Negative binomial regression models counts whose variance exceeds their mean, by estimating the spread as a parameter instead of forcing it to equal the average. Poisson regression assumes variance and mean are identical. When they are not, its coefficients stay roughly right and its standard errors are far too small — so the estimates look fine and every interval and p-value built on them is wrong.
What a 95% interval really covers
The usual way to describe overdispersion is that the Poisson standard errors are “too small”, which sounds like a correction of the second decimal. It is worth putting a number on instead.
Two hundred datasets were simulated at each of three settings, with a known true coefficient of 0.7. For each, a Poisson model was fitted and its nominal 95% interval was checked: did it contain the truth?
When the Poisson assumption genuinely held, it did so 96.5% of the time — correct behaviour. At a dispersion of 0.5, 61.5%. At a dispersion of 1.2, 38.5%. The negative binomial interval covered 97.0%, 96.0% and 95.0% across the same three settings.
An interval labelled 95% that contains the truth in fewer than two of five samples is not a conservative interval or a slightly optimistic one. Every significance test built on it is operating at a false-positive rate several times what it claims. And nothing about the fit signals this: the coefficients are consistent under overdispersion, so they land in the right place, and only the uncertainty around them is wrong.
The preset shows the mechanism directly. Mean count 3.40, variance 20.87 — six times larger, where the Poisson model requires them equal. The resulting standard errors are 2.279 times too small, so every interval is under half its proper width.
Three ways to notice
The first is the crudest and usually enough: compare the mean and the variance of the counts. A Poisson model insists they are equal, and on the preset they are 3.40 and 20.87. That comparison ignores the predictors, so it is a screening check rather than a test, but a factor of six is not subtle.
The second is the Pearson dispersion statistic, which is the sum of squared standardised residuals divided by the residual degrees of freedom. It should be about 1 under a correct Poisson fit and comes back at 5.33 here. Unlike the raw comparison it accounts for the predictors, so it is the one to quote.
The third is the likelihood-ratio test between the two fits, reported with its p-value halved. That halving is not a fudge: testing whether the dispersion parameter is zero puts the null on the boundary of the parameter space, because dispersion cannot go negative, and the usual chi-square reference distribution is then the wrong one. The correct reference is a mixture, which works out as halving the ordinary p-value.
A fourth signal is on the tool for a different reason: observed zeros against Poisson-predicted zeros. On the preset a Poisson fit predicts 9.9 accounts with no tickets and 49 had none. Excess zeros are a symptom of overdispersion, but they can also be a separate phenomenon entirely, which the next section covers.
Overdispersed, or zero-inflated
Two different things produce more zeros than a Poisson expects, and they call for different models.
Overdispersion means the rate varies across units. Some accounts raise many tickets, some raise almost none, and the spread of rates produces both a long tail and extra zeros. The negative binomial handles this by construction — it is a Poisson whose rate is itself random — and on the preset it predicts 54.6 zeros against 49 observed, close enough.
Zero inflation means some units could never have had an event at all. Accounts on a plan without support access do not raise tickets at any rate, and they are a different population rather than the low end of one. A zero-inflated model splits them out explicitly.
The predicted-zero row is how to tell. If the negative binomial matches the observed zero count, the spread explains them and there is nothing further to do. If it still falls well short, there is a separate source of zeros that varying the rate cannot reach, and the zero-inflated model is the right shape. The distinction is substantive as much as statistical: it asks whether a zero means “had the opportunity and no events” or “never had the opportunity”.
The other fixes, and when they are enough
Quasi-Poisson is the common alternative: keep the Poisson fit and multiply the standard errors by the square root of the estimated dispersion. It is simple, it fixes the intervals, and it is often sufficient. What it does not give you is a likelihood, so information criteria and likelihood-ratio tests are unavailable, and it does not model the count distribution — so it says nothing about predicted zeros or any other feature of the shape. The same arithmetic is laid out on the Poisson regression page.
Robust or sandwich standard errors are another route to roughly the same place, and carry the same limitation: corrected uncertainty around a model of the mean, with no claim about the distribution.
The negative binomial does more work than either. It is a genuine probability model, so it supports likelihood comparisons, predicts the distribution of counts including the zeros, and gives a dispersion parameter that is interpretable in its own right. The cost is one extra parameter, and the second preset shows what that costs when it is not needed: dispersion 0.0025, coefficients identical to four decimals, standard error ratio 1.005. Nothing is lost by fitting it.
Reporting it
Report the dispersion parameter and the Pearson statistic, not just the coefficients. They are what justify the model choice, and a count model reported without any dispersion diagnostic leaves a reader unable to judge whether the intervals mean anything.
Give rate ratios rather than raw coefficients where the audience is not statistical — exp(β) is a multiplicative effect on the expected count and reads directly. Report observed against predicted zeros, since that is the check that distinguishes overdispersion from zero inflation.
If you fitted a Poisson and found no overdispersion, say so and give the number. “Dispersion 0.0025, likelihood-ratio p = 0.46” is a much stronger statement than the silence that usually stands in for it, and it costs one line.
Sources and methodology
References for the model and for detecting overdispersion.
Method. Coefficients are fitted by iteratively reweighted least squares at a fixed dispersion, alternating with a golden-section search on the profile likelihood for the dispersion itself. The likelihood-ratio p-value is halved because the null sits on the boundary of the parameter space. The coverage figures quoted above come from 200 simulated datasets at each setting with a known true coefficient, counting how often each nominal 95% interval contained it. The suite separately confirms the model collapses onto the Poisson when there is no overdispersion — dispersion below 0.05, coefficients agreeing and standard errors within 1.15 — and that the dispersion parameter is recovered on average at true values of 0.3, 0.8 and 1.5. That engine is verified on every change against 490 assertions. The count and the per-case breakdown are published on the formula verification page.