Lagged Marker Values in the Event Hazard

Why Put a Lag in the Marker Link?

A marker may affect event risk only after some delay. A marker measured or inferred at the current time might therefore be a poor summary of the biologically relevant exposure. A fixed-lag link instead relates the hazard at time \(t\) to the latent marker value \(\delta\) time units earlier. The lag \(\delta\) is a specified, nonnegative duration; for example, if time is measured in years, \(\delta=2\) means that the hazard at year 5 is linked to the marker at year 3.

The marker in this link is the subject-specific latent trajectory, not a single noisy measurement. Write that trajectory as \(m_i(s)\) for individual \(i\) at time \(s\). In a shared-random-effects model, it is determined by the longitudinal model and its individual random effects. The event model then conditions on those same random effects, so the marker trajectory and event risk are connected through the individual-level latent process.

Model and Hazard-Ratio Interpretation

A clear way to define the link at all nonnegative times is to carry the baseline marker value forward until the lagged time is available. Let \(w_i\) denote event-model covariates, let \(\gamma\) be their coefficient vector, let \(h_0(t)\) be the baseline hazard, and let \(\alpha\) be the association parameter. The conditional hazard is

Thus, for \(0\leq t<\delta\), the marker term is \(\alpha m_i(0)\); for \(t\geq\delta\), it is \(\alpha m_i(t-\delta)\). This convention makes the marker value defined at early times and joins continuously at \(t=\delta\), provided the trajectory itself is continuous. It is a modeling choice, not a mathematical necessity: an alternative is to set the marker association to zero before \(\delta\), so that the marker link only becomes active once \(t-\delta\geq0\). The chosen rule must be stated because the two rules imply different early hazards.

For two individuals \(i\) and \(j\) at the same time \(t\), conditional on their random effects, the hazard ratio is obtained by dividing their hazards. If they have the same event covariates, the baseline hazard and covariate term cancel:

When their lagged marker values differ by one marker unit, the conditional hazard ratio is \(\exp(\alpha)\). If \(\alpha>0\), a higher lagged marker value corresponds to a higher conditional hazard; if \(\alpha<0\), it corresponds to a lower one. This is an association conditional on the model and random effects, not evidence by itself that changing the marker would cause a change in event risk. If covariates differ, the hazard ratio also includes \(\exp\{\gamma^\top(w_i-w_j)\}\).

What the Lag Does to a Trajectory

The lag shifts the time at which the trajectory is evaluated; it does not change the longitudinal trajectory itself. For a linear latent trajectory \(m_i(s)=a_i+b_i s\), the association term after the lag becomes \(\alpha[a_i+b_i(t-\delta)]\). For a quadratic trajectory \(m_i(s)=a_i+b_i s+c_i s^2\), it becomes \(\alpha[a_i+b_i(t-\delta)+c_i(t-\delta)^2]\). Before the lag is available, the carry-forward convention uses \(m_i(0)=a_i\) in either case.

Times Before the Lag Is Available

When \(t<\delta\), the literal time \(t-\delta\) is negative. If the longitudinal trajectory is only defined from time zero, \(m_i(t-\delta)\) is not available without an additional convention. The carry-forward rule gives \(m_i((t-\delta)_+)=m_i(0)\), so the early hazard depends on the baseline latent marker. This is coherent when the intended assumption is that the baseline marker represents the relevant marker value throughout the initial interval.

A different scientific assumption is that there is no marker association until a full lag has elapsed. One can express that choice as

where the marker term is defined to be zero for \(t<\delta\), and \(\mathbf{1}(t\geq\delta)\) is one when \(t\geq\delta\) and zero otherwise. This version may create a change in the marker contribution at \(t=\delta\). Its meaning differs from carrying the baseline value forward, which includes the marker association from time zero. Neither convention is universally correct; the choice should follow the scientific question and be reported with the model.

Justification, Assumptions, and Diagnostics

The hazard-ratio result follows directly from the model. At a fixed time, write \(q_i=m_i((t-\delta)_+)\). For equal covariates, \(h_i=h_0(t)\exp(\gamma^\top w+\alpha q_i)\) and \(h_j=h_0(t)\exp(\gamma^\top w+\alpha q_j)\). Dividing cancels the shared terms, leaving

This interpretation depends on a correctly specified conditional hazard and longitudinal model, a well-defined latent trajectory over the times used, and a lag \(\delta\) expressed on the same time scale as the event process. The usual shared-random-effects joint-model assumptions are also relevant: the event process is modeled conditional on the random effects, and the latent marker values used in the event model are those implied by the longitudinal model rather than error-prone observed values. The coefficient \(\alpha\) is assumed constant over time in this formulation.

Diagnostics should focus on whether the lagged values are supported by the observed longitudinal follow-up. If many event times require \(m_i(t-\delta)\) at times with little or no marker information, the association may rely heavily on model-based extrapolation. Inspect the fitted longitudinal trajectories and residual behavior, check whether the lag and early-time convention are scientifically defensible, and assess whether the estimated association is driven by sparse regions of follow-up. A plausible fit to observed marker measurements does not automatically guarantee reliable estimates of latent values at all lagged times.

How This Link Differs From Other Association Structures

A current-value link uses \(m_i(t)\), whereas a fixed-lag link uses \(m_i(t-\delta)\) when available. A slope link concerns the rate of marker change, and a cumulative link summarizes marker history over an interval. The lagged link instead asks whether the marker level at one specified earlier time is associated with the current hazard. These alternatives answer distinct scientific questions; the lagged model is not simply a computational substitute for the current-value model.

A single fixed lag can be too restrictive if the relevant delay differs across individuals or if hazard depends on marker values across a range of earlier times. Choosing a lag also changes which part of each trajectory informs the association. These limitations matter especially when follow-up is short or early events are common. The model should therefore be interpreted as a specific delayed association structure, not as a general description of every possible biological delay.

Common Pitfall

A frequent error is to insert \(m_i(t-\delta)\) at every time and silently evaluate it at negative times. That assumes an unmentioned extension of the marker trajectory before observation began. The correction is to define the early-time rule explicitly—such as baseline carry-forward or zero marker contribution before the lag—and then calculate hazards consistently with that rule. A second error is to interpret \(\exp(\alpha)\) as the hazard ratio for any two individuals. It applies to a one-unit difference in lagged marker value at the same time, conditional on equal covariates and the stated model; other differences require the full hazard-ratio expression.