Lagged Biomarker Values and Delayed Risk

Why Use a Lagged Biomarker Value?

A biomarker measured close to an event may reflect an impending event, rather than a biological process that preceded it. If a change in the biomarker is expected to affect risk only after a delay, linking the hazard to the biomarker’s value at the same time may not represent the scientific question. A lagged association instead asks whether the biomarker value at an earlier time is associated with the event hazard now.

The lag is a modeling choice, not a claim that the biomarker causes the event. Comparing several plausible lags can reveal whether the estimated association is sensitive to the assumed delay. It does not, by itself, identify the true biological delay.

Definition: Let \(m_i(t)\) denote the latent longitudinal biomarker trajectory for individual \(i\), and let \(\delta \geq 0\) be a prespecified lag, measured in the same time units as \(t\). A fixed-lag association links the event hazard at time \(t\) to \(m_i(t-\delta)\), the latent biomarker value \(\delta\) time units earlier. The case \(\delta=0\) is the current-value association.

Model and Assumptions

One common joint-model specification has a longitudinal submodel and a time-to-event submodel. For measurements \(Y_{ij}\) at times \(t_{ij}\), a simple longitudinal model is

Y_{ij}=m_i(t_{ij})+\epsilon_{ij}, \qquad m_i(t)=x_i(t)^\mathsf{T}\beta+z_i(t)^\mathsf{T}b_i.

Here \(x_i(t)\) and \(z_i(t)\) are specified design vectors, \(\beta\) is a vector of fixed effects, \(b_i\) is a vector of individual random effects, and \(\epsilon_{ij}\) is measurement error. The event submodel with a fixed lag is

h_i(t\mid b_i)=h_0(t)\exp\left\{\gamma^\mathsf{T}w_i+\alpha m_i(t-\delta)\right\}.

In this expression, \(h_0(t)\) is the baseline hazard, \(w_i\) is a vector of event-model covariates, \(\gamma\) is its coefficient vector, and \(\alpha\) is the association parameter. Conditional on the random effects and covariates, the hazard at \(t\) depends on the latent biomarker value at \(t-\delta\), not on the biomarker’s current value unless \(\delta=0\). If the biomarker is measured in units such as mg/L, then \(\alpha\) is a log-hazard change per 1 mg/L difference in the lagged value.

Interpretation of this model relies on several conditions. The trajectory model must describe the biomarker values needed at the lagged times; the chosen time scale and lag must have a meaningful scientific interpretation; and the event model must adequately represent the conditional hazard, including the baseline hazard and relevant covariates. The lagged value must be available from the modeled trajectory over the risk interval being analyzed. Standard joint-model likelihood inference also relies on an appropriate model for the random effects and measurement errors, and on censoring being conditionally independent of the event process given the modeled information.

For an individual with observed biomarker measurements \(y_i\), event or censoring time \(T_i\), and event indicator \(d_i\), the event contribution conditional on \(b_i\) is

h_i(T_i\mid b_i)^{d_i} \exp\left\{-\int_0^{T_i} h_i(u\mid b_i)\,du\right\}.

The joint likelihood combines this contribution with the longitudinal measurement model and integrates over the random-effects distribution. Thus, the lagged value is a model-based latent trajectory value, not necessarily the most recent observed measurement carried forward. Uncertainty in that trajectory is part of the joint analysis.

What the Lag Changes

The hazard ratio has a direct interpretation when two individuals have the same modeled covariates and differ by a fixed amount in their lagged biomarker values. If their values at \(t-\delta\) differ by \(\Delta\), their conditional hazard ratio at \(t\) is

\frac{h_i(t\mid m_i(t-\delta)+\Delta)} {h_i(t\mid m_i(t-\delta))} =\exp(\alpha\Delta).

The contrast concerns the earlier values, with the other event-model terms held fixed. It is a conditional association, not a causal effect. Changing \(\delta\) changes which part of each trajectory enters the hazard, so estimates from different lags need not agree even when the same people, biomarker scale, and covariates are used.

