Influential Events In The Survival Component

Why Event Influence Needs More Than a Large Residual

A subject may have an unusual event time without materially changing the fitted survival model. Conversely, a subject with a modest-looking event residual may affect many event-time comparisons because that subject remains in several risk sets. Assessing influence therefore requires asking not only whether an event history is surprising under the fitted model, but also whether the survival-component estimates change when that history is removed.

This tutorial focuses on the survival component of a joint model. The longitudinal measurements matter because the fitted event hazard may depend on a subject’s latent trajectory, such as a current underlying marker value. The diagnostic question is narrower: does a particular subject’s observed event or censoring history have disproportionate impact on the estimated survival parameters, given the longitudinal part of the fitted model?

Event Contributions and Residual Evidence

For subject \(i\), let \(T_i\) be the observed follow-up time, the minimum of the event time and censoring time, and let \(\delta_i=1\) if the event is observed and \(\delta_i=0\) if the observation is right-censored. Let \(Y_i\) denote the longitudinal measurements and \(b_i\) the subject-specific random effects. A common joint-model survival component specifies a hazard conditional on those random effects:

Here \(h_0(t)\) is the baseline hazard, \(w_i\) contains survival covariates, and \(m_i(t;b_i)\) is a longitudinal-model quantity, often the latent marker value at time \(t\). The parameter \(\alpha\) describes the association between that quantity and the event hazard. Conditional on \(b_i\), the event-history likelihood contribution is

This expression assumes the specified hazard model is appropriate and that censoring is non-informative conditional on the modeled information: the censoring mechanism must not add unmodeled information about the event time. Delayed entry, if present, must also be represented in the risk sets or likelihood. In a joint model, \(b_i\) is unobserved; the event contribution is integrated over its distribution in the fitted model, rather than treating a posterior mean of \(b_i\) as known without uncertainty.

A useful residual summary is the event martingale residual, defined here using a fitted cumulative hazard:

A positive value means the subject experienced an event when the fitted model expected fewer cumulative events for that subject; a negative value means the fitted cumulative hazard exceeded the observed event count. For censored subjects, \(\delta_i=0\), so the residual is \(-\widehat H_i(T_i)\). In a joint model, \(\widehat H_i\) should account for uncertainty about \(b_i\), for example through the model’s integration or posterior-predictive calculation. A large residual is a reason to inspect a subject, not proof that the subject is influential.

Risk-Set Contributions

At an event time, the event’s information depends on who else was still under observation and event-free. In a Cox-type risk-set calculation, let \(z_i(t)\) be a scalar covariate or fitted longitudinal association quantity, and let \(\beta\) be its coefficient. For a single event at time \(t_j\), the event-time score contribution is

where \(z_{(j)}\) is the event subject’s value and \(R(t_j)\) is the risk set immediately before \(t_j\). The second term is the risk-set weighted mean. A subject can affect this score in two ways: by having the event at \(t_j\), or by being in the risk set and changing its weighted mean. Thus a censored subject can influence estimates at earlier event times, even though that subject has no event residual term \(\delta_i=1\).

The Cox calculation is a useful way to understand risk-set leverage, but a joint-model survival likelihood may instead be based on a parametric or semiparametric hazard conditional on random effects. In either case, risk-set influence must respect the actual follow-up: a subject belongs to a risk set only while that subject is under observation and event-free, with entry and censoring handled according to the design and model. Tied events require the tie-handling method used in the fit; changing that method for a diagnostic comparison would confound the comparison.

Leave-One-Subject-Out Comparisons

A direct diagnostic is to remove one subject’s event history, refit the survival component, and compare the resulting estimates with the full-data estimates. For a survival parameter \(\theta\), write \(\widehat\theta\) for the full-data estimate and \(\widehat\theta_{(-i)}\) for the estimate after deleting subject \(i\). A simple standardized change is

The denominator is the full-data standard error, so \(D_i\) reports the deletion shift in standard-error units. Its sign indicates the direction of the shift; its magnitude indicates how large that shift is relative to estimated sampling uncertainty. There is no universal cutoff that makes a subject influential in every application. Consider the size of the shift, the parameter’s scientific role, uncertainty, and whether the change is driven by a defensible feature of the observation or by a model/data problem.

