{-\lambda t}$. For usual care $e^{-0.35}$ is 0.705, so the probability is 0.295, and for the service $e^{-0.25}$ is 0.779, so it is 0.221. Both are below the rates because some patients are admitted more than once.

7. Interpret. The service lowers the admission rate by about 29%, with a 95% interval for the rate ratio of 0.53 to 0.96 once overdispersion is allowed for. In probabilistic sensitivity analysis the log rate ratio would be drawn from a normal distribution with mean minus 0.336 and standard error 0.152, since the uncorrected 0.114 would understate decision uncertainty.

Poisson regression in cost-effectiveness and HTA work

Mihaylova and colleagues, reviewing methods for health care resources and costs, describe generalised linear models used for resource use with Poisson and negative binomial specifications. Jones notes that the Poisson model and its extensions are often used in health economics to model the number of visits to a doctor.

The estimator also serves for outcomes that are not counts. Jones explains that Poisson software can fit an exponential mean model for skewed costs by quasi-maximum likelihood, consistent if the mean is correctly specified, with sandwich standard errors. In a comparison by Hill and Miller that he reports, using US medical expenditure survey data, Poisson regression was among the best performing models.

The model links to survival analysis too. Rodríguez shows that treating deaths as Poisson counts with log exposure time as the offset gives the same estimates and standard errors as an exponential distribution for censored survival times. He adds that Holford (1980) and Laird and Olivier (1981) showed a piecewise proportional hazards model to be equivalent to a Poisson regression on one record per patient per interval, so piecewise models with constant hazards can be fitted with count software.

Choosing between Poisson regression and neighbouring models

Each neighbouring model changes a different part of the specification. The choice depends on the outcome and on whether the analysis needs only the mean rate or the whole distribution of counts.

1. Negative binomial regression. Keeps the log link and rate ratios and adds a parameter for variation between patients.
2. Hurdle models. Cameron notes that when zeros are treated as a separate process, the mean is no longer the Poisson mean and the Poisson estimator is inconsistent.
3. Logistic regression for binary outcomes. Logistic regression gives odds ratios. Zou proposed Poisson regression with a robust error variance to estimate the relative risk directly from binary data in prospective studies.
4. Least squares on log counts. Cameron calls this less desirable, since zero counts need ad hoc fixes and the exponential of a predicted log count is not the predicted count.
5. Repeated counts per patient. Correlated counts can be handled by generalised estimating equations or a mixed-effects model.

Common misreadings and limits

Most errors in economic evaluations come from the variance assumption or from mixing rates, counts and probabilities. Four recur.

1. Trusting model-based standard errors. With overdispersed counts they are too small, narrowing intervals.
2. Leaving out follow-up time. Without an offset, patients who leave early look like low users.
3. Reading a large Pearson statistic as dispersion alone. It may reflect a wrong model for the mean, which no correction to standard errors repairs.
4. Using a rate as a cycle probability. A rate of 0.35 per person-year is a one-year probability of 0.295 under a constant rate.

Sources

• Cameron AC, Trivedi PK. Regression Analysis of Count Data. 2nd ed. Cambridge University Press; 2013. Chapter 3, summarised in Cameron AC, Count data regression made simple. https://cameron.econ.ucdavis.edu/racd2/basic_count_regression.pdf
• Dias S, Welton NJ, Sutton AJ, Ades AE. NICE DSU Technical Support Document 2: A generalised linear modelling framework for pairwise and network meta-analysis of randomised controlled trials. Decision Support Unit; last updated September 2016. Sections 3.1 and 3.2. https://www.sheffield.ac.uk/sites/default/files/2022-02/TSD2-General-meta-analysis-corrected-2Sep2016v2.pdf
• Jones AM. Models for health care. HEDG Working Paper 10/01. University of York; 2010. https://www.york.ac.uk/media/economics/documents/herc/wp/10_01.pdf
• Mihaylova B, Briggs A, O'Hagan A, Thompson SG. Review of statistical methods for analysing healthcare resources and costs. Health Economics. 2011;20(8):897-916. https://doi.org/10.1002/hec.1653
• Rodríguez G. Lecture Notes on Generalized Linear Models. Princeton University. Chapter 4, Poisson models for count data (2007), https://grodri.github.io/glms/notes/c4.pdf; addendum, Models for count data with overdispersion (2022), https://grodri.github.io/glms/notes/c4a.pdf; chapter 7, Survival models (2010), https://grodri.github.io/glms/notes/c7.pdf
• StataCorp. poisson, Poisson regression. Stata Base Reference Manual. StataCorp LLC; 2025. https://www.stata.com/manuals/rpoisson.pdf
• Zou G. A modified Poisson regression approach to prospective studies with binary data. American Journal of Epidemiology. 2004;159(7):702-706. https://doi.org/10.1093/aje/kwh090