Q47. Overdispersion and quasi-Poisson inference

Topic: Underdispersion, positive responses, and quasi-likelihood · more questions on this topic
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[10 marks]

For independent counts, suppose \mathbb{E}(Y_i)=\mu_i=\exp(\mathbf x_i^\top\boldsymbol\beta), \qquad \operatorname{Var}(Y_i)=\phi\mu_i.

(a) Explain what \phi>1 means relative to a Poisson model. [2 marks]

(b) Using the quasi-score U(\boldsymbol\beta) =\sum_i\frac{Y_i-\mu_i}{\operatorname{Var}(Y_i)} \frac{\partial\mu_i}{\partial\boldsymbol\beta}, show why the coefficient estimates coincide with the Poisson estimates when the data and mean model are unchanged. [3 marks]

(c) A coefficient has Poisson standard error 0.10. If \hat\phi=4, find its quasi-Poisson standard error. Explain the effect on a Wald statistic. [2 marks]

(d) Does specifying the mean and variance above determine a complete probability distribution? Explain why an ordinary likelihood-ratio test cannot automatically be applied. [3 marks]

Worked answer (click to show)

Definitions. Let \mu_i=e^{\mathbf x_i^\top\boldsymbol\beta}, where \mathbf x_i is a fixed predictor column vector. The dispersion \phi>0 scales variance to \phi\mu_i. A quasi-score weights residuals by the inverse assumed variance and the derivative of the mean; it is an estimating equation, not necessarily the derivative of a full log-likelihood. Standard error is the square root of an estimated coefficient variance.

Assumptions. Compare fits to the same data using the same unpenalised log-linear mean, conditionally independent observations, and a common dispersion \phi. The design and fitted solution must support finite, identifiable coefficient estimates and the stated covariance calculation. Equality of estimates is not asserted for different mean models, penalties, or observation-specific dispersion.

(a) Meaning of \phi>1. A Poisson model has \operatorname{Var}(Y_i)=\mathbb{E}(Y_i)=\mu_i. Here \operatorname{Var}(Y_i)=\phi\mu_i, so \phi>1 means overdispersion: at the same expected count, the variance is \phi times the Poisson variance.

(b) Equality of the coefficient estimates. For \mu_i=e^{\mathbf x_i^\top\boldsymbol\beta}, \frac{\partial\mu_i}{\partial\boldsymbol\beta}=\mu_i\mathbf x_i. Substitution into the supplied quasi-score gives \begin{aligned} U(\boldsymbol\beta) &=\sum_i\frac{Y_i-\mu_i}{\phi\mu_i}\mu_i\mathbf x_i\\ &=\frac1\phi\sum_i\mathbf x_i(Y_i-\mu_i) =\boxed{\frac1\phi S_{\mathrm{Poisson}}(\boldsymbol\beta)}. \end{aligned} Because \phi>0 is a common scalar, U(\boldsymbol\beta)=\mathbf0 \quad\Longleftrightarrow\quad S_{\mathrm{Poisson}}(\boldsymbol\beta)=\mathbf0. The same data and mean model therefore give the same fitted coefficients, when the estimates exist and are well defined. Estimating the common dispersion does not change these estimating equations’ roots.

(c) Standard error and Wald statistic. The usual model-based covariance matrix is multiplied by \hat\phi: \widehat{\operatorname{Cov}}_{\mathrm{quasi}}(\widehat{\boldsymbol\beta}) =\hat\phi(X^\top\widehat WX)^{-1}, \qquad \widehat W=\operatorname{diag}(\widehat\mu_i). Thus standard errors are multiplied by \sqrt{\hat\phi}, not by \hat\phi itself: \boxed{\operatorname{SE}_{\mathrm{quasi}}=\sqrt4(0.10)=0.20.} For a fixed null value \beta_{j,0}, the signed Wald statistic becomes z_{\mathrm{quasi}} =\frac{\widehat\beta_j-\beta_{j,0}}{0.20} =\frac12z_{\mathrm{Poisson}}. Its squared version is one quarter as large. Against the same reference distribution, this gives weaker evidence against the null; it does not necessarily change a particular reject/non-reject decision.

(d) Is a full probability distribution specified? No. A mean and variance alone do not determine all the count probabilities. Quasi-Poisson inference specifies these moments and an estimating equation, not a unique probability mass function. There is therefore no ordinary full likelihood supplied by these assumptions from which to construct an ordinary likelihood-ratio test automatically. A valid test needs an appropriate quasi-likelihood calibration or an additional fully specified distributional model; a Poisson likelihood-ratio test cannot simply ignore the overdispersion.