Influential Subjects And Joint-Model Stability

Why Examine Individual Subjects?

A joint model combines each subject’s longitudinal biomarker history with that subject’s event or censoring information. Because these data contribute together to estimation, a small number of subjects can sometimes have a large effect on the estimated biomarker–event association, the random-effects distribution, or the numerical behavior of the fit. A subject-level deletion analysis asks a direct sensitivity question: how does the fitted model change when one subject’s entire contribution is removed?

This is not a procedure for deciding which observations to discard. A large change can indicate a data error, a poorly supported model feature, or a genuinely informative but unusual subject. The diagnostic identifies a dependency of the fit on the observed sample; it does not, by itself, show that the subject is invalid or that the model is wrong.

Model and Definition of a Case Deletion

Let \(y_{ij}\) be the biomarker measurement for subject \(i\) at time \(t_{ij}\), for \(j=1,\ldots,n_i\). Let \(T_i\) be the observed event or censoring time and \(\delta_i\) indicate whether the event was observed (\(\delta_i=1\)) or the observation was censored (\(\delta_i=0\)). A common shared-random-effects specification has a longitudinal model

and a conditional event hazard such as

Here \(\beta\) describes the population longitudinal trajectory, \(b_i\) is a subject-specific random effect with covariance matrix \(D\), \(\epsilon_{ij}\) is a measurement error, \(\lambda_0(t)\) is a baseline hazard, and \(\alpha\) is the current-value association parameter. In this specification, \(\exp(\alpha)\) is the hazard ratio for a one-unit increase in the underlying biomarker value, holding the other modeled covariates fixed. The exact link and longitudinal structure should be those of the fitted model; the diagnostic does not require this particular link.

The interpretation of a deletion analysis depends on the model being fitted appropriately. Standard assumptions include independent subjects; a correctly specified longitudinal and event model; the stated random-effects distribution; and conditional independence of the longitudinal measurement errors and event process given the modeled random effects and covariates. The observation and censoring processes must also be ignorable under the model, or be modeled when that assumption is not reasonable. These assumptions do not guarantee numerical stability, but without them the full-data fit itself may not represent the intended analysis.

If \(L_i(\theta)\) is the observed-data likelihood contribution for subject \(i\), with the random effects integrated out, the full log likelihood and the log likelihood after deleting subject \(i\) are

The notation \(\hat\theta\) denotes the full-data estimate and \(\hat\theta_{-i}\) the estimate after deletion, when that refit converges satisfactorily. The comparison should use the same model specification, parameterization, integration settings, convergence criteria, and relevant covariate definitions. Otherwise a change attributed to the subject may instead reflect a change in the fitting procedure.

Quantifying Changes in Estimates

For a scalar parameter \(\psi\), a simple diagnostic is the deletion difference \(\Delta_i=\hat\psi_{-i}-\hat\psi\). To put the difference on a scale that accounts for the precision of the full-data estimate, one can also calculate

This is a sensitivity measure, not a formal test statistic: it does not have a general reference distribution under the null hypothesis of no influence, and there is no universal cutoff that classifies a subject as influential. A large absolute value signals a change worth investigating, with its importance judged in context. For a vector of parameters, a scale-adjusted overall comparison can use the full-data estimated covariance matrix \(V\):

This summary depends on the parameterization and the quality of \(V\); it should not replace examination of the individual parameters. For random-effect standard deviations, compare standard deviations directly or compare their logarithms. A percentage change in a variance is also interpretable. Correlations and other constrained parameters require similarly careful scales; raw differences can be misleading near their boundaries.

Why a Subject Can Affect a Joint Fit

A subject can be influential through a distinctive biomarker trajectory, an unusual event time, or the combination of the two. For example, an early event paired with a high estimated biomarker level may contribute substantial information about the association. A subject with a long, atypical measurement history may affect the estimated random-effect distribution even if the event is censored. These are possible mechanisms, not diagnoses: because the joint likelihood combines the two outcome processes through shared latent effects, a whole-subject deletion cannot isolate the effect of the biomarker history from the effect of the event record.

The relationship to likelihood curvature gives a useful justification for looking at deletions. Let \(U(\theta)\) be the full-data score, the derivative of \(\ell(\theta)\), and let \(U_i(\theta)\) be subject \(i\)’s observed-data score contribution. At an interior full-data maximum, \(U(\hat\theta)=0\). The score after deleting \(i\), evaluated at the full-data estimate, is therefore

Write \(H_{-i}\) for the derivative of the deleted-data score, evaluated at \(\hat\theta\). A first-order Taylor expansion of the deleted-data score around \(\hat\theta\) gives

If \(J_{-i}=-H_{-i}\) is the deleted-data observed information matrix, the same approximation is \(\hat\theta_{-i}-\hat\theta\approx-J_{-i}^{-1}U_i(\hat\theta)\). A large subject score, or weak curvature in a parameter direction, can therefore lead to a large deletion change. This is a local approximation, not a substitute for refitting: joint-model likelihoods can be nonlinear, and a deletion can change the curvature or move the estimate to a different region.

Convergence Is Also Part of the Diagnostic

A deletion can affect not only parameter values but whether the model can be fitted reliably. Record whether each refit converges under the prescribed criteria, whether the reported solution is reproducible from reasonable starting values, and whether its gradient and uncertainty calculations are acceptable under the course’s established diagnostics. A failed refit has no trustworthy \(\hat\theta_{-i}\) to compare. Do not treat its last iterate as a deletion estimate.

Practical Comparisons and Failure Modes

A useful deletion analysis keeps the subject as the unit of removal and compares the same quantities across refits: the association parameter and its uncertainty, important random-effect standard deviations and correlations, and the convergence status. It is often informative to rank subjects by the absolute standardized change in a key parameter, then inspect the records and refit results for the largest changes. Such a ranking is a screening device, not a decision rule.

Several pitfalls deserve particular attention:

• Calling an influential subject an outlier. A large deletion change says the fit is sensitive to the record, not that the record is incorrect. Check the source data and context before considering any exclusion.
• Deleting visits but retaining the event record. That is a different sensitivity analysis and may break the intended subject-level likelihood contribution. The case-deletion unit here is the complete subject.
• Comparing unverified fits. A failed or materially different numerical solution should not be treated as a valid estimate. Keep fitting settings fixed and assess convergence before interpreting changes.
• Using a universal influence cutoff. A standardized change is descriptive, not a test with a universal threshold. Consider scientific importance, uncertainty, model behavior, and the full pattern of deletions.
• Attributing the change to only one data source. Removing a subject removes both the longitudinal and event information. The deletion alone cannot determine which component caused the shift.

Case deletion is especially useful when an association estimate, random-effect variance, or convergence result seems fragile. It has limits: many refits can be computationally expensive; small samples can make each deletion consequential; and deleting a subject changes the data rather than simulating a new sample. It is a targeted sensitivity analysis, not a cure for misspecification or weak identification. The earlier material on Bootstrap Instability In Joint-Model Estimates addresses a different question: variability across resampled data sets rather than the effect of removing a particular observed subject.