Poisson Distribution

Poisson Distribution for Event Counts in Health Economics

The Poisson distribution gives the probability that a patient or a population has 0, 1, 2 or more events, such as hospital admissions, exacerbations or adverse reactions, over a stated length of follow-up. It needs only the expected count, which is the event rate multiplied by the time at risk, and that one number is both the mean and the variance of the count. Health economists use it to turn event rates into expected resource use and costs, and to write the likelihood for rate data in evidence synthesis. Resource-use counts often vary more between patients than the Poisson allows, so a check for overdispersion belongs with every use of it.

Probability of a given number of events

The distribution is defined by its probability mass function. In health economic work the expected count is usually written as a rate per unit of person-time multiplied by the exposure, because patients are followed for different lengths of time.

$$P(Y = y) = \frac{e^{-\mu} \mu^{y}}{y!}, \qquad \mu = \lambda t, \qquad y = 0, 1, 2, \ldots$$

where $Y$ is the random number of events, $y$ is a particular count, $\mu$ is the expected number of events, $\lambda$ is the event rate per unit of person-time, $t$ is the exposure in the same time units, $y!$ is the factorial of $y$ and $e$ is the base of the natural logarithm.

The NIST/SEMATECH e-Handbook of Statistical Methods lists the main properties. The standard deviation is the square root of the mean and the skewness is one over that square root, so the distribution is strongly right skewed when the expected count is small and close to symmetric when it is large.

$$E(Y) = \mu, \qquad \operatorname{Var}(Y) = \mu$$

where $E(Y)$ is the expected count and $\operatorname{Var}(Y)$ is its variance. Rose and colleagues, in a Cochrane tutorial on count data, state that the rate and variance of a Poisson distribution are equal, and they use this equality to suggest that one trial's counts were overdispersed. In Excel, POISSON.DIST(x, mean, cumulative) returns the probability of exactly x events when the last argument is FALSE and of between zero and x events inclusive when it is TRUE.

Rates, exposure and the Poisson process

Gelman and colleagues, in Bayesian Data Analysis, explain that the Poisson and exponential distributions arise as the number of counts and the waiting times for events that occur independently in time with a constant rate of occurrence. The waiting times are covered on the exponential distribution page and are how a discrete event simulation usually times events that have a fixed rate.

Rose and colleagues define a rate as the average count per unit of exposure, point out that rates, unlike risks, can be greater than 1, and state that count outcomes cannot be analysed unless exposures are measured or imputed. The Poisson formula with $y = 0$ links a rate to a probability. The probability of no events in time $t$ is $e^{-\lambda t}$, so the probability of at least one event is as follows.

$$P(Y \ge 1) = 1 - e^{-\lambda t}$$

where $\lambda$ is the constant event rate and $t$ is the length of the period. Gidwani and Russell give this conversion for a model node with two possible transitions, and the NICE Decision Support Unit's Technical Support Document 2 (TSD 2) derives it by assuming a Poisson process in which the time to an event is exponential. It underlies rate to probability conversion and many a transition probability in a Markov model. It gives the chance of one or more events, not the expected number, which matters when events can repeat.

Estimating an event rate and its uncertainty

With total events $D$ observed over total person-time $T$, the maximum likelihood estimate of the rate is $D/T$, and Gelman and colleagues show that the likelihood depends on the data only through the total count. For a frequentist confidence interval, Ulm described a simple way to obtain the exact Poisson limits through the link between the chi-square and Poisson distributions.

$$\lambda_L = \frac{\chi^2_{0.025}(2D)}{2T}, \qquad \lambda_U = \frac{\chi^2_{0.975}(2D + 2)}{2T}$$

where $\lambda_L$ and $\lambda_U$ are the lower and upper 95% limits for the rate, $D$ is the total number of events, $T$ is the total person-time and $\chi^2_{q}(k)$ is the $q$ quantile of the chi-square distribution with $k$ degrees of freedom. With no events the upper limit is 3.69 events divided by the person-time, so a count of zero still bounds the rate.

A Bayesian analysis treats the rate itself as uncertain. Gelman and colleagues show that the gamma distribution is the conjugate prior for the Poisson rate, so a $\text{Gamma}(\alpha, \beta)$ prior with $D$ events in exposure $T$ gives a $\text{Gamma}(\alpha + D, \beta + T)$ posterior. The ISPOR-SMDM task force on parameter estimation recommends gamma or log normal distributions for right-skewed parameters in probabilistic sensitivity analysis. The Poisson describes chance variation in counts for a known rate, while the gamma describes parameter uncertainty about the rate.

Poisson likelihoods for rate data in evidence synthesis

Trials of chronic disease often report events over person-years with follow-up that differs between trials. TSD 2 states that for counts over a time period a Poisson likelihood and a log link are used, with total person-years at risk in place of a denominator, which allows for censored observations. The model for arm $k$ of trial $i$ is written below.

$$r_{ik} \sim \text{Poisson}(\lambda_{ik} E_{ik}), \qquad \log(\lambda_{ik}) = \mu_i + \delta_{i,bk} I_{\{k \ne 1\}}$$

where $r_{ik}$ is the number of events in the arm, $E_{ik}$ is its exposure in person-years, $\lambda_{ik}$ is its event rate, $\mu_i$ is the log rate in the trial's reference arm, $\delta_{i,bk}$ is the log rate ratio of the treatment in arm $k$ against the reference treatment $b$, and $I_{\{k \ne 1\}}$ is 1 for arms other than the first and 0 otherwise. This is the rate model for pairwise and network meta-analysis, and its exponentiated treatment effects are the incidence rate ratio. Patient-level covariates on the same log link give Poisson regression.

