Direct comparison
Quasi-Poisson vs Negative Binomial Regression
Quasi-Poisson scales SEs by a dispersion factor; negative binomial models overdispersion via a mixing distribution. Compare the two and when each fits.
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How do Quasi-Poisson, Negative Binomial compare side by side?
The table below compares Quasi-Poisson, Negative Binomial across 10 procurement-relevant dimensions, from core mechanism through when it's preferred.
Side-by-side comparison
| Dimension | Quasi-Poisson | Negative Binomial |
|---|---|---|
| Core mechanism | Keeps the Poisson mean structure; inflates standard errors by an estimated dispersion multiplier (Pearson chi-square ÷ residual df). | Models the extra variance directly as a Poisson-Gamma mixture — a random effect on the mean drawn from a Gamma distribution. |
| Distributional assumption | None beyond a mean-variance relationship, Var(Y) = φ·μ. It is a quasi-likelihood method, not a full probability model. | Full parametric distribution (negative binomial), typically Var(Y) = μ + α·μ² (the NB2 parameterization). |
| Regression coefficients | Numerically identical to a plain Poisson fit — only the standard errors change. | Estimated jointly with the dispersion parameter α via maximum likelihood; typically very close to the Poisson/quasi-Poisson estimates but not numerically identical. |
| Standard errors vs. plain Poisson | Uniformly scaled by √(dispersion) — every coefficient's SE grows by the same multiplicative factor. | Grow to reflect the modeled overdispersion, but the scaling isn't forced to be uniform across coefficients the way quasi-Poisson's is. |
| Extra parameter estimated | One scalar dispersion factor φ, estimated post hoc from the Poisson fit's residuals — not part of the likelihood. | One dispersion parameter α, estimated jointly with the regression coefficients as part of the model likelihood. |
| AIC / BIC / likelihood-ratio tests | Not valid — there is no proper likelihood to compute them from. Use quasi-AIC (QAIC) or an F-test for nested model comparison instead. | Valid — AIC, BIC, and likelihood-ratio tests against a nested Poisson model all work normally. |
| Underdispersion (variance < mean) | Handles it directly — the dispersion multiplier can be estimated below 1 with no change to the method. | The standard NB model only adds variance (α ≥ 0); underdispersion needs a different model (e.g., Conway-Maxwell-Poisson). |
| Predictive distribution / simulation | Not available in a principled way — there is no full distribution to simulate new counts from. | Available — the fitted NB distribution can generate prediction intervals and simulate new count outcomes directly. |
| Common software | R: glm(y ~ x, family = quasipoisson). Python: statsmodels GLM with Poisson family, scaled by Pearson chi-square. | R: MASS::glm.nb(y ~ x) or glm(family = negative.binomial(theta)). Python: statsmodels NegativeBinomial / GLM with an NB family. |
| When it's preferred | You mainly need valid coefficient tests and confidence intervals, don't want to commit to a specific overdispersion mechanism, or dispersion might be below 1. | You need model comparison via AIC/likelihood-ratio tests, a real predictive distribution, or plan to extend to a zero-inflated/hurdle model built on the same likelihood framework. |
Common questions
Common questions about Quasi-Poisson vs Negative Binomial
Do quasi-Poisson and negative binomial give the same regression coefficients?
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Quasi-Poisson's coefficients are numerically identical to a plain Poisson fit, since it only rescales the standard errors. Negative binomial's coefficients come from a separate maximum-likelihood fit and are typically very close but not numerically identical — in a reproducible simulation with a known Gamma-Poisson data-generating process (n=300, true slope 0.35, true NB dispersion α=0.8), the fitted slope was 0.3747 under Poisson/quasi-Poisson and 0.3727 under negative binomial, both close to the true value.
Can I compare a quasi-Poisson model to a plain Poisson model using AIC?
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No. Quasi-Poisson has no proper likelihood, so AIC and BIC are undefined for it. Use an F-test for nested model comparison, or quasi-AIC (QAIC), which substitutes the estimated dispersion into the AIC-style penalty instead of relying on a genuine likelihood.
How do I know I actually have overdispersion before choosing either method?
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Fit a plain Poisson model first and check the ratio of the Pearson chi-square statistic to its residual degrees of freedom. A ratio near 1 is consistent with equidispersion (Poisson is fine); a ratio well above 1 indicates overdispersion. Formal dispersion tests (e.g., the Cameron-Trivedi test, available as AER::dispersiontest() in R) test this ratio against the null of no overdispersion rather than relying on eyeballing it.
My count data has a lot of zeros — is that the same problem as overdispersion?
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Not necessarily. Overdispersion means the overall variance exceeds the mean across the whole distribution; excess zeros can be a distinct mechanism where some units are structurally incapable of a nonzero count. Compare the model's predicted proportion of zeros to the observed proportion — if observed zeros substantially exceed what even a negative binomial model predicts, a zero-inflated or hurdle model addresses a different problem than either quasi-Poisson or plain negative binomial does.
