Regression & Correlation

Overdispersion Check

Checks for overdispersion in count data models using the Pearson χ²/df ratio.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Regression > Poisson Regression > Results (Goodness-of-Fit)

Description

The overdispersion check evaluates whether the observed variance in a Poisson or negative binomial GLM exceeds the modelled variance. The Pearson χ² statistic is computed from the Pearson residuals and divided by the degrees of freedom (n − p). A ratio close to 1 indicates correct dispersion, values > 1.5 indicate overdispersion, and values > 3 indicate severe overdispersion. In these cases the Poisson model's standard errors are too small, and an alternative model (quasi-Poisson, negative binomial) should be used.

Formulas

Pearson-Residuum

r_{P, i} = \frac{y _{i} - μ ^ _{i}}{V ( μ ^ _{i} )}

Pearson residual: difference between observed and fitted value, standardised by the square root of the variance function

Pearson χ²

χ_{P}^{2} = i = 1 \sum n r_{P, i}^{2}

Pearson χ² statistic: sum of squared Pearson residuals

Dispersionsschätzer

\overset{φ}{^} = \frac{χ _{P}^{2}}{n - p}

Dispersion estimate: Pearson χ² divided by degrees of freedom

Erwartungswert unter Poisson

E (\overset{φ}{^}) \approx 1 falls Modell korrekt

Under a correctly specified Poisson model the expected dispersion estimate is approximately 1

Entscheidungsregeln

\overset{φ}{^} > 1, 5 \Rightarrow \ddot{U} berdispersion, \overset{φ}{^} > 3 \Rightarrow schwere \ddot{U} berdispersion

Rule-of-thumb thresholds for assessing overdispersion

Assumptions

Pearson residuals come from a count data model (Poisson or NegBin GLM)
Correct mean structure (misspecified mean inflates χ²)
Independent observations

Limitations

Rule-of-thumb thresholds (1.5 / 3) — no formal hypothesis test provided
Underdispersion (ratio << 1) may also indicate model misspecification
Does not distinguish overdispersion from zero-inflation
Small samples: χ²/df estimate has high variance

References

Cameron, A. C., Trivedi, P. K. (2013), Regression Analysis of Count Data, 2nd Ed., Cambridge University Press, Ch. 3.4
Hilbe, J. M. (2014), Modeling Count Data, Cambridge University Press, Ch. 5