Influential Cases and False Dispersion Signals

Why Inspect Influence Before Interpreting Dispersion

A Pearson dispersion ratio can look large because many observations vary more than the fitted GLM predicts. It can also look large because one observation contributes an unusually large residual. Those situations call for different interpretations: broad extra variation may affect uncertainty throughout the data, while a single unusual case may instead point to a data issue, a local model failure, or an influential observation.

The total statistic alone does not say how its evidence is distributed. A useful diagnostic therefore combines residual inspection with case deletion: remove a case, refit the model, and examine how the fitted coefficients and dispersion diagnostic change. A case can matter because its response is unusual, because its design point has high leverage, or because both features occur together.

Residual Contributions and Case Deletion

For independent observations with fitted means \(\hat{\mu}_i\) and model variances \(V(\hat{\mu}_i)\), the Pearson residual and Pearson statistic are

For grouped binomial data with \(y_i\) successes out of \(m_i\) trials, \(V(\hat{\mu}_i)=m_i\hat{p}_i(1-\hat{p}_i)\). For Poisson counts, \(V(\hat{\mu}_i)=\hat{\mu}_i\). Each squared Pearson residual is a nonnegative contribution to the total. A large contribution identifies a case worth investigating, but it does not establish why that case is unusual.

There are two distinct deletion calculations. One can omit a contribution while keeping the original fitted values fixed, or remove the case and refit the model. The first calculation is a simple accounting check. The second is the relevant assessment of influence, since deleting a case may change every fitted mean and therefore every remaining residual.

To verify the fixed-fit identity, write the original statistic as \(X_P^2=r_{Pi}^2+\sum_{j\ne i}r_{Pj}^2\). Omitting only the \(i\)th term leaves \(\sum_{j\ne i}r_{Pj}^2=X_P^2-r_{Pi}^2\). Under refitting, the remaining residuals become \(r_{Pj}^{(-i)}\), calculated using new fitted means \(\hat{\mu}_j^{(-i)}\). Their sum of squares need not equal the old remaining sum. This distinction is essential when a case has substantial influence.

Leverage and the Residual That Looks Too Small

A case can influence a fit even when its raw residual is not large. Leverage describes how unusual a design point is relative to the other design points, in the geometry of the fitted model. In a GLM, the working-weight hat matrix is

where \(X\) is the model matrix and \(W\) is the diagonal matrix of fitted working weights. This expression assumes \(X^\mathsf{T}WX\) is invertible. The diagonal entry \(h_{ii}\) is the leverage for case \(i\); large values indicate that the case has substantial potential to affect the fitted values.

In a linearized GLM fit, the weighted working residuals are approximately projected by \(I-H\). If the standardized working errors have covariance matrix \(I\), the covariance of these residuals is \(I-H\), since \(H\) is symmetric and idempotent: \((I-H)(I-H)^\mathsf{T}=I-H\). In particular, the approximate residual variance for case \(i\) is \(1-h_{ii}\). This motivates examining Pearson residuals adjusted by \(\sqrt{1-h_{ii}}\), especially when leverage differs substantially across cases. The approximation is a diagnostic guide, not an exact finite-sample distributional result for every GLM.

A high-leverage observation can pull the fitted mean toward itself, reducing its own raw residual while materially changing the fitted coefficients. Conversely, a large residual at an ordinary design point may have limited effect on the coefficient estimates. Residual size and leverage therefore answer different questions and should be considered together.

Approximating the Effect of Deleting a Case

A refit is the clearest way to assess influence, but the score equation explains why unusual cases can matter. Let \(U(\beta)=\sum_i u_i(\beta)\) be the full-model score, with a finite maximum-likelihood estimate \(\hat\beta\) satisfying \(U(\hat\beta)=0\). Let \(U_{-i}(\beta)=U(\beta)-u_i(\beta)\) be the score after case \(i\) is removed, and suppose the deleted fit \(\hat\beta_{-i}\) exists.

To derive this approximation, set \(\delta=\hat\beta_{-i}-\hat\beta\). Taylor expansion of the deleted score around \(\hat\beta\) gives \(0=U_{-i}(\hat\beta_{-i})=U_{-i}(\hat\beta)-J_{-i}(\hat\beta)\delta+R(\delta)\), where differentiability implies \(R(\delta)=o(\|\delta\|)\) as \(\delta\) tends to zero. Since \(U(\hat\beta)=0\), the deleted score at the original estimate is \(U_{-i}(\hat\beta)=U(\hat\beta)-u_i(\hat\beta)=-u_i(\hat\beta)\). Substitution yields \(0=-u_i(\hat\beta)-J_{-i}(\hat\beta)\delta+o(\|\delta\|)\). Ignoring only this higher-order remainder and multiplying by the inverse information matrix gives the stated first-order approximation. It is not an exact deletion formula; a full refit is preferable when the change is large, the information matrix is poorly conditioned, or the fit is near a boundary. \(\square\)

The expression combines the case's score contribution with the information available to estimate the coefficients. A large case score can be influential, but so can a design that makes the information matrix sensitive in a particular direction. In practice, compare the approximate signal with an actual refit rather than treating the approximation as a replacement.

Interpreting the Diagnostic Pattern

A few practical checks help keep the interpretation disciplined:

• Inspect the contribution distribution. List or plot squared Pearson residuals to see whether the total is broadly shared or dominated by a small number of cases.
• Check both response and design. Review unusual outcomes, data collection, exposures, and predictor combinations. A high residual and a high-leverage design point are distinct features.
• Refit after deletion. Recalculate coefficients, fitted means, residuals, and the dispersion ratio using the reduced data. Do not subtract one old residual contribution and call that a refitted analysis.
• Track degrees of freedom. The residual degrees of freedom generally change when a case is removed. Compare the recomputed ratios with that change in view.
• Retain the scientific context. Do not discard a case simply because its removal improves a diagnostic. Establish whether it is erroneous, outside the target population, or a valid observation that the model fails to describe.

A common mistake is to reason, “The dispersion ratio is high, so the model has broad extra variation,” without checking which observations generated the statistic. The correction is to examine case-level residual contributions and conduct refitted deletion comparisons. A second mistake is to conclude that a high-leverage case is harmless because its raw residual is small. Leverage can pull the fit toward the case; check the leverage-adjusted residual and the change in coefficients after deletion. Neither check alone proves a cause, and deleting a case is a sensitivity analysis rather than a routine data-cleaning rule.