Diagnosing the Fit of a Poisson Count Model

Why Diagnose a Poisson Model?

A Poisson model can be a useful description of counts, but its assumptions matter. A model might give reasonable probabilities for a typical count while missing important features of the data, such as clusters of unusually high counts or many more zeros than expected. Checking the fit means looking for signs that the model’s assumptions do not match how the counts were generated.

For example, suppose a clinic records the number of calls received during many comparable observation periods. A Poisson model may be a reasonable starting point if each period has the same length, the underlying call rate is stable, and the events occur independently. If the rate varies over time or calls arrive in bursts, the model may not describe the observed pattern well. Comparing the sample mean and variance is one useful first check, but it is not enough on its own.

The Model and Its Assumptions

The assumptions are substantive, not just mathematical conveniences. Counts should refer to the same kind of event, measured over comparable opportunities or exposure. Their underlying rate should be reasonably stable across observations, and the count in one observation should not systematically predict the count in another. Events that occur in clusters, or a rate that changes between observations, can undermine these assumptions.

If exposures differ, raw counts need not have the same expected value: a longer observation period can produce a larger count even when the event rate is unchanged. The simplest mean–variance comparison is therefore clearest for comparable observations with equal exposure. If exposure or conditions differ, those differences must be accounted for before treating all counts as draws from one common Poisson distribution.

Comparing the Sample Mean and Variance

For \(n\) observed counts \(x_1,\ldots,x_n\), the sample mean is \(\bar{x}\), and the sample variance is

The Poisson model’s mean–variance property suggests a diagnostic: for a sufficiently informative collection of comparable observations, \(\bar{x}\) and \(s^2\) should be in the same general range. A variance much larger than the mean is called overdispersion relative to the Poisson model; a variance much smaller than the mean is called underdispersion. Neither comparison is an exact pass-or-fail rule. Sample statistics fluctuate, especially when \(n\) is small, and a close match does not establish that the full distribution is Poisson.

The reason changing rates can create overdispersion can be shown with a short variance decomposition. Suppose the rate differs across observations, represented by a random quantity \(\Lambda\), and that, conditional on \(\Lambda\), the count is Poisson with mean \(\Lambda\). Then the law of total variance gives

Here the first term is the average Poisson variance, while the second is the extra variation due to differing rates. If the rates vary, \(\operatorname{Var}(\Lambda)>0\), so the overall variance exceeds the overall mean \(E(\Lambda)\). This provides one explanation for overdispersion; clustering or dependence can also produce it.

Look for Structure, Not Just a Summary

Counts can have a plausible mean and variance and still depart from a Poisson distribution. Useful diagnostic questions include: Are unusually high counts close together in time or grouped among particular units? Do counts change systematically with the observation period or setting? Are zeros more common than the model predicts? Do the observations appear dependent? These checks are guided by how the data were collected as well as by the numerical summaries.

Check Zeros and the Shape of the Counts

For a Poisson distribution with mean \(\lambda\), the probability of a zero is \(P(X=0)=e^{-\lambda}\), obtained by setting \(x=0\) in the probability mass function. If a common-rate model is plausible, the observed proportion of zero counts should not be wildly at odds with that probability, allowing for ordinary sample variation. It is also useful to inspect the frequencies of other counts: matching the mean and variance does not guarantee that those frequencies are well described.

What a Diagnostic Can and Cannot Tell You

A diagnostic is evidence about whether a model is a sensible description, not an automatic verdict. A small sample may show a large gap between \(\bar{x}\) and \(s^2\) just by chance. Conversely, several different count patterns can share similar values of those summaries. Context helps distinguish possible explanations and suggests what to inspect next.

These are possible explanations, not conclusions to assume from a numerical warning. For instance, excess zeros can result from different conditions among observations, while a time pattern can reflect a genuine change in rate. The right response is to check how observations were selected, what exposures they represent, and whether the suspected mechanism is plausible.

Common Mistake

Wrong reasoning: “The sample mean and variance are almost equal, so the data must follow a Poisson distribution.” Equality of the population mean and variance is a property of the Poisson model, but it does not uniquely identify that distribution. A dataset can have a similar mean and variance while having an unusual zero frequency, dependence, or a systematic pattern in observation order.

Correction: Treat the mean–variance comparison as one screening check. Also consider whether observations have comparable exposure, whether the rate appears stable, whether counts may be dependent or clustered, and whether the observed frequencies—including zeros—look plausible. A diagnostic concern should prompt investigation rather than an unsupported declaration that the model either fits or fails.