What a Local Influence Plot Does—and Does Not—Show
A local influence plot can identify perturbations under which a fitted likelihood or a target estimate is unusually sensitive. That is useful in missing-not-at-random (MNAR) analyses, where a subject’s observed outcome history or selection-model contribution may have substantial leverage on an inferential result. But a plot is not a general ranking of “important people.” Its interpretation depends on what was perturbed, how the perturbations were scaled, and which diagnostic was plotted.
Three tempting readings are especially misleading: that high curvature means a large residual, that an influential subject is causally important, or that high curvature proves the model is wrong. Each confuses a sensitivity calculation with a different statistical question. The calculations below use hypothetical fitted models and data; they illustrate interpretation, not evidence about an actual trial.
The standard calculation assumes the fitted likelihood is sufficiently smooth near the baseline, the estimate is an interior local maximum, and the observed information can be inverted. In this setting \(J\) is positive definite, so \(A=\Delta'J^{-1}\Delta\) is positive semidefinite: for any vector \(d\), \(d'Ad=(\Delta d)'J^{-1}(\Delta d)\geq 0\). The curvature formula therefore describes local sensitivity in the chosen perturbation coordinates. If the fit is on a boundary, the information is singular, or numerical derivatives are unstable, the usual calculation may not be reliable.
For a plot of coordinate directions, \(d=e_i\), where \(e_i\) changes only coordinate \(i\), the displayed subject-level value is \(C_{e_i}=2A_{ii}\). Other plots may show curvature in a selected direction or a target-specific diagnostic, such as one aimed at a treatment-effect contrast rather than the entire parameter vector. These are not interchangeable. A high eigenvalue can correspond to a combination of subjects, not to any one subject’s coordinate. Always identify the plot’s perturbation scheme and target before interpreting its labels.
Misreading 1: High Curvature Means a Large Residual
A residual describes a discrepancy between an observation and a model-based value, using a particular residual definition and scale. Local curvature instead describes how the fitted likelihood changes locally when a specified perturbation is applied. The two quantities can be related in some models, but there is no general rule that the largest residual must have the largest curvature, or vice versa. In an MNAR analysis, subject contributions can affect the outcome and selection parts of the likelihood, and the relevant parameter information and perturbation design also matter.
Worked Example: The Largest Residual Is Not the Most Curved Subject
Suppose a hypothetical three-subject diagnostic perturbs one standardized subject contribution at a time. At the fitted model, numerical differentiation gives \(J=I_2\), the \(2\)-by-\(2\) identity matrix, and the columns of \(\Delta\) are \(v_1=(0.1,0.2)'\), \(v_2=(0.6,0)'\), and \(v_3=(0.2,0.1)'\). Assume the differentiability and interior-fit conditions above hold. For coordinate \(i\), \(C_{e_i}=2v_i'J^{-1}v_i=2v_i'v_i\).
For subject 1, \(C_{e_1}=2(0.1^2+0.2^2)=2(0.05)=0.10\). For subject 2, \(C_{e_2}=2(0.6^2+0^2)=2(0.36)=0.72\). For subject 3, \(C_{e_3}=2(0.2^2+0.1^2)=2(0.05)=0.10\). Thus subject 2 has the largest coordinate curvature.
Suppose the same hypothetical analysis reports standardized outcome residuals \(0.4\), \(0.2\), and \(1.8\), respectively. Subject 3 has the largest residual, but subjects 1 and 3 have equal coordinate curvature, and subject 2 has the greatest curvature. These rankings answer different questions. A large residual may motivate residual diagnostics; the curvature ranking indicates sensitivity to the defined perturbations.
This disagreement is not a contradiction. A residual is often a one-observation discrepancy, whereas curvature incorporates how the perturbation propagates through the likelihood and the information matrix. A subject can fit poorly without strongly changing the chosen target, and a subject with a modest residual can be influential because its likelihood contribution is especially informative for a parameter or contrast.