Worked Example: Interpreting a Six-Month Lag

Suppose a fitted model uses \(\delta=6\) months and estimates \(\alpha=0.040\) per 10 mg/L. At a particular event time, two otherwise comparable individuals have modeled biomarker values six months earlier of 60 and 50 mg/L. What is their conditional hazard ratio?

The difference is \(10\) mg/L, one unit on the coefficient’s stated scale. Therefore,

HR=\exp(0.040\times 1)=\exp(0.040)=1.041.

At that event time, the model associates a 10 mg/L higher biomarker value six months earlier with a hazard about 1.041 times as large, or about 4.08% higher, conditional on the modeled covariates and random effects. This does not say that raising the biomarker would cause the hazard to increase.

A simple derivation shows why a lag can alter the apparent association even for a smooth trajectory. If an individual’s trajectory is linear, \(m_i(t)=a_i+c_i t\), then

m_i(t-\delta)=a_i+c_i(t-\delta)=m_i(t)-c_i\delta.

Consequently, the lagged linear predictor can be written as

\alpha m_i(t-\delta)=\alpha m_i(t)-\alpha\delta c_i.

Thus, for a linear trajectory, a fixed lag shifts the value according to the individual’s rate of change as well as their level. This identity is a description of the model’s algebra; it does not mean that the lagged-value model estimates an unrestricted separate slope effect. With nonlinear trajectories, the relation depends on the trajectory’s shape over the interval from \(t-\delta\) to \(t\).

Worked Example: A Rising Trajectory at Two Lags

Consider a hypothetical individual whose biomarker follows \(m_i(t)=40+2t\) mg/L, where \(t\) is measured in months from baseline. At month 12, compare the model covariate for lags of 0 and 3 months.

At lag 0, the value is \(m_i(12)=40+2(12)=64\) mg/L. At lag 3, it is \(m_i(9)=40+2(9)=58\) mg/L. The lagged value is 6 mg/L lower because the trajectory rises by 2 mg/L per month over 3 months.

If the association is \(\alpha=0.03\) per mg/L, the lag-0 linear predictor contribution is \(0.03(64)=1.92\), whereas the 3-month-lag contribution is \(0.03(58)=1.74\). Holding all other terms fixed, the hazard ratio for the lag-0 contribution relative to the lag-3 contribution is

\exp(1.92-1.74)=\exp(0.18)=1.197.

This comparison follows from the specified trajectory and link. It is not evidence that either lag is biologically correct; it illustrates how a changing trajectory makes the modeled covariate depend on the lag.

Comparing Alternative Lags

A sensitivity analysis fits otherwise comparable models at a set of scientifically plausible fixed lags. For an interpretable comparison, keep the biomarker scale, covariates, trajectory specification, baseline-hazard approach, and analysis population consistent where possible. Report each lag and its estimate, uncertainty, and hazard-ratio interpretation. A lag should be chosen for scientific reasons where possible, rather than selected solely because it gives the largest association estimate or smallest p-value.

Worked Example: Reading a Lag-Sensitivity Table

Suppose four hypothetical models use the same biomarker scale, covariates, participants, and event-model specification. The association estimates are per 10 mg/L difference in the lagged value.

Lag \(\delta\) (months)	\(\hat{\alpha}\)	\(SE(\hat{\alpha})\)
0	0.012	0.010
3	0.025	0.011
6	0.031	0.012
9	0.018	0.013

For the 6-month lag, a Wald 95% confidence interval for \(\alpha\) is

0.031\pm1.96(0.012) =0.031\pm0.02352 =(0.00748,\ 0.05452).

