Why Plugging Observed Marker Values Into the Hazard Fails

Why the Hazard Needs the Latent Trajectory

A marker measured over time is often intended to represent a subject’s changing health status. But the value recorded at a visit is not necessarily the subject’s underlying marker value: it may contain measurement error, and it says nothing directly about what the marker was between visits. These distinctions matter when the marker enters a time-to-event model through a nonlinear function such as an exponential hazard.

The central issue is not that observed measurements are useless. They provide information about the trajectory. The issue is treating one noisy, intermittent measurement as if it were the exact marker value governing the hazard at every time. In a joint model, the hazard is linked to the latent trajectory, while the longitudinal measurements are modeled as imperfect observations of that trajectory.

The hazard is an instantaneous event rate, conditional here on the latent trajectory and covariates. If the marker is measured in points, \(\alpha\) has units of inverse points, so \(\alpha m_i(t)\) is dimensionless. A plug-in approach instead inserts a recorded value \(Y_{ij}\), or a value carried forward or interpolated from measurements, in place of \(m_i(t)\). That substitution changes the model: a recorded value contains error and is observed only at selected times.

A basic assumption behind the displayed measurement equation is that the error has mean zero conditional on the latent trajectory, and that the hazard model describes event risk conditional on that trajectory and the stated covariates. Often the measurement errors are also assumed independent across visits and independent of the event process conditional on the latent trajectory. Those assumptions are substantive, not automatic. For example, error that increases as a patient becomes sicker or a visit schedule driven by unrecorded deterioration may require additional modeling.

Measurement Error Inside an Exponential

Even mean-zero measurement error does not disappear when the measurement is put inside an exponential. For a simple calculation, suppose that at a particular time the true marker is \(m\), the observed value is \(Y=m+\epsilon\), and \(\epsilon\sim N(0,\sigma^2)\). Hold the baseline hazard and other covariates fixed, and write \(C=h_0(t)\exp(\gamma^\top w)\). The latent-trajectory hazard is \(C\exp(\alpha m)\); the observed-value plug-in hazard is \(C\exp\{\alpha(m+\epsilon)\}\).

The normal moment-generating function gives the average plug-in hazard over repeated measurement errors:

Thus, under these assumptions, the error-averaged plug-in hazard exceeds the latent-trajectory hazard by the factor \(\exp(\alpha^2\sigma^2/2)\) whenever \(\alpha\ne 0\) and \(\sigma^2>0\). This follows from the convexity of the exponential function: averaging exponentiated noisy values is not the same as exponentiating their average. The factor describes an average over measurement errors at a fixed latent value. It does not say that every observed value produces an overestimate; a particular error may be positive or negative.

Sparse Visits Add a Timing Problem

Even if measurement error were negligible, measurements are usually recorded at visits rather than continuously. A plug-in hazard needs a marker value at each time a subject is at risk. A common shortcut is to carry the last observation forward. Another is to interpolate between visits. Either can differ from the latent value at the time of risk, particularly when the trajectory changes quickly, has curvature, or varies substantially between subjects.

The time discrepancy has a direct effect on the hazard. If a value used in place of the latent marker differs by \(\delta\), then, holding other terms fixed, the ratio of the plug-in hazard to the latent hazard is \(\exp(\alpha\delta)\). This identity makes clear why the same marker discrepancy can have a larger impact when the association parameter has greater magnitude.

What a Valid Comparison Requires

The latent-trajectory model and an observed-value plug-in model answer different modeling questions. In the joint model, longitudinal measurements inform \(m_i(t)\), while the event hazard is specified as a function of that latent trajectory. This is appropriate when the scientific target is the association of the underlying marker course with event risk and the measurement process is imperfect. The earlier material on the observed-data likelihood explains how marker records contribute when some measurements are missing; missingness and measurement error are distinct issues.

A plug-in analysis may be tempting because it can resemble an ordinary time-dependent survival model with a recorded marker as a covariate. But it treats the supplied marker value as if it were the relevant value at that risk time. If it is a noisy measurement, that ignores the error distribution; if it is carried forward, it additionally assumes the marker stays constant until the next visit. If it is interpolated, the result depends on the interpolation rule and does not automatically account for uncertainty in the trajectory.

A useful comparison is:

The joint model is not assumption-free. Its trajectory form may be too rigid, the error distribution may be misspecified, and the visit process may be informative. Sparse data can also leave substantial uncertainty about the latent marker between observations. Modeling the trajectory does not create information that was never collected; it makes explicit how the available information is being used.

Diagnostics and Common Pitfalls

Diagnostics should examine both the longitudinal fit and the event-model implications. Check whether fitted trajectories describe the observed marker patterns, including systematic residual trends, changing variability, and outlying measurements. Examine whether residual behavior differs over time or across risk groups; such patterns can signal that a simple independent, constant-variance error model is inadequate. For sparse visits, assess how much the inferred marker between visits depends on the assumed trajectory shape and compare predictions with observed values at later visits where possible.

For the hazard component, compare conclusions under plausible trajectory specifications and measurement-error assumptions. Large changes in the estimated association or fitted risk patterns are a warning that the data do not strongly determine the latent trajectory. Also consider whether visit timing is related to health status or impending events. If visits are scheduled more frequently when a subject worsens, treating the schedule as irrelevant can leave important information unmodeled.

• Wrong reasoning: “The measurement errors have mean zero, so plugging in the observed marker gives the same hazard on average.”
• Correction: The hazard is exponential in the marker. Even when \(E(\epsilon)=0\), generally \(E\{\exp(\alpha\epsilon)\}\ne1\); under normal error it equals \(\exp(\alpha^2\sigma^2/2)\).
• Wrong reasoning: “The latest observation is the marker value for all times until the next visit.”
• Correction: That is a piecewise-constant approximation, not a consequence of the measurement data. It can be especially misleading when the marker changes appreciably between visits.

A final caution is that the direction of distortion for an individual subject or an estimated association coefficient is not determined by the simple average-hazard calculation alone. Positive and negative errors occur, the visit pattern can be related to the trajectory, and fitting a survival model involves more than averaging hazards at a fixed latent value. The calculation establishes why the substitution is not generally justified; it does not supply a universal adjustment for every plug-in analysis.