Outlying Trajectories and Link-Sensitivity Results

Why Look for Influential Trajectories?

Two association links can give different conclusions even when they are fitted to the same longitudinal and event-time data. Sometimes that difference reflects a broad pattern in the cohort. Sometimes it depends heavily on a few subjects whose biomarker histories are unusual, especially when those histories coincide with event times that strongly inform the hazard. A trajectory plot can reveal unusual histories; case-deletion refits assess whether those histories matter to the fitted comparison.

The central distinction is between outlying and influential. A subject can have an extreme biomarker history but little effect on the association estimates. Conversely, a history that does not look extreme may be influential if its measurements and event time provide information that is unusual relative to the rest of the sample. Neither a plot nor a deletion result alone establishes that a subject is erroneous or should be excluded.

What Is Deleted, and What Is Compared?

Let \(Y_{ij}\) be the biomarker measurement for subject \(i\) at visit time \(t_{ij}\), \(T_i\) the event or censoring time, and \(\delta_i\) the event indicator. Let \(m_i(t)\) denote the subject-specific underlying biomarker trajectory in the shared random-effects model. A joint model combines a longitudinal model for the \(Y_{ij}\) values with an event-time model whose hazard depends on a specified function of \(m_i(t)\), possibly alongside covariates.

Suppose the analysis compares candidate link specifications indexed by \(k\). Examples include a current-value link, which uses the current \(m_i(t)\), and a cumulative-exposure link, which uses an accumulated history. The deletion diagnostic does not change those definitions. It asks whether their fitted results are driven by particular subjects.

A fair comparison keeps the longitudinal and event-time components, covariates, baseline-hazard specification, random-effects structure, and estimation settings the same across full-data and deletion fits, except for the candidate association link being compared. A deletion refit that changes other modeling choices at the same time cannot isolate the influence of that subject.

For link \(k\), let \(\hat\alpha_k\) be the full-data association estimate and \(\hat\alpha_{k(-i)}\) its estimate after deleting subject \(i\). A basic deletion change is \(\Delta_{ik}=\hat\alpha_{k(-i)}-\hat\alpha_k\). Because association parameters can have different scales under different links, compare \(\Delta_{ik}\) within a link, for example by dividing by that link’s full-data standard error. Do not interpret raw coefficients from different links as directly comparable just because they are both called association parameters.

To assess whether deletion changes the difference between links, define a common-scale contrast in advance. For example, specify two biomarker histories \(A\) and \(B\), hold other covariates fixed, and calculate the fitted log-hazard contrast between them under each link. Denote that contrast by \(C_k\). Then compare \(C_k\) across links in the full-data fit and in each deletion fit. The difference between those link comparisons is itself a deletion diagnostic. The histories and scale must be chosen so the contrast has the same scientific meaning for every link.

Trajectory Displays That Support the Diagnosis

A useful display shows each subject’s observed measurements against time, along with a fitted trajectory \(m_i(t)\) and a marker for the event or censoring time. Where subjects overlap, show a subset of histories or use a display that identifies subjects without obscuring the group pattern. A second panel can show standardized residuals from the longitudinal model, so that a high or low trajectory is distinguished from a history that is poorly described by the model.

For a clean comparison of biomarker histories, the fitted trajectories can be obtained from the common longitudinal submodel, using the same rule for every subject and link. If trajectories are instead estimated conditional on all the data in each joint model, the event-time information can alter them and the displays may differ by link. That can be useful, but it should be stated; it is not a link-neutral picture of the observed biomarker pattern.

A plot should help identify what to inspect, not decide the analysis by eye. A subject with sparse measurements may have a highly uncertain fitted trajectory. A smooth model-based curve is not a direct observation between visits. Also check whether an apparent extreme reflects a data-entry or measurement issue, a plausible clinical course, or a limitation of the trajectory model. These possibilities call for different responses.

Why a Case-Deletion Change Can Be Large

A short derivation explains why an influential subject can have a large deletion effect. Let \(\theta\) collect all parameters in one candidate joint model, and write its log likelihood as a sum of subject-level contributions, \(\ell(\theta)=\sum_i\ell_i(\theta)\). The score is \(U(\theta)=\sum_i U_i(\theta)\), where \(U_i(\theta)\) is subject \(i\)’s score contribution. At the full-data maximum likelihood estimate \(\hat\theta\), \(U(\hat\theta)=0\).

After deleting subject \(i\), the score at the old estimate is \(U_{(-i)}(\hat\theta)=-U_i(\hat\theta)\). Let \(J_{(-i)}\) be the observed information matrix for the remaining subjects, the negative derivative of their score. A first-order Taylor expansion around \(\hat\theta\) gives

Rearranging yields the approximate deletion shift

This approximation shows two ingredients: the subject’s score contribution and the information available from the remaining subjects. A large contribution can matter more when the remaining data provide limited information in the relevant parameter direction. Since joint models estimate nuisance parameters as well as association parameters, the exact effect can involve changes in the longitudinal, random-effects, and event-time components. The approximation is an explanation and screening aid, not a substitute for refitting the model.

Interpreting and Checking the Results

There is no universal cutoff at which a deletion change becomes “influential.” Report the change in the parameter’s own units, its size relative to the full-data standard error as a scale aid, and any change in a common-scale contrast between links. Inspect the trajectory and event-time marker for the subjects with the largest changes. Repeat the refits for all subjects when feasible; selecting only visually unusual cases can miss influential subjects whose histories do not look extreme.

A useful summary table lists, for each candidate link, the full-data estimate, the largest absolute deletion changes, the corresponding subjects, and the deletion estimates. A companion trajectory plot helps explain the pattern. If a refit fails to converge or lands at a different solution, do not record its estimate as an ordinary deletion effect. Investigate starting values, numerical integration, boundary estimates, and whether the model is identifiable in the reduced sample; report the failure rather than silently omit it.

Compare case deletion with other approaches carefully. Residual plots can reveal a poor fit to the biomarker histories but do not measure the effect on link estimates. A trajectory plot displays patterns but cannot establish influence. Excluding a subject permanently or replacing observed values with trimmed values changes the analysis and can introduce bias; such steps require a defensible data-quality or scientific rationale. More elaborate robust-model or resampling approaches answer different questions and are not interchangeable with this targeted diagnostic.

Common Pitfalls

• Calling every unusual curve influential. Unusualness is not a deletion effect. Refit the model and quantify the change.
• Deleting only a visit. This leaves the subject’s other longitudinal and event-time information in the model. For subject-level influence, remove the whole subject contribution.
• Comparing unlike coefficients directly. Link parameters can use different exposure scales. Compare within-link shifts or use a clearly defined common-scale hazard contrast.
• Treating a large shift as a reason to exclude. First investigate data quality and model fit. A plausible influential history is part of the evidence, not automatically an error.
• Ignoring instability in the refit. Nonconvergence or boundary estimates can make a deletion comparison uninterpretable; diagnose and report the problem.