Measurement Error in Last-Observed Biomarkers

Why the Latest Measurement May Not Be the Biomarker Value We Need

A landmark prediction uses information available for people who are still event-free at a specified time. Often, one of those predictors is the most recently recorded biomarker. But a laboratory result is not the patient’s exact underlying biomarker value: it can vary because of assay imprecision, sample handling, or other measurement conditions. Consequently, a model may estimate risk using a noisy proxy for the biomarker level relevant at the landmark.

There are two distinct discrepancies to keep separate. Measurement error is the difference between a recorded assay result and the underlying biomarker value at the time of that assay. Staleness is the difference between the underlying value at the assay time and the value at the landmark. If the latest measurement was taken before the landmark, both can matter. This tutorial focuses on measurement error; it does not model how visit timing is generated.

For a time-to-event application, \(Y_i\) could indicate whether the event occurs within a specified horizon after \(s\). The landmark risk is conditional on having survived event-free to \(s\), and is based on the available prediction information at that time. In practice, the outcome may instead be modeled using a survival regression. The linear result below clarifies one mechanism by which error can affect estimates; it is not a general correction formula for survival models.

What Classical Measurement Error Does in a Linear Risk Model

Suppose the target outcome is a fixed-horizon event indicator and, among landmark survivors, its conditional mean given the true biomarker is \(E(Y_i\mid X_i)=a+\beta X_i\). Assume the classical-error conditions in the definition, finite variances, and \(\operatorname{Var}(X_i)>0\). A linear regression using \(W_i\) instead of \(X_i\) has population slope equal to the linear-projection slope of \(Y_i\) on \(W_i\).

The equalities require more than an error with overall mean zero. Independence of \(\epsilon_i\) and \(X_i\) gives \(\operatorname{Cov}(X_i,\epsilon_i)=0\). The conditional-mean assumption gives \(\operatorname{Cov}(\epsilon_i,Y_i)=0\): since \(E(Y_i\mid X_i,\epsilon_i)=a+\beta X_i\), the error carries no additional outcome information after \(X_i\) is known. Also, \(\operatorname{Cov}(X_i,Y_i)=\beta\operatorname{Var}(X_i)\). Substitution yields the displayed slope.

The quantity \(\lambda\), between zero and one when the error variance is positive, is the reliability ratio under this model. Thus, the linear-projection slope is pulled toward zero. This is often called regression dilution or attenuation. The result concerns the slope under these particular assumptions; it does not imply that every risk prediction is simply multiplied by \(\lambda\), or that an error-contaminated estimate is always closer to zero in a more complex model.

Replicates and Validation Information

When independent replicate assays are available from the same sample or time point, averaging them can reduce technical measurement error. Suppose \(W_{ir}=X_i+\epsilon_{ir}\) for replicate \(r=1,\ldots,k\), with each error having mean zero and variance \(\sigma_\epsilon^2\). If the replicate errors are mutually independent and independent of \(X_i\), their average has error variance \(\sigma_\epsilon^2/k\). This gain does not apply to shared systematic error or to genuine short-term biological variation that is part of the target quantity.

Validation information can also help. For example, a subset of patients might have both the routine assay \(W_i\) and a more accurate reference measurement of \(X_i\). Such data can help estimate the error variance, reliability, or conditional distribution of \(X_i\) given \(W_i\). A correction then depends on the outcome model and the information collected; the reliability ratio alone is not a universal recipe for individual risk prediction.

Why Survival-Model Effects Need More Care

A landmark Cox model might relate the post-landmark hazard to a noisy biomarker through a term such as \(h_i(t\mid T_i>s,W_i,Z_i)=h_{0,s}(t)\exp\{\gamma W_i+\theta^\mathsf{T}Z_i\}\), for \(t>s\). Here \(Z_i\) denotes other predictors and \(h_{0,s}\) is the landmark-specific baseline hazard. Replacing a true biomarker with a noisy one can weaken or otherwise distort its estimated association and alter predicted risks.

However, the reliability-ratio formula derived above is not generally the correction for \(\gamma\) in a Cox model. Survival probabilities involve a nonlinear transformation of the hazard, and averaging over uncertainty in \(X_i\) does not generally preserve the same proportional-hazards form. Consequently, there is no general rule that a Cox coefficient is attenuated by exactly \(\lambda\), or that a patient’s risk estimate can be corrected by dividing it by \(\lambda\). The appropriate method depends on the assumed error distribution, the survival model, and the prediction target.

One practical alternative is to use replicate or validation data to estimate the measurement process and incorporate that information into an error-adjusted analysis. Another is to model a latent biomarker trajectory and connect the event process to that underlying trajectory, rather than treating one assay as error-free. A latent-trajectory approach can represent measurement error and the time course jointly, but requires stronger modeling choices and adequate longitudinal data. It may be unnecessary when assay error is small or when the prediction target is deliberately defined using the routinely observed value.

Assumptions, Diagnostics, and Common Pitfalls

The classical-error calculation applies to the eligible landmark cohort, not automatically to the original study population. Its conditions must be credible among people with \(T_i>s\). For the simple derivation, the error must be centered and unrelated to the true biomarker, and the event outcome must not depend on the error once the true biomarker is known. If assay error varies with biomarker level, depends on batch or health status, or shares a cause with the outcome, the displayed reliability ratio need not describe the bias. Likewise, a biomarker measured before \(s\) is not converted into the value at \(s\) simply by correcting technical assay error.

A single assay per patient generally cannot separate the true between-patient variation from measurement-error variation. Useful diagnostics therefore rely on additional information: replicate assays, reference measurements, laboratory quality-control data, or comparison of error patterns across batches and biomarker ranges. Inspect whether replicate differences are centered near zero and whether their spread is roughly constant. Large or systematic differences, floor or ceiling effects, and evidence of shared replicate errors are warnings against the simplest model. Without replicates, validation data, or credible external assay information, a reliability estimate is not identified from the observed biomarker values alone.

A common mistake is to say, “The measurement error has mean zero, so the noisy biomarker is unbiased and its regression coefficient is unbiased.” Mean-zero error does not make the regression slope unbiased: under classical error, the denominator \(\operatorname{Var}(W_i)\) includes error variance, producing the reliability-ratio result. Worse, an overall mean of zero alone does not imply \(\operatorname{Cov}(X_i,\epsilon_i)=0\) or \(\operatorname{Cov}(\epsilon_i,Y_i)=0\). The correction is to state the independence and conditional-mean assumptions explicitly, and then apply the result only to the model for which it was derived.

The practical choice is therefore not simply “correct” versus “ignore.” Replicates reduce independent technical noise; validation data can identify features of the measurement process; a latent-trajectory model can represent the underlying biomarker over time; and a routine-value landmark model may be reasonable if the target is explicitly prediction from routinely available measurements. Each choice has limitations. Replicates may be costly or unavailable, validation measurements may be imperfect, and latent models can be sensitive to their trajectory and error assumptions. The analysis should state which biomarker quantity the prediction is intended to use.