Exponentiating gives a 95% interval for the hazard ratio per 10 mg/L: \(\big(\exp(0.00748),\exp(0.05452)\big)=(1.0075,\ 1.0560)\). Under this fitted model, a 10 mg/L higher biomarker value six months earlier is associated with a hazard about 1.031 times as large; the interval expresses uncertainty in that model’s association estimate. The 6-month estimate is largest among these four, but that fact alone does not establish that six months is the correct delay. The estimates are based on related trajectory values, and choosing the largest after examining the results adds selection uncertainty not represented by an ordinary single-model interval.

Boundary Choices, Diagnostics, and Alternatives

The lagged value may not be available throughout the risk period. For example, if the trajectory is only defined from baseline \(t=0\), then \(m_i(t-\delta)\) is undefined for \(t<\delta\). This is a modeling boundary, not a minor computational detail. Depending on the scientific design, options include having pre-baseline biomarker data to support the required values, defining a justified baseline-history rule, or beginning the risk analysis at a prespecified landmark when the lagged value is defined. Restricting risk time changes the target population and estimand; silently extrapolating the trajectory before baseline can be misleading.

Check whether the relevant lagged times are supported by the measurement schedule. If visits are sparse, values between observations depend more strongly on the trajectory model; if the lag requires values outside the observed time range, they may be extrapolations. Useful checks include plotting fitted trajectories against observed measurements, inspecting how many event times require interpolation or extrapolation, and checking whether conclusions change under plausible trajectory specifications. Also examine the estimated association and its uncertainty across the prespecified lag range, rather than reporting only a selected lag.

A current-value link uses \(m_i(t)\), while a fixed-lag link uses \(m_i(t-\delta)\). A cumulative exposure link summarizes values over a time window, and a slope link relates risk to the trajectory’s rate of change. These alternatives represent different scientific questions; a lagged value is not a general substitute for an exposure history or a slope. If the lagged models are compared using fit statistics, remember that the lag is part of the association specification and that a better in-sample fit does not prove a biological delay.

A common mistake is to interpret the lagged coefficient as the effect of the biomarker measured at the event time, or to say that the lag with the smallest p-value proves when the biomarker begins to affect risk. The correction is to state the time contrast explicitly: the coefficient describes the modeled conditional association between risk at \(t\) and the latent biomarker at \(t-\delta\). Different lags provide sensitivity evidence, not a causal timing conclusion. If the association changes sharply across lags, that may reflect a genuine timing pattern, but it can also reflect sparse visits, trajectory misspecification, or strong correlation among nearby lagged values.

Key takeaway: A lagged-value model links the hazard at time \(t\) to the latent biomarker value at a prespecified earlier time. Interpret the coefficient on the lagged-value scale, justify the time boundary, and treat differences across lags as evidence about sensitivity to the association structure—not as automatic identification of a causal delay.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

With \(\hat{\alpha}=0.05\) per 5 units and a 5-unit difference in the biomarker value at the selected lag, what is the fitted hazard ratio?
A 4-month lag is specified, but the trajectory is only defined from baseline. What problem occurs for event times before month 4, and name one defensible response.
Why is the model with the largest estimated association across several lags not necessarily the preferred model?
For \(m_i(t)=30+1.5t\), with time in months, calculate \(m_i(10-2)\) and \(m_i(10)\). What does their difference show?

Solutions. (1) \(HR=\exp(0.05\times1)=\exp(0.05)=1.051\), approximately a 5.13% higher conditional hazard for the stated lagged-value contrast. (2) The model requests \(m_i(t-4)\) at a time before baseline, where it is undefined under the stated trajectory domain. Pre-baseline measurements, a scientifically justified history rule, or a prespecified landmark after month 4 are possible responses; the choice should be explicit. (3) Selecting the largest estimate after examining the models creates selection uncertainty, and the lagged values are related. A maximum estimate does not establish the true delay or a causal effect. (4) \(m_i(8)=30+1.5(8)=42\), while \(m_i(10)=30+1.5(10)=45\). The 2-month lagged value is 3 units lower, consistent with a rise of 1.5 units per month.