TSD 2 is explicit about the assumptions. The hazard in each arm must be constant over follow-up, which holds only in homogeneous populations. With constant but varied rates, the average hazard must decrease over time as high-risk patients reach the end point first. For repeated events the arm total can be modelled, with the added assumption that a person who has had one event is no more likely to have another.

Overdispersion and the negative binomial alternative

Counts of health care use often have a variance larger than the mean. Rose and colleagues note that one event may change the rate of later events in the same participant, which leads to overdispersion, counts that vary more than the Poisson allows. Keene and colleagues reanalysed exacerbation data from the TRISTAN and ISOLDE trials in chronic obstructive pulmonary disease. Poisson regression did not satisfactorily account for variability between patients, while a negative binomial model, which gives each patient a separate Poisson parameter, gave higher estimated exacerbation rates and wider confidence intervals for treatment comparisons, with the trials' conclusions unchanged.

Gelman and colleagues derive the negative binomial as a mixture of Poisson distributions whose rates follow a gamma distribution, and the regression form is covered under negative binomial regression. Two quick checks compare the variance of patient counts with their mean, and the observed share of patients with no events with $e^{-\mu}$.

Worked example with hospital admissions in a heart failure cohort

All numbers in this example are illustrative. A cohort of 150 patients with heart failure is followed for 0.8 years each, a total of 120 person-years, and has 42 emergency admissions. A cost-effectiveness model with an annual cycle needs the admission rate, its uncertainty and the expected cost.

1. Estimate the rate. The rate is 42/120 = 0.35 admissions per person-year.

2. Exact 95% interval. The chi-square quantiles are $\chi^2_{0.025}(84) = 60.54$ and $\chi^2_{0.975}(86) = 113.54$. Halving them gives 30.27 and 56.77 admissions, and dividing by 120 person-years gives a rate of 0.252 to 0.473. A normal interval on the log rate, with $SE = 1/\sqrt{42} = 0.154$, gives 0.259 to 0.474, close to the exact one because 42 events is not a small count.

3. Admissions per patient-year. With $\mu = 0.35$, $P(0) = e^{-0.35} = 0.705$, $P(1) = 0.35 \times 0.705 = 0.247$ and $P(2) = (0.35^2/2) \times 0.705 = 0.043$, leaving 0.006 for three or more.

4. Expected cost. At an illustrative GBP 3,000 per admission, the expected cost is 0.35 × 3,000 = GBP 1,050 per patient-year. A model that applies the probability of at least one admission, 1 minus 0.7047 or 0.2953, and costs a single admission gives 0.2953 × 3,000 = GBP 886, which is GBP 164 or about 16% too low because repeat admissions are lost.

5. Distribution for probabilistic sensitivity analysis. With a vague gamma prior whose parameters are close to zero, the posterior for the rate is close to Gamma(42, 120), with mean 42/120 = 0.35 and standard deviation $\sqrt{42}/120 = 0.054$. Its 2.5th and 97.5th percentiles are 0.252 and 0.464. The lower point equals the exact lower limit, and the exact upper limit of 0.473 is the 97.5th percentile of Gamma(43, 120).

6. Check the dispersion. Each patient has a mean of 42/150 = 0.28 admissions, and the Poisson predicts $e^{-0.28} = 0.756$ of patients, or 113, with none. Suppose 126 had none and the variance of the patient counts is 0.62, a variance-to-mean ratio of 0.62/0.28 = 2.21. A negative binomial with the same mean and variance has shape $r = 0.28^2/(0.62 - 0.28) = 0.231$ and predicts $(0.231/0.511)^{0.231} = 0.833$ of patients, or 125, with none.

The rate and the expected cost of GBP 1,050 per patient-year are the same under both models. The extra variance widens the interval for the rate, and that wider uncertainty is what should reach the probabilistic analysis.

Where the Poisson sits among count models

The binomial distribution counts patients with at least one event among a fixed number followed, so no count can exceed the number of patients, while the Poisson counts events over person-time and allows any number per patient. When many patients each have a small probability of the event, a binomial with mean $np$ is close to a Poisson with the same mean. Rose and colleagues note that large counts are approximately normal but that the approximation is poor for small counts, which is where exact Poisson methods matter. Counts are compared with other outcome types under discrete data.

Limitations and common misreadings

Many errors with the Poisson distribution come from forgetting either the time at risk or the variation between patients. Each one shifts the estimated rate or understates the uncertainty around it.

1. Treating a rate as a probability. A rate of 1.2 admissions per person-year cannot be a probability and needs converting before it enters a Markov model.
2. Dividing events by patients when follow-up differs. Arms followed for longer then appear to have more events per patient.
3. Fitting a Poisson model to overdispersed counts. Confidence intervals come out too narrow, as the Keene reanalysis shows for exacerbations.
4. Assuming one rate in a mixed population. Varied patient rates make the average hazard fall over time, which a single constant rate cannot reproduce.
5. Using the Poisson to describe parameter uncertainty. Uncertainty about the rate belongs in a gamma distribution, and the Poisson describes chance variation in counts once the rate is known.
6. Costing one event per cycle. The probability of at least one event understates use when events repeat, as step 4 shows.

Sources

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