Endoscopic Activity and Surgery in Crohn’s Disease

Why the Assessment Schedule and Endpoint Matter

Endoscopic activity scores summarize what is seen during an examination; bowel surgery records an event that may occur months or years later. A clinical question might be whether patients with persistently greater activity have a higher instantaneous rate of surgery. Answering it is not simply a matter of matching each patient’s last score to their surgery time. Examinations may be irregularly spaced, a patient’s disease activity changes between examinations, and an operation is a clinical decision as well as an outcome.

The example below is hypothetical. Let time zero be a defined baseline examination, let \(Y_{ij}\) be patient \(i\)’s endoscopic activity score at examination time \(t_{ij}\), and let \(T_i\) be time from baseline to the first qualifying bowel surgery. The event indicator \(\delta_i\) is 1 if that surgery occurs during follow-up and 0 if follow-up ends first for a specified reason. The endpoint must say which procedures qualify and how deaths or other events are handled.

For a first illustration, suppose the score is treated as approximately continuous and higher values indicate greater activity. A simple longitudinal submodel is a linear mixed model:

Here \(m_i(t)\) is patient \(i\)’s latent mean score at time \(t\); \(b_{0i}\) and \(b_{1i}\) are patient-specific deviations in level and slope, and \(\epsilon_{ij}\) is measurement error. For example, if time is measured in years, \(\beta_1\) and \(b_{1i}\) are in score points per year. A current-value survival submodel is

In this expression, \(h_0(t)\) is the baseline hazard, \(X_i\) contains specified baseline covariates, and \(\alpha\) describes association between latent current activity and the instantaneous hazard of the defined surgery endpoint. The hazard is a rate among patients who have not yet had the endpoint; it is not the probability of surgery by time \(t\). A positive \(\alpha\) means greater latent activity is associated with a higher hazard, conditional on the model and covariates. It does not establish that activity causes surgery.

Assumptions and What the Link Represents

The model is useful only when its assumptions are plausible enough for the intended analysis. The linear score model assumes that the mean trajectory is adequately represented by the chosen time function, that the random effects describe relevant between-patient differences, and that the remaining measurement errors have mean zero and an appropriate variance structure conditional on the model. A linear trend may be a poor description if scores change nonlinearly, have a strong floor or ceiling, or are intrinsically ordinal. In those cases, a flexible time function or an ordinal/count longitudinal model may be more appropriate.

The event model assumes that, conditional on the specified covariates and latent trajectory, the hazard is represented adequately by the chosen association structure and baseline hazard. A current-value link is not the only possible link: a scientific question might instead concern the latent slope, accumulated activity, or a combination. These choices define different associations and should be set before interpreting results. The model also requires a coherent event-time origin, a clear surgery definition, and correct recording of event and censoring times.

The usual joint likelihood assumes that the longitudinal measurements and event time are conditionally independent given the shared patient-specific effects and included covariates. This link can account for association induced by those shared effects. It does not, on its own, make the timing of endoscopies non-informative. If examinations are prompted by worsening symptoms, a patient’s observation process may itself depend on unobserved activity or impending surgery. Ignoring that process can distort the estimated trajectory and its association with surgery.

A further assumption concerns censoring. For a standard single-event survival model, censoring must be independent of the event time conditional on the model variables. If death prevents a later bowel surgery, it is a competing event, not ordinary loss to follow-up. Treating death as independent censoring can misrepresent the probability of surgery; the target must distinguish a cause-specific surgery hazard from the cumulative probability of surgery in the presence of competing events.

How the Joint Likelihood Connects Scores and Surgery

The reason for linking the two model parts can be seen from the likelihood contribution for one patient. Conditional on their random effects \(b_i\), the score data contribute their longitudinal density. The event-time contribution is the hazard at the observed time raised to the event indicator, multiplied by the probability of remaining free of the endpoint up to that time:

If surgery occurs, \(\delta_i=1\), and the event contribution is \(h_i(T_i\mid b_i)S_i(T_i\mid b_i)\), the event-time density. If follow-up is censored before surgery, \(\delta_i=0\), and the contribution is only \(S_i(T_i\mid b_i)\). The patient-specific effects are not observed, so the contribution used for estimation averages over their distribution:

This integration matters. A patient’s observed scores are noisy measurements of a latent trajectory, and that same trajectory enters the hazard. The event information can therefore inform the estimate of the trajectory, while the observed scores inform the patient-specific activity associated with the event. It is not equivalent to treating a single observed score as error-free.

Assessment Timing Is Part of the Analysis

Endoscopic examinations need not occur at evenly spaced visits. Suppose a patient is assessed at baseline and month 4, then not again until month 15. A model can use the recorded times directly; it should not pretend that the month-4 score was measured at month 12. Between examinations, the estimated \(m_i(t)\) is model-based interpolation or extrapolation, and its uncertainty generally grows as the gap from observed data increases.

Timing can also be informative. If symptoms lead clinicians to schedule an earlier examination, patients with high or worsening activity may have more measurements near a period of clinical deterioration. A standard joint score-and-surgery model accounts for the association between the measured score trajectory and surgery through its shared effects, but does not automatically model why each examination occurred when it did. Useful checks include plotting assessment times by follow-up and event status, comparing visit frequency across observed activity levels, and examining whether the schedule changes near surgery. If the observation process plausibly depends on unobserved activity, a model for visit intensity or a sensitivity analysis may be needed.

Surgery also ends the opportunity to observe later scores. This is not ordinary missingness that can be fixed by carrying the last score forward. In a joint model, the event contribution uses the patient’s survival history up to surgery, while their observed scores contribute information on the trajectory before the event. Post-surgery scores should not be treated as if they were pre-surgery observations of the same untreated course.

Diagnostics, Alternatives, and Limitations

A credible analysis checks both model components and the connection between them. For the score model, inspect observed-versus-fitted trajectories, residual patterns over time, the distribution of residuals, and whether fitted values systematically miss high or low scores. Look for floor or ceiling effects and score distributions inconsistent with a Gaussian error model. For the event component, assess whether the baseline hazard and covariate form are adequate, and check whether the chosen current-value association is plausible. Compare predicted and observed event patterns at clinically relevant times where possible.

The assessment schedule deserves its own diagnostic review: tabulate the number and timing of examinations, compare them by eventual surgery status, and identify large gaps or clusters of visits. A model may converge and still be poorly identified if there are few surgeries, little score variation, or sparse scores close to the event. Wide uncertainty intervals, sensitivity to the time function or endpoint definition, and unstable estimates are warnings against strong conclusions. Extrapolating a latent trajectory far beyond a patient’s last examination is particularly uncertain.

A joint model is most helpful when repeated scores are noisy and their underlying trajectory is meaningfully associated with event risk. It is not automatically preferable in every dataset. A simple time-dependent model may be adequate for a question about the association with the most recently observed score, provided its measurement and timing limitations are explicit. A joint model can also be fragile when there are few events relative to model complexity. Neither approach turns an observational association into a causal effect: treatment changes, clinician decisions, comorbidities, and unmeasured severity may affect both scores and surgery.

Finally, endoscopic activity and bowel surgery are different kinds of information. The score is an intermittently observed assessment; surgery is a time-to-event outcome influenced by disease, patient preferences, treatment response, access to care, and clinical judgment. A conclusion should name the modeled score, time scale, surgery endpoint, follow-up population, and association scale. “Higher latent activity was associated with a higher hazard of the specified surgery endpoint” is more defensible than saying that the score predicts or causes surgery without qualification.