Diagnosing Time-Local Association Misspecification

Why Look for Association Misspecification in Time Windows?

A joint model can fit the overall relationship between a biomarker and event risk reasonably well while missing when that relationship is most apparent. For example, a model using the current fitted biomarker value might understate risk after a recent biomarker deviation, or the assumed association might work early in follow-up but not later. A time-local diagnostic asks whether recent biomarker residuals and event residuals align unusually within a specified part of follow-up.

The diagnostic is a targeted check, not a search for a causal effect. Its interpretation depends on the fitted longitudinal and event components, the construction of the residual summary, and the risk set at each time. The earlier material on association-link residuals provides the broader residual-association context; here the emphasis is on localizing possible lack of fit in time.

Model and Time-Local Residual Summary

Let \(Y_{ij}\) be the biomarker measurement for subject \(i\) at time \(t_{ij}\), and let \(m_i(t)\) be the fitted latent mean trajectory under the longitudinal component. A standardized longitudinal residual at a measurement time can be written \(r_{ij}=\{Y_{ij}-\widehat m_i(t_{ij})\}/\widehat\sigma\), where \(\widehat\sigma\) is the fitted residual standard deviation. Thus \(r_{ij}\) is measured in residual standard-deviation units.

For a diagnostic at time \(t\), summarize only measurements observed strictly before \(t\). One example is an exponentially weighted recent residual average:

where \(\kappa>0\) controls how quickly older measurements lose weight, \(\Delta>0\) defines the recent-history window, and \(I(\cdot)\) is the indicator function. Define \(q_i(t)\) only when at least one eligible measurement is available. A positive value means that recent biomarker observations tend to be above the fitted trajectory; a negative value means they tend to be below it. Other prespecified summaries, such as a recent residual slope, are possible, but they answer different diagnostic questions.

Let \(N_i(t)\) count subject \(i\)'s events by time \(t\), and let \(Y_i(t)\) indicate whether that subject is still under observation and at risk just before \(t\). Under a fitted joint model, write the conditional event hazard as

where \(b_i\) denotes the subject-specific latent effects, \(w_i\) denotes event-component covariates, and \(g_i(t)\) is the fitted biomarker association representation, such as the current latent value. This diagnostic does not replace that association term. It checks whether adding a local contribution from \(q_i(t)\) would improve the event component in a selected time window.

Derivation and Assumptions

For the derivation, temporarily treat the latent effects and model parameters as known. Suppose the alternative conditional hazard is \(h_i(t\mid b_i)\exp\{\beta_W a_W(t)q_i(t)\}\). The event-process log likelihood, omitting terms not involving \(\beta_W\), is

Differentiating with respect to \(\beta_W\) and setting \(\beta_W=0\) gives

The expression in braces is the event martingale increment under the correctly specified null model. In this known-parameter, conditional formulation, its mean is zero given the relevant past history. A positive score is evidence of positive weighted alignment: for example, positive \(q_i(t)\) paired with an event excess, and/or negative \(q_i(t)\) paired with an event deficit. A negative score reflects the reverse alignment. The score's sign alone does not say which pattern produced it; inspect its time- and subject-specific contributions.

This derivation relies on a correctly specified null hazard and risk process, conditionally independent censoring given the modeled history, and a predictable \(q_i(t)\): its value at time \(t\) must be computable from information available before \(t\). The longitudinal residual summary should also be meaningfully standardized and its construction prespecified. Using future biomarker measurements to calculate a residual summary at an earlier time can break predictability and create misleading alignment.

In practice, latent effects and parameters are estimated, and subject-specific hazards are not directly observed. A marginal joint-model score accounts for latent effects by integrating over their distribution; equivalently, an observed-data score can be formed by taking the conditional expectation of the complete-data score given the observed data. Plugging in estimated effects and hazards is a useful exploratory approximation, but its uncertainty must account for parameter estimation, latent-effect uncertainty, and the fact that \(q_i(t)\) is itself constructed from data.

Worked Calculations

Interpreting Patterns and Checking the Diagnostic

A useful assessment starts with a plot over follow-up of the recent residual summaries and event-residual contributions, alongside the chosen window boundaries. Individual martingale residual increments can be positive or negative; the score weights them by \(q_i(t)\), so a cumulative score curve can reveal when alignment accumulates. Examine whether the signal is concentrated in a few subjects, whether it changes sign, and whether the chosen window has adequate risk-set support.

The window should have a scientific or modeling rationale. If many windows, decay rates, or residual summaries are tried and only the largest score is reported, chance selection can make the apparent evidence too strong. Prespecify a small set of windows or calibrate the maximum over the search, for example by simulating or bootstrapping under the fitted null and repeating the entire search and model-fitting procedure.

Common warning signs include too few at-risk subjects in part of the window, sparse biomarker measurements, unstable residual summaries, and abrupt changes in observation or censoring patterns. Also check whether a broad time-varying hazard problem could explain the apparent association signal: a misspecified baseline hazard or omitted time-varying covariate can induce event residual patterns that correlate with biomarker residuals. The event-process assumptions and censoring model therefore matter, not only the longitudinal residuals.

How This Check Differs From Alternatives

The methods are complementary. A local score provides a focused direction and time window; broader residual checks help identify competing explanations, while joint predictive checks assess whether the fitted model reproduces larger-scale features of both processes.

Common Mistakes and Limitations

• Reading a positive score as only “more events after positive residuals.” Negative \(q_i(t)\) paired with fewer events than predicted also contributes positively. Inspect the product \(q_i(t)\{dN_i(t)-Y_i(t)\widehat h_i(t)dt\}\).
• Calling the bracket the variance. The integrated squared score weight is a predictable quadratic variation. Its expectation is the martingale's unconditional variance under the stated conditions; the observed bracket is commonly used as an information or variance estimate, not an identity for that variance.
• Using residuals that include future information. A residual summary at \(t\) based on biomarker observations after \(t\) is not a predictable history summary. Use past-only residuals for this construction, or treat the resulting analysis as a different, non-martingale diagnostic and calibrate it accordingly.
• Treating an exploratory score as a p-value. A plug-in score and bracket do not automatically account for estimated parameters, latent effects, generated residuals, or searching across windows. Use an appropriate marginal score calculation or a null resampling procedure that refits the model and repeats the diagnostic.
• Over-interpreting the result. A detected local alignment can arise from misspecified hazard structure, unmeasured time-varying predictors, residual-model problems, or chance. It does not establish that biomarker changes cause events.

This diagnostic is limited by its chosen history summary and time window: it can miss associations that operate through a different lag, a nonlinear transformation, or a longer-term biomarker history. It is also less informative when measurement schedules are sparse or the risk set is small. A weak score is not proof of adequate association specification.