Lagged Covariates and Exposure Histories

Why the Timing of a Covariate Matters

In a survival analysis, a covariate attached to a person-period record must describe information available for that person at the time represented by the record. A measurement made after that time cannot be used as if it were known earlier. This is particularly important when a person’s treatment or exposure changes during follow-up, or when covariates are measured intermittently.

A lagged covariate refers to an earlier value of a variable, with the lag defined by a time rule. A prior exposure summary condenses information about exposure before a specified prediction time, often over a stated window. These are distinct constructions: a lag may select one earlier value, while a summary may combine several earlier observations or describe whether exposure occurred during a prior period.

The central design decision is not just what to calculate, but which records are eligible to contribute. State the prediction time, the lag or look-back window, and the treatment of measurements at the boundary before calculating anything.

Time Indexing and the Information Boundary

For start-stop survival data, subject \(i\) may contribute one or more records for intervals \([a_{ik},b_{ik})\), during which the subject is at risk. A covariate used for the interval beginning at \(a_{ik}\) should be defined from information available by that prediction time under a clear convention. A conservative convention is to use measurements with timestamps strictly less than \(a_{ik}\). If a measurement taken exactly at \(a_{ik}\) is available before the interval’s risk prediction is made, it may instead be included, but that ordering must be justified and used consistently.

Write \(\mathcal{F}_{i,t-}\) for the information about subject \(i\) recorded strictly before time \(t\). A valid prior-history covariate \(Z_i(t)\) is a function of that information: \(Z_i(t)\) is determined by \(\mathcal{F}_{i,t-}\). This is sometimes described as a predictable covariate. The idea is simple: at a risk time \(t\), the model may condition on the past, not on a future observation whose value would not yet be available.

For example, a time-dependent Cox model can be written as

Here \(\lambda_0(t)\) is the baseline hazard and \(\beta\) is a log hazard ratio per unit of \(Z_i(t)\). Interpreting this as a proportional-hazards model requires the proportional-hazards specification to be suitable and the covariate history to be aligned correctly with risk time. The temporal construction does not itself establish those modeling assumptions; it ensures that the covariate does not use unavailable future information.

Fixed Lags Are Not the Same as “Last Observed” Values

Measurements in practice are often irregular. Two different constructions are easily confused:

• Fixed-time lag: select the value measured at a specified time before \(t\), such as \(t-7\) days. If no such measurement is available, the lag is unavailable.
• Most-recent-prior value: select the value at the latest measurement time strictly before \(t\). The elapsed time between that measurement and \(t\) can vary across records.

The second construction can be useful, but it must be named and its maximum acceptable age considered. It is not equivalent to a fixed lag. Treating the most recent value as current without qualification can conceal long gaps between measurement and prediction. The earlier material on why last observation carried forward fails in survival data explains why automatically carrying a value forward is not a general solution to missing or intermittent measurements.

Constructing Prior-Exposure Summaries

A history summary is interpretable only when its inputs and time window are explicit. Examples include an indicator of any exposure during the previous 30 days, the number of recorded exposed visits in a prior window, or the mean of a measured exposure over eligible earlier observations. These summaries answer different questions. For instance, “any exposure in the prior month” does not retain frequency or intensity, while the number of exposed visits depends on the measurement schedule.

A general form for a window-based summary is

where \(E_i(s_{ij})\) is an exposure record, \(w>0\) is the look-back duration, and \(g\) is the prespecified summary function. The left endpoint is included and the prediction time is excluded in this convention. Other endpoint rules are possible, but changing them can change the result when a measurement occurs exactly on a boundary.

Do not label an arbitrary summary “cumulative exposure” unless its units and construction truly represent accumulated exposure. For example, a count of exposed visits is measured in visits, whereas a time-weighted exposure integral would combine exposure level and duration. Constructing time-updated cumulative exposures requires additional choices about the exposure process and is a separate task.

Derived Histories in Multiple Imputation

When a lag or history summary is calculated from a covariate that has missing values, it should normally be derived after that source variable has been imputed in each completed data set. If a missing source value is imputed differently across completed data sets, the corresponding lag or summary should reflect that uncertainty and can differ as well. For a deterministic rule, calculate the derived variable from the completed source values rather than imputing a separate, unconstrained version of the derived variable.

This requires the imputation and analysis data to preserve the subject identifier, measurement time, interval boundaries, and the agreed timing rule. A derived value can be wrong even when every imputed value is plausible if rows have been sorted incorrectly, linked to the wrong subject, or assigned to the wrong risk interval.

Checks, Alternatives, and Failure Modes

Before using derived histories in a survival model or an imputation model, check their construction against the actual timestamps. The following checks often reveal errors that a model-fitting command will not detect:

• For every record, verify that each contributing measurement belongs to the same subject and satisfies the time-window rule.
• Check whether any lag or summary uses a measurement at or after its prediction time.
• Tabulate the age of the measurement used for a most-recent-prior value; unusually old values may indicate gaps or mistaken carry-forward.
• Check the number of eligible measurements per window. A visit-based mean based on one visit has a different precision and interpretation from one based on many.
• Distinguish an undefined lag because no scheduled prior measurement exists from a scheduled measurement that is missing. These have different meanings and should not automatically receive the same treatment.
• Inspect values at interval starts, event times, censoring times, and window boundaries to confirm the endpoint convention was applied consistently.

The main alternatives serve different scientific aims. A contemporaneous covariate uses information current at the prediction time, if it is actually available then. A fixed lag targets a value at a defined earlier time. A most-recent-prior value accommodates irregular timing but may have variable age. A window summary compresses several earlier records and may be less sensitive to one noisy measurement, at the cost of discarding detail. A full time-varying history retains more information but can require more modeling and more complete measurement histories.

No construction repairs a poorly timed measurement process. A prior-window average of sparse, selectively observed records can be biased as a representation of the underlying exposure. A fixed lag may be unavailable for many subjects. A summary may discard the pattern of change that matters for the outcome. These choices should be justified by the scientific question and the data-collection schedule, and reported so the meaning of each covariate is reproducible.