Check the Poisson Mean–Variance Assumption

Identify the level at which the assumption applies

The R Poisson distribution documentation states that a Poisson variable has equal mean and variance. In a regression or rate model, that statement concerns the distribution conditional on the modeled mean and exposure. It does not require the pooled variance of every observed count to equal the pooled mean when expected counts differ across records.

Start by defining the observation unit, event count, exposure duration, and important predictors. A clinic observed for ten months is expected to have a different count from the same clinic observed for one month, even if its underlying event rate is unchanged. Comparing raw counts without their exposure can create a misleading dispersion signal.

Use a small descriptive check carefully

For a fictional set of four equally exposed units with counts 0, 0, 4, and 4, the mean is 2. The sample variance is 16/3, approximately 5.3333, and the variance-to-mean ratio is approximately 2.6667. These numbers justify looking more closely at the data structure; four observations do not establish a stable dispersion parameter or diagnose a particular mechanism.

Now consider two equal-sized groups with Poisson means 1 and 9. Within each group, the variance equals its mean. Across the mixture, the mean is 5 while the theoretical variance is 5+16 = 21: average within-group variance plus variance of the group means. The pooled variance-to-mean ratio is 4.2 even though each group's conditional model is Poisson.

Investigate plausible causes

Check duplicate episodes, inconsistent observation windows, repeated events from the same person, center clustering, and heterogeneous baseline risk. Determine whether zero counts represent genuine observed absence or periods when events could not have been detected. These explanations imply different modeling changes and should not be collapsed into a single automatic correction.

After fitting an appropriate model, inspect residual behavior relative to fitted means and exposure. A model-based dispersion estimate requires the relevant residual definition and degrees of freedom; it cannot be reconstructed from two aggregate event totals alone. Avoid selecting a more flexible count model solely because it yields a preferred significance result.

Connect the check to interval interpretation

The independent Poisson rate-ratio tool uses a log-Wald variance based on independent Poisson counts. It cannot test dispersion from its four aggregate inputs. If the assumed variance is inadequate, the displayed interval can misrepresent uncertainty even when the rate ratio itself is calculated correctly.

Use the rate comparison worksheet to document the assessment and the rationale for any model that handles heterogeneity or dependence. Preserve both the event counts and the exposure definition. Report the chosen variance assumptions explicitly rather than describing every count analysis simply as a Poisson analysis. The useful endpoint is a defensible uncertainty calculation, not a rule that all raw count variances must match their means.

Sources checked on 5 November 2026; all numerical examples are fictional and independently recomputed.