Why Delete a Subject From a Joint Model?
A subject in a joint longitudinal and time-to-event study contributes two connected kinds of information: repeated biomarker measurements and an event or censoring history. A subject may have an unusual trajectory, an early event, or a combination of both. Case-deletion diagnostics ask how much the fitted model changes when that subject’s entire contribution is removed. This differs from deleting only one measurement or only one event record: the question is whether the subject’s combined history has disproportionate influence.
A large change is a diagnostic signal, not proof that the subject is erroneous or that the model is wrong. It indicates that conclusions may depend materially on that case. The useful follow-up is to inspect the data, the model’s fit to the subject, and the scientific context, then report whether key estimates or risks are sensitive to the deletion.
Likelihood Contributions and the Deletion Approximation
Let subject \(i\) have longitudinal measurements \(Y_i\), event or censoring information \((T_i,\delta_i)\), covariates \(X_i\), and random effects \(b_i\). Write \(\theta\) for the joint-model parameter vector. Under the usual independent-subject likelihood, the marginal log-likelihood is a sum of subject contributions. A typical contribution integrates over the subject-specific random effects:
The event contribution must reflect the observed event or censoring status, and the longitudinal contribution includes the measurements actually observed. This formulation assumes independent subjects, a correctly specified likelihood for the joint data, and a valid treatment of censoring, such as independent censoring conditional on variables included in the model. The same fitted model and likelihood approximation should be used in every deletion refit.
For a likelihood-based fit, define the subject score \(u_i(\theta)=\partial\ell_i(\theta)/\partial\theta\), and define the observed information from the full data as \(J(\theta)=-\partial^2\ell(\theta)/\partial\theta\partial\theta^\mathsf{T}\). At an interior maximum, the full-data score is zero. Removing subject \(i\) therefore leaves score \(-u_i(\hat{\theta})\) at the full-data estimate. A first-order Taylor expansion of the deleted-data score around \(\hat{\theta}\) gives the local approximation below.
Here \(J_i(\theta)=-\partial^2\ell_i(\theta)/\partial\theta\partial\theta^\mathsf{T}\) is the information contribution of subject \(i\). To see the sign, let \(\Delta=\hat{\theta}_{-i}-\hat{\theta}\). The deleted-data score at \(\hat{\theta}\) is \(-u_i(\hat{\theta})\); its first-order expansion is approximately \(-u_i(\hat{\theta})-J_{-i}(\hat{\theta})\Delta\). Setting this to zero yields the displayed approximation. When the deleted contribution is small relative to the total information, \(J_{-i}\) is sometimes approximated by \(J\), but the resulting estimate is only a screening device. Direct refitting is needed to assess actual sensitivity.
The inverse information matters. For equal score components, a direction with weaker information generally permits a larger parameter shift, while a direction with stronger information generally permits a smaller one. A large score can still produce a notable shift in any direction after the inverse-information scaling. The approximation depends on the product \(J_{-i}^{-1}u_i\), not on score size alone.
Worked Example: Reading the Score Approximation
Suppose a hypothetical scalar association parameter has full-data estimate \(\hat{\theta}=0.40\). For one subject, the score at that estimate is \(u_i(\hat{\theta})=1.80\), and the deleted-data observed information is \(J_{-i}(\hat{\theta})=120\). The local estimate of the deletion shift is
If the same score were scaled by information of \(30\), the approximate shift would instead be \(-1.80/30=-0.0600\). Thus, equal score components can imply larger shifts in the less-informed direction. In a hypothetical actual refit, suppose the estimate is \(0.386\); the observed change is \(0.386-0.400=-0.0140\), close to the first-order approximation. This agreement supports its use as a rough screen here, but does not make the approximation a substitute for refitting.
Compare Parameter Changes on Meaningful Scales
Raw changes in different parameters cannot be compared directly when their units or scales differ. For parameter \(r\), one simple standardized change is
The denominator is the full-data standard error. This expresses the deletion shift in units of the full-data standard error; it is a sensitivity measure, not a test statistic with a universal cutoff. For a vector of parameters, a scale-adjusted summary is \(\{(\hat{\theta}_{-i}-\hat{\theta})^\mathsf{T}\widehat{\mathrm{Var}}(\hat{\theta})^{-1}(\hat{\theta}_{-i}-\hat{\theta})\}^{1/2}\), provided the covariance estimate is nonsingular and the parameters are on a suitable unconstrained scale. Such a summary can flag cases for closer inspection, but the individual changes and their scientific meaning should still be examined.
Worked Example: Standardizing a Two-Parameter Change
Suppose a hypothetical full-data fit estimates an intercept and a biomarker association parameter as \((0.40,0.12)\). Their full-data standard errors are \(0.10\) and \(0.02\), respectively. After deleting one subject, the estimates are \((0.34,0.15)\). The changes are \(-0.06\) and \(0.03\), so the standardized changes are
The deletion moves the intercept downward by \(0.60\) full-data standard errors and the association estimate upward by \(1.50\) full-data standard errors. If the two parameter estimates are treated as uncorrelated for a simple illustrative summary, the Euclidean length of these standardized changes is \(\sqrt{(-0.60)^2+(1.50)^2}=\sqrt{2.61}=1.616\). In a real fit, parameter correlations should be retained when calculating a covariance-adjusted multivariate distance. This example identifies a more pronounced relative change in the association parameter; it does not establish that the parameter is statistically significant or that the case should be removed.
