Sensitivity to the Cumulative Exposure Window

Why the Exposure Window Matters

A cumulative-exposure association asks whether a person’s biomarker history over a specified period is associated with the event hazard. But “history” depends on how far back the calculation reaches. A one-month window and a one-year window summarize different portions of the trajectory, and a person’s cumulative burden can change substantially when the window changes. The estimated association may change too.

The aim of a window-sensitivity analysis is not to find whichever window gives the strongest association. It is to assess how conclusions change across clinically justified, prespecified windows, while keeping the other parts of the joint model fixed. A window is a modeling choice about which part of the trajectory contributes to the association; it is not, by itself, evidence that the biomarker causes the event.

Definition and Hazard Interpretation

If \(t\geq w\), the integration interval has length \(w\). If \(t<w\), it begins at time zero, so only the observed follow-up period contributes. This convention avoids requiring biomarker history before the defined start of follow-up. When the scientific question concerns mean level rather than total area, one can instead use the cumulative average \(A_i(t;w)/\min(w,t)\). That is a different exposure summary and has different units.

For a window-specific association model, write the hazard as

Here \(h_0(t)\) is the baseline hazard, \(\mathbf{W}_i\) is the vector of included covariates, and \(\alpha_w\) is the association parameter for window length \(w\). Holding other covariates and the baseline hazard fixed, a difference \(\Delta A\) in cumulative exposure corresponds to a hazard ratio of \(\exp(\alpha_w\Delta A)\). The units of \(\alpha_w\) are the reciprocal of biomarker-units multiplied by time-units.

This interpretation is conditional on the specified model. It compares hazards at time \(t\) for subjects whose cumulative exposures differ by \(\Delta A\), with the other modeled quantities held equal. It does not assert that changing the biomarker would cause that hazard difference.

How Window Length Changes the Exposure

For two windows \(w_2>w_1\) and a time \(t\geq w_2\), the longer window adds the earlier segment from \(t-w_2\) to \(t-w_1\):

Thus the difference between window summaries depends on the biomarker values in the additional segment, not merely on the two window lengths. If the biomarker is nonnegative, the longer-window total is at least as large; if the biomarker has been centered around a reference value, that ordering need not hold. Near the beginning of follow-up, a longer nominal window may add no exposure at all because both integrals start at zero.

A useful local justification follows from changing the lower limit. When \(t>w\) and \(m_i\) is continuous, the fundamental theorem of calculus gives

Increasing the window slightly adds a thin slice at its older boundary, so the rate of increase is the biomarker value at that boundary. This explains why longer windows do not have a uniform effect across subjects: their older biomarker histories can differ.

Making Window-Specific Associations Comparable

Each prespecified window generally has its own fitted coefficient, denoted \(\alpha_w\). A smaller coefficient for a longer window does not automatically mean a weaker clinical association. The cumulative exposure itself usually changes scale with window length, and its distribution may change as well. A one-unit increase in total exposure is not necessarily a clinically comparable contrast across windows.

One useful comparison is the hazard ratio for a specified sustained biomarker difference. Suppose the trajectory is higher by \(c\) biomarker units throughout the full window, at a time \(t\geq w\). The cumulative-exposure difference is then \(cw\), and the corresponding hazard ratio is

For \(t<w\), the available interval has length \(t\), so the same sustained difference gives \(\exp(\alpha_w c t)\). Alternatively, investigators may report hazard ratios for a common, explicitly chosen increment in cumulative exposure or use a cumulative-average summary. The comparison must state which contrast it uses.

Calculation From Visit Data and Early Follow-Up

In a joint model, the exposure is defined using the latent trajectory \(m_i(s)\), not by treating noisy visit measurements as the trajectory itself. If a fitted trajectory is represented by piecewise-linear interpolation between visits, its integral over each segment is the trapezoid area: segment length multiplied by the average of the two endpoint values. Summing the segment areas gives the window total. The fitted trajectory must cover the requested interval; extrapolation beyond well-supported data can make a nominally precise window summary unreliable.

Assumptions, Diagnostics, and Alternatives

A meaningful sensitivity comparison requires more than calculating several integrals. The windows should be selected in advance for scientific reasons, and each model should use the same longitudinal trajectory specification, covariates, baseline-hazard approach, and event-time information, changing only the exposure window and its associated coefficient. This is the controlled comparison emphasized in the earlier material on link-sensitivity analysis. The hazard model must also be adequate for the data, and the estimated longitudinal trajectory must represent the relevant biomarker history well enough over each window.

Check whether the data support each window. A long window may reach into periods with sparse visits or require substantial extrapolation, while short windows may be dominated by measurement noise or a small number of observations. Compare fitted trajectories and exposure distributions across windows, inspect visit coverage near the older boundaries, and check whether a few subjects account for much of the added area. A window-specific result that changes sharply when poorly supported history is included warrants caution.

The main alternatives summarize different features of the trajectory. A current-value link uses the biomarker level at \(t\); a cumulative-average link summarizes average level over the available window; a cumulative-total link uses area and therefore incorporates duration; and an observed sum uses measured values rather than integrating the fitted latent trajectory. These are not interchangeable estimands. The earlier material on current value versus cumulative exposure addresses the distinction between an instantaneous level and a history summary; here the question is how the specified history interval changes that summary.

A common mistake is to say, “The longer-window coefficient is smaller, so long-term exposure matters less.” This reasoning compares coefficients without accounting for the exposure scale. The correction is to report each coefficient’s units and compare hazard ratios for a common, clinically interpretable contrast, such as a stated sustained biomarker difference. Another mistake is to treat the largest estimated association as proof that its window is the true biological window. Window sensitivity assesses robustness to a modeling choice; it cannot identify the correct biological mechanism on its own.

Limitations remain even when the analysis is carefully conducted. Results depend on the trajectory model, the window choices, and the hazard model. Longer windows can be weakly supported by the visit schedule, and different windows can summarize highly overlapping histories. A change in association across windows may reflect genuine differences in relevant history, but it can also reflect estimation uncertainty or model misspecification. Interpret the pattern alongside the data support and clinical rationale, not as a ranking based only on point estimates.