In a joint model, a survival-component deletion should be defined explicitly. One useful conditional diagnostic removes subject \(i\)’s event/censoring record from the survival likelihood and risk sets while holding the fitted longitudinal model and its representation of that subject’s trajectory fixed. For example, the event likelihood can be averaged over the fitted distribution of \(b_i\) given \(Y_i\). This asks how the survival estimates respond to that event history, conditional on the longitudinal component. A full deletion that also refits the longitudinal model and updates the joint distribution answers a broader question; it is not the same diagnostic.

Why the Deletion Approximation Works

A local score argument explains why a modest deletion effect can be approximated from a case’s contribution to the score. Let \(U(\theta)\) be the full-data score, with \(U(\widehat\theta)=0\), and let \(U_{(-i)}(\theta)\) be the score after deleting subject \(i\). Define the deletion score difference as \(u_i^{\mathrm{del}}(\theta)=U(\theta)-U_{(-i)}(\theta)\). At the full estimate, \(U_{(-i)}(\widehat\theta)=-u_i^{\mathrm{del}}(\widehat\theta)\). A first-order Taylor expansion around \(\widehat\theta\) gives

where \(I_{(-i)}\) is the observed information after deletion, the negative derivative of the deleted score. Substituting \(U_{(-i)}(\widehat\theta)=-u_i^{\mathrm{del}}(\widehat\theta)\) and solving yields

This is a local approximation, not a replacement for refitting. In risk-set models, deleting one person can alter score contributions at multiple event times, so \(u_i^{\mathrm{del}}\) is not generally just that subject’s event-time residual or a single independent likelihood term. If deletion causes a large estimate shift, changes the effective risk sets substantially, or moves the fit toward a boundary, the first-order approximation is especially unreliable.

A Practical Diagnostic Sequence

Diagnostics, Alternatives, and Limitations

Useful displays include event residuals against follow-up time, deletion changes indexed by subject, and summaries of which event-time risk sets contained each flagged subject. Repeatedly large deletion shifts concentrated in a short follow-up interval may point to sparse risk sets or a time-local hazard problem. A shift concentrated in the association parameter may warrant checking the longitudinal quantity entering the hazard and the assumed association structure. Convergence failures after deletion are themselves informative, but they do not automatically prove that the omitted subject was influential: the reduced data may simply provide too little information to identify the model.

Residual screening and case deletion answer different questions. Residuals describe discrepancy between an observed event history and its fitted expectation; risk-set calculations reveal how a subject participates in event comparisons; leave-one-out refits assess actual estimate sensitivity. Alternatives include local score/information approximations, which are faster but less reliable for large changes, and posterior predictive checks, which assess model-data compatibility rather than the effect of removing a particular subject. These approaches complement one another rather than substitute for one another.

Deletion diagnostics have important limits. They are sensitive to model specification, risk-set construction, and the chosen definition of deletion. A finite-sample change can reflect sparse information rather than a data error. Many subject-by-parameter comparisons can also produce apparently striking changes by chance, so avoid interpreting a single threshold as a universal decision rule. Finally, a conditional survival-component deletion does not measure the effect of removing the subject’s longitudinal observations or allowing the full joint model to re-estimate every component.

• Common mistake: “This subject has the largest event residual, so it must be the most influential.”
• Correction: A residual measures event discrepancy, whereas influence concerns the estimate change under deletion. Examine risk-set participation and refit the specified survival component.
• Common mistake: “A censored subject cannot affect event estimates because that subject had no event.”
• Correction: A censored subject may have been in risk sets at earlier event times and can change their denominators and score contributions.