Misreading 2: A High-Curvature Direction Identifies One Dominant Subject
A local influence plot may summarize coordinate perturbations, but the largest curvature over all unit directions is an eigenvalue problem. The corresponding eigenvector can involve several coordinates at once. Its signs and magnitudes describe a perturbation pattern: for example, increasing one subject’s contribution while decreasing another’s. That pattern is not the same as a list of independently influential subjects.
Worked Example: A High-Curvature Combination
In a hypothetical two-subject analysis, suppose the curvature matrix is \(A=\Delta'J^{-1}\Delta=\begin{pmatrix}0.30&0.25\\0.25&0.30\end{pmatrix}\). Assume \(J\) is positive definite, the baseline fit is regular, and both perturbation coordinates have been standardized comparably. For the first subject alone, \(d=e_1\), so \(C_{e_1}=2(0.30)=0.60\). For the second alone, \(C_{e_2}=2(0.30)=0.60\).
Now consider the unit direction \(d=(1,1)'/\sqrt{2}\). The quadratic form is \(d'Ad=0.55\), so \(C_d=2(0.55)=1.10\). In the contrasting unit direction \(d=(1,-1)'/\sqrt{2}\), \(d'Ad=0.05\), giving \(C_d=0.10\). The eigenvalues are \(0.55\) and \(0.05\), with the largest-curvature direction proportional to \((1,1)'\).
The largest directional curvature, \(1.10\), is greater than either single-coordinate value, \(0.60\). It describes a joint perturbation, not evidence that either subject alone has curvature \(1.10\). The opposite-sign direction is comparatively insensitive. A plot that shows the maximum direction should therefore be read as a pattern of sensitivity, not as a stand-alone subject ranking.
Scale matters as well. Curvature is defined for a unit direction in the coordinates used for \(\omega\); changing the units or relative scaling of perturbations changes what “unit” means and can change the reported ranking. A subject-level plot is interpretable only relative to its specified perturbation scheme. Standardization can make perturbations more comparable, but it does not make different perturbation schemes equivalent. The earlier material on scaling perturbations and competing high-curvature directions is relevant when comparing such plots.
Misreading 3: Influence Is Causal Importance
A perturbation is a mathematical change to the fitted analysis, not an intervention assigned to a subject. High curvature does not say that the subject caused the treatment effect, caused dropout, or would have changed an outcome under a real-world intervention. Even when a perturbation resembles a plausible data change, the local diagnostic describes model sensitivity under that formal change. Causal conclusions require a causal estimand, suitable design or identification assumptions, and an analysis directed at that estimand.
Worked Example: Sensitivity of a Treatment Contrast Is Not a Subject Effect
Consider a hypothetical longitudinal trial fit with a baseline estimated treatment contrast of \(-2.00\) points. Under a defined, standardized local perturbation of one subject’s observed outcome contribution, refitting at perturbation values \(t=-0.10\) and \(t=0.10\) gives estimated contrasts of \(-2.08\) and \(-1.92\) points. The centered finite-difference slope is \( [\hat{\psi}(0.10)-\hat{\psi}(-0.10)]/[0.10-(-0.10)] = [-1.92-(-2.08)]/0.20 = 0.16/0.20 = 0.80\) points per perturbation unit.
This calculation describes local sensitivity of the fitted contrast to that specified perturbation. It does not mean the subject’s treatment effect is \(0.80\) points, that changing the subject’s outcome would causally change other participants’ outcomes, or that the subject generated the observed treatment contrast. The perturbation unit is a feature of the diagnostic definition, not a clinical unit unless explicitly constructed that way.
Also distinguish curvature from the first derivative of a target estimate. Curvature in the likelihood and sensitivity of an estimate are related diagnostic ideas, but they are not generally the same numerical quantity. If a plot is target-specific, its definition should state which target is being assessed and how nuisance parameters are handled. A high full-parameter curvature need not imply high curvature for the treatment contrast of interest; conversely, a direction can matter for a target even if it is not the leading full-parameter direction.
