Masking And Swamping In Multiple-Case Deletion

Why Multiple-Case Deletion Matters

A case-by-case influence analysis can miss an influential group. If several observations pull the fitted model in a similar direction, each may appear modest when deleted alone even though deleting them together changes the fit substantially. The reverse can also happen: a genuinely unusual group can distort the fitted model enough that ordinary observations look influential by comparison. These are called masking and swamping.

The central diagnostic question is therefore not only “What changes when case \(i\) is deleted?” but also “What changes when a proposed set \(S\) is deleted, and how do the individual and joint changes compare?” The answer depends on the fitted model, the parameter scale, and the cutoff used to flag a change. A flag is a screening signal, not a proof that an observation is erroneous or should be removed.

Subset Deletion and a Coefficient-Change Measure

To compare changes across coefficient scales, use the observed coefficient information matrix at the full-data fit, \(J(\hat{\beta})=-\partial^2\ell(\beta)/\partial\beta\,\partial\beta^\mathsf{T}\vert_{\hat{\beta}}\). Assume it is positive definite. One possible dimensionless measure is \(D(S)=\Delta_S^\mathsf{T}J(\hat{\beta})\Delta_S\), with \(\sqrt{D(S)}\) reporting the change in information-scaled units. This tutorial uses that fixed full-fit reference for comparisons. Other diagnostics may use a different scaling; comparisons are meaningful only when the same scaling rule is applied consistently.

The cutoff is a practical screening choice, not a universal constant. In particular, a collection of modest individual changes need not add up to a large joint change: their directions may differ, and the information matrix changes when observations are deleted. Conversely, several changes in a similar direction can accumulate. For that reason, inspect candidate subsets and their complements rather than treating a ranked list of single-case diagnostics as a complete analysis.

Why the Joint Change Is Not a Sum of Single-Case Changes

The likelihood for independent observations is additive, which gives a useful first-order approximation for deleting a set. Write \(U(\beta)=\partial\ell(\beta)/\partial\beta\) for the full score and \(U_i(\beta)\) for observation \(i\)’s score contribution. Let \(U_{(-S)}(\beta)=\sum_{i\notin S}U_i(\beta)\). At an interior full-data maximum, \(U(\hat{\beta})=0\), so the deleted-data score at the original estimate is the negative of the contributions from the deleted cases.

Proof. Additivity of the score and \(U(\hat{\beta})=0\) give \[ U_{(-S)}(\hat{\beta})=U(\hat{\beta})-\sum_{i\in S}U_i(\hat{\beta}) =-\sum_{i\in S}U_i(\hat{\beta}). \] By Taylor expansion around \(\hat{\beta}\), the deleted score at the refitted estimate satisfies \[ 0=U_{(-S)}(\hat{\beta}+\Delta_S) =U_{(-S)}(\hat{\beta})-J_{(-S)}(\hat{\beta})\Delta_S+r(\Delta_S). \] When the derivative of the score is locally Lipschitz, \(\|r(\Delta_S)\|\leq C\|\Delta_S\|^2\) in a sufficiently small neighborhood, for some finite \(C\). Rearranging and using nonsingularity yields \[ \Delta_S=-[J_{(-S)}(\hat{\beta})]^{-1}\sum_{i\in S}U_i(\hat{\beta}) +[J_{(-S)}(\hat{\beta})]^{-1}r(\Delta_S). \] The last term is second order in the deletion change, which proves the stated first-order approximation. The approximation need not be accurate for a large deletion or a poorly conditioned deleted fit. \(\square\)

This result gives a useful interpretation: subset deletion depends on the sum of the deleted cases’ score contributions, scaled by the information remaining after deletion. It is not generally obtained by adding single-case deletion changes. The deleted information matrix differs from the full information matrix, and refitting is nonlinear in general. Thus a one-case approximation is a screening aid, not a substitute for a joint refit when the subset may have a large effect.

An Exact Formula in the Intercept-Only Gaussian Model

Proof. The full-data estimate is \(\bar{y}=n^{-1}\sum_{i=1}^n y_i\), so the sum of the responses not in \(S\) is \(n\bar{y}-\sum_{i\in S}y_i\). The deleted-data mean is therefore \[ \bar{y}_{(-S)}=\frac{n\bar{y}-\sum_{i\in S}y_i}{n-m}. \] Subtracting \(\bar{y}\), putting both terms over the denominator \(n-m\), and simplifying gives \[ \bar{y}_{(-S)}-\bar{y} =\frac{m\bar{y}-\sum_{i\in S}y_i}{n-m} =-\frac{\sum_{i\in S}(y_i-\bar{y})}{n-m}. \] This proves the formula. \(\square\)

The numerator is the sum of the deleted observations’ deviations from the full mean. Similar deviations can accumulate, producing masking. Deviations with opposite signs can cancel, so a set can have a small joint effect even when its members have large individual effects.

A Practical Subset-Comparison Workflow

Joint deletion is most useful when guided by a diagnostic question: do several cases share a design pattern, a response discrepancy, or a data-collection feature that could cause their effects to reinforce or cancel? A broad search over every possible subset can be computationally expensive and can produce chance flags. Start with plausible groups suggested by the data and subject matter, then verify the effect by refitting.

Common Pitfall and Interpretation

A common mistake is to conclude that a group is harmless because none of its members exceeds a single-case cutoff. That reasoning ignores masking: several modest, similarly directed effects may produce a substantial joint change. The correction is to perform a joint deletion and refit for plausible groups. The opposite mistake is to assume every case flagged in the full model is genuinely influential. Swamping can make ordinary cases look influential while a more extreme case distorts the comparison; check whether those flags persist after removing the suspected influential set.

Neither comparison justifies automatic deletion. The cases may be valid observations, and a large change can reveal a model that is sensitive to an important part of the data. Report the subset considered, the refitting procedure, the coefficient or fitted quantities that changed, and the scale used for comparison. This keeps a diagnostic finding distinct from a decision about data validity or model choice.