Delayed Entry And Left Truncation in Multi-Omics Integration

Why Entry Time Changes the Analysis

In many multi-omics studies, follow-up does not begin at the biological or clinical time origin. A participant may enter a cohort years after diagnosis, or a specimen may be collected only when a patient reaches a clinic. If the outcome is time from diagnosis, that participant is observed only after having remained event-free long enough to enroll. This is delayed entry, also called left truncation: people who experience the event before they could be observed are absent from the data.

This is different from right censoring. Right censoring means an enrolled participant's event time is not observed beyond their last follow-up. Delayed entry means a participant is not at risk of being observed in the study before their entry time. An analysis that treats everyone as observed from the original time origin puts people into risk sets before they were eligible to be seen. The resulting risk sets and likelihood are wrong, and associations with omics measurements can be distorted by selection into the observed cohort.

The time origin must match the scientific question. For example, time since diagnosis and time since enrollment are distinct scales and define distinct survival quantities. The earlier material on choosing survival estimands addresses that choice; here, assume the estimand uses a common origin before or at entry. Let \(X_i\) denote the omics covariates and any other modeled covariates. Write \(S(t\mid X_i)=P(T_i>t\mid X_i)\), \(f(t\mid X_i)\) for the event-time density, and \(\lambda(t\mid X_i)=f(t\mid X_i)/S(t\mid X_i)\) for the hazard.

Conditional Likelihood After Delayed Entry

Because a subject appears in the data only after surviving to \(L_i\), their event-time contribution must be conditional on \(T_i>L_i\). For a subject with an observed event at \(Y_i\), the conditional density is \(f(Y_i\mid X_i)/S(L_i\mid X_i)\). For a subject censored at \(Y_i\), the contribution is the conditional probability of surviving beyond \(Y_i\), namely \(S(Y_i\mid X_i)/S(L_i\mid X_i)\). These expressions presume that censoring is independent of the event time conditional on the modeled covariates and relevant observation history.

Here \(\theta\) denotes model parameters. The denominator is not an optional adjustment: it accounts for the fact that entry requires event-free survival through \(L_i\). In hazard form, the same contribution is \(\lambda(Y_i\mid X_i;\theta)^{\delta_i}\exp\{-\int_{L_i}^{Y_i}\lambda(u\mid X_i;\theta)\,du\}\). The exponent integrates hazard only over the interval the person was actually observed at risk.

The entry assumption is often called independent truncation. In its simple form, conditional on \(X_i\), entry time \(L_i\) is independent of event time \(T_i\) in the source population. More generally, analyses rely on an appropriate quasi-independence condition over combinations that could be observed, \(T_i>L_i\). If entry depends on unmeasured prognosis even after conditioning on \(X_i\), the observed survivors need not represent the event process the model claims to describe. Correcting risk sets alone cannot repair that problem.

Risk Sets for Regression Models

For a Cox proportional-hazards model, the hazard at time \(t\) is \(\lambda(t\mid X_i)=\lambda_0(t)\exp(\beta^\mathsf{T}X_i)\), where \(\lambda_0(t)\) is an unspecified baseline hazard. With delayed entry, the risk set just before event time \(t\) contains only subjects who have entered and have not yet had an event or been censored: \(R(t)=\{j:L_j<t\leq Y_j\}\). The strict inequality reflects that subjects must enter before the event time to be at risk for it; when data record entry and event at the same time, the protocol must specify their ordering. Under the proportional-hazards model, the conditional probability that subject \(i\) is the one event at \(t\), among the risk set, is its relative hazard divided by the sum of relative hazards in that set. The baseline hazard cancels.

Thus delayed entry changes who belongs in the denominator at each event time. It does not mean that entry time is itself an event or that follow-up should automatically be reset to zero at enrollment. The expression above is for an event with no tied failures; ties require a suitable tied-event method, and the risk set must still respect entry.

Diagnostics, Interpretation, and Alternatives

A practical analysis should first audit the time scale and the observation intervals. Confirm that \(L_i\), \(Y_i\), and the event indicator use the same origin and units; verify \(L_i<Y_i\) under the study's event-time convention; and inspect whether the risk-set software actually excludes not-yet-entered subjects. Compare entry-time distributions across omics groups and clinical covariates. Large differences do not by themselves disprove independent truncation, but they signal that selection and covariate adjustment deserve careful consideration.

Ignoring delayed entry can bias an omics-survival association because observed subjects are survivors to their entry times. The resulting selection can induce or alter associations between an omics feature and prognosis, especially when entry depends on the feature, correlated clinical factors, or survival. The direction of bias is not universal: it depends on the event-time distribution, entry mechanism, covariates, and model. Omics assays measured at enrollment create an additional interpretation issue: treating a measurement obtained after the time origin as if it were a baseline exposure may misrepresent when the predictor was present and can involve survivor selection. A delayed-entry likelihood accounts for observation beginning late; it does not turn a later measurement into a valid measurement at time zero.

The main alternative is to define time zero as enrollment and analyze time since enrollment. That can be appropriate if the scientific question concerns prognosis from enrollment, and if the cohort and covariates are defined consistently at that time. It does not estimate survival since diagnosis. Conversely, resetting the clock to enrollment merely to avoid modeling delayed entry changes the estimand rather than correcting the original-time analysis. If entry is informative, a model for the joint entry and event processes, a defensible sampling design, or sensitivity analyses may be needed; standard delayed-entry methods alone are insufficient.

Other warning signs include few subjects in risk sets at late times, little overlap in entry histories across omics groups, and estimates that change markedly when the time origin or entry assumptions are varied. These can indicate weak information or sensitivity to selection assumptions. Standard errors do not account for an incorrect entry model. Report the time origin, entry definition, censoring assumptions, and the population to which the association is intended to apply.

Common Mistake

Wrong reasoning: “Everyone's survival time is measured from diagnosis, so everyone belongs in the risk set from diagnosis; later enrollment is only a shorter follow-up period.” This treats unobserved, pre-entry time as if the subject had been observed and at risk in the study. It can add people to denominators before their entry and omit the conditioning on survival to enrollment.

Correction: Retain the original time origin if it defines the estimand, record each \(L_i\), and use delayed-entry likelihood contributions or risk sets. If the research question is instead survival from enrollment, explicitly redefine time zero and interpret the result as prognosis from enrollment. Neither choice should be made solely because it is computationally convenient.