Misreading 4: High Curvature Proves the Model Has Failed
A large curvature value is a warning to investigate sensitivity, not a formal test of model adequacy. It does not by itself establish a bad likelihood, incorrect missingness assumptions, an anomalous observation, or invalid inference. Nor does a small value certify that a model is correct: the perturbation may not probe the assumption that is most consequential, and the diagnostic is local to the fitted model and baseline.
A useful check separates three questions: (1) Is the calculation numerically stable? (2) What feature of the data or likelihood produces the sensitivity? (3) Does the substantive conclusion change under a defensible analysis? Numerical checks can compare analytic and finite-difference derivatives, vary step size, and confirm that the baseline is reproduced at \(\omega_0\). Data checks can inspect the subject’s observed trajectory, dropout history, and relevant covariates. Inferential checks can refit under a clearly specified perturbation or compare a suitable target estimate and uncertainty measure. These steps investigate a signal; they do not convert curvature into a model test.
Record whether the plot shows coordinate curvature, a maximum direction, or a target-specific quantity, and identify exactly what changes under the perturbation.
Verify the baseline, derivatives, information matrix, and perturbation units. A ranking can be an artifact of inconsistent scaling or unstable numerical differentiation.
Inspect relevant subject data and refit a defensible perturbation. Report whether the target and its uncertainty materially change, without treating the diagnostic as a causal or model-validity verdict.
How Local Influence Differs From Other Diagnostics
Local influence complements, rather than replaces, residual checks, deletion diagnostics, and model assessment. The distinction is especially important in selection models, where an observed-data discrepancy and sensitivity to a selection-equation perturbation are not identical phenomena.
| Diagnostic | Primary question | What it cannot establish alone |
|---|---|---|
| Residual diagnostic | How discrepant is an observed value from a model-based value, on the chosen residual scale? | Whether that discrepancy strongly affects a particular fitted target. |
| Local influence | How curved or sensitive is the fitted likelihood or stated target under a specified small perturbation? | Causal importance, global robustness, or model failure. |
| Case-deletion analysis | How does a refitted analysis change when a specified subject or set of subjects is removed? | The local behavior between the baseline and that finite deletion, or why the change occurred. |
| Model-fit assessment | Are the model’s distributional and structural assumptions compatible with observed data and the analysis purpose? | A definitive MNAR assumption check from observed data alone. |
Local curvature is attractive because it can expose sensitive directions without requiring a full refit for every infinitesimal change. Its limitation is the same feature: it is local. A small perturbation approximation need not predict the effect of a large change or deletion. A plot also cannot reveal sensitivity to perturbations that were never considered. In MNAR settings, observed-data diagnostics cannot by themselves verify assumptions about unobserved outcomes; sensitivity analysis remains necessary for conclusions that depend on those assumptions.
Check Your Understanding
Answer each question, then compare with the brief solutions.
- A subject has the largest standardized residual but the smallest coordinate curvature. Is this logically inconsistent? Why?
- A maximum-curvature eigenvector has nonzero entries for four subjects. What does that direction represent?
- Why can rescaling perturbation coordinates change a subject-level curvature ranking?
- A high-curvature subject is a participant with early dropout. Does the plot establish that the participant caused the treatment effect or that the selection model is wrong?
- Name one numerical check and one substantive check to perform after finding high curvature.
Solutions. (1) No. Residual size measures observed discrepancy on a residual scale, while curvature measures sensitivity under a defined perturbation and likelihood. (2) It is a joint perturbation pattern involving those subjects, not necessarily a ranking of their separate coordinate effects. (3) Unit-length directions depend on coordinate units, so rescaling changes the perturbations represented by equal-length directions. (4) No. It is evidence of local sensitivity under the chosen perturbation, not causal attribution or proof of model failure. (5) For example, compare numerical derivatives across step sizes; then inspect the subject’s observed trajectory and assess whether a defensible refit changes the stated inferential target.