Compare Subject-Level Risks as Well
Parameter sensitivity and prediction sensitivity answer different questions. A parameter may change without materially altering clinically relevant predictions, while a modest parameter change may matter for patients near an important risk threshold. Define a prediction target before comparing fits. For example, at landmark time \(s\) and horizon \(\Delta\), a conditional event risk for subject \(k\) can be written
This risk conditions on survival to \(s\) and on the biomarker history available by \(s\); it does not condition on future measurements or the future event outcome. The random-effects uncertainty is integrated according to the fitted model, including the information in the subject’s permitted history. For each deletion, compare \(R_k(s,\Delta;\hat{\theta}_{-i})\) with \(R_k(s,\Delta;\hat{\theta})\) using the same target subjects, landmark, horizon, and available histories. A deleted subject can still be a prediction target if the purpose is to assess how the training fit predicts that subject from information available at the landmark; deleting a case from fitting does not mean silently changing the prediction definition.
Worked Example: Risk Change After Deletion
Suppose a hypothetical model gives subject A a conditional event risk of \(0.18\) over the next year, given survival to the landmark and the biomarker history observed by then. After deleting subject B and refitting, the same risk for subject A is \(0.27\). The full and deleted fits therefore imply conditional survival probabilities of \(1-0.18=0.82\) and \(1-0.27=0.73\), respectively. The risk change is
or an increase of 9 percentage points in the estimated one-year conditional event risk for subject A. This is a model-sensitivity comparison, not evidence that deleting subject B changes subject A’s actual outcome. The landmark, horizon, and history used in both calculations must remain identical for the comparison to be interpretable.
Practical Workflow, Diagnostics, and Alternatives
A disciplined analysis separates screening from confirmation. A useful sequence is:
Confirm convergence, plausible parameter estimates, and a well-behaved information matrix before using deletion changes as diagnostics.
Use subject scores, residual evidence, or scientific concerns to identify cases for inspection. Record the selection rule; avoid presenting a post hoc selection as if it were prespecified.
Remove the selected subject’s full joint contribution and use consistent optimization and integration settings. Check convergence and compare the direct refit with any score-based approximation.
Report raw and standardized parameter shifts, plus changes in prespecified conditional risks for a fixed prediction target and horizon.
Review data quality, the subject’s longitudinal and event histories, model assumptions, and the consequences for the scientific conclusions.
The score approximation is computationally useful for screening many subjects, but it can fail when a case has a large contribution, the likelihood is strongly nonlinear, or the deleted fit is near a boundary or another local optimum. A change in convergence status, large variation across starting values, or a nearly singular information matrix makes the comparison difficult to interpret. Check that numerical integration or approximation error is not being mistaken for case influence.
These diagnostics complement, rather than replace, component-specific influence checks. A longitudinal-only deletion can reveal sensitivity in the repeated-measurement fit, and an event-component diagnostic can reveal sensitivity in survival fitting; joint deletion assesses the combined contribution under the joint likelihood. The joint result need not equal the sum of two component-specific results, because the shared random effects and association structure connect the components. Residual diagnostics address patterns of model-data disagreement; deletion diagnostics address the effect of removing a case. Neither alone establishes the source of a problem.
A deletion analysis also has limits. Large influence is not the same as an outlier, a bad observation, or a causal effect. A single-case result may reflect sparse data, limited event information, or a legitimately unusual patient. Deleting multiple selected subjects can produce a different result from adding their individual shifts, because the likelihood and information change jointly. If a substantial fraction of subjects is influential, consider broader model adequacy and data structure rather than repeatedly excluding cases. For Bayesian fits, the likelihood-score expansion above does not directly describe a posterior summary; compare posterior fits after deletion or use a method justified for the chosen posterior computation.
Common Mistake
Wrong reasoning: “This subject has a large score, so the estimate must move a lot in every parameter direction; alternatively, high information amplifies that score.” Both claims omit the inverse-information scaling. The local shift is approximately \(-J_{-i}^{-1}u_i\): for equal score components, weaker-information directions generally have larger shifts, while stronger information generally dampens them. A sufficiently large score can still matter after this scaling, and the actual refit is the check on the approximation.
Check Your Understanding
Answer each question, then compare with the brief solutions.
- If a subject score component is unchanged but the information in that parameter direction doubles, what does the local score approximation imply about the size of the deletion shift?
- A one-year conditional risk changes from \(0.31\) to \(0.24\) after deleting a case. State the change in percentage points and give a careful interpretation.
- Why should a subject’s full longitudinal and event contribution be deleted for a joint case-deletion diagnostic?
- Name two checks to make when an actual deletion refit differs substantially from the score-based approximation.
Solutions. (1) With the score component fixed, doubling information approximately halves the shift magnitude, since the scalar approximation divides the score by information. (2) The change is \(0.24-0.31=-0.07\), a decrease of 7 percentage points in the model-estimated one-year conditional risk; it is a sensitivity comparison, not a change in the subject’s realized outcome. (3) The target is the influence of that subject’s combined contribution to the joint likelihood, whose longitudinal and event components are connected through the shared model structure. (4) Check convergence and sensitivity to starting values, and verify that the same model specification and numerical integration settings were used; also inspect whether nonlinearity or a boundary estimate makes a first-order approximation unreliable.