Predictable Covariate Processes

Why Covariate Timing Matters

In event-history analysis, a covariate can change over time: a person may begin treatment, a biomarker may be updated, or a participant may move into a new state. The key question is not only what value was recorded, but also when that value became available. To describe an event’s rate at time \(t\), the covariate must be determined by information available immediately before \(t\), not by information revealed by the event itself or afterward.

This timing requirement is formalized through predictability relative to the filtration for the event history. Filtrations were introduced in the earlier material on Filtrations for Event History Data: \(\mathcal{F}_t\) represents the information available up to time \(t\). The distinction between \(\mathcal{F}_{t-}\), information available just before \(t\), and information learned at \(t\) is essential when covariates and events can change together.

Definition: The predictable sigma-field associated with a filtration \(\{\mathcal{F}_t\}\) is the sigma-field on time and outcome generated by all adapted, left-continuous processes. A process \(Z(t)\) is predictable if it is measurable with respect to this sigma-field. In particular, an adapted left-continuous process is predictable. Predictability means that the value used at time \(t\) is determined by the information available just before \(t\) (with the initial value treated as \(\mathcal{F}_0\)-measurable).

For a process with right-continuous paths and left limits, its left-limit process \(Z(t-)\) is predictable when the original process is adapted. Thus, if \(Z(t)\) records a state that changes at an event time, \(Z(t-)\) describes the state immediately before that change. The distinction matters at the change time itself: \(Z(t)\) may include information revealed at \(t\), whereas \(Z(t-)\) does not.

A standard schematic event-intensity model illustrates the role of a predictable covariate:

\lambda(t \mid \mathcal{F}_{t-})=Y(t)\lambda_0(t)\exp\{\beta^\mathsf{T}Z(t)\}, \qquad Z(t)\text{ predictable}.

Here \(Y(t)\) indicates whether the subject is under observation and at risk, \(\lambda_0(t)\) is a baseline rate, and \(\beta\) is a coefficient vector. The formula is schematic: its purpose here is to show that the covariate entering the rate must be available before the event at \(t\). Predictability is a condition on timing and information, not a claim that the covariate is unconfounded, measured without error, or statistically independent of the event process.

How to Recognize Predictability

A useful building block is a process that holds a value \(H\) over an interval \((a,b]\), where \(H\) is already known at time \(a\). In process notation this is \(H\,1\{a<t\leq b\}\). The value is fixed using \(\mathcal{F}_a\), before every time \(t\) in that interval. Such processes are predictable, and they help explain how predictable processes represent covariate histories assembled from information known in advance of each interval.

A covariate need not be constant for a long period to be predictable. It may change repeatedly, provided each new value is available before the time at which it is used. Conversely, a variable can be recorded numerically and measured accurately yet fail to be predictable at the time of an event if its value is learned only at, or after, that event.

Worked Example: Treatment Begins at an Observed Time

Hypothetical data. A participant is untreated until treatment begins at month \(T=6\). The treatment-start time is observed as it occurs. Let \(A(t)=1\{T\leq t\}\) denote the usual right-continuous treatment-status record. What treatment value should be used for an event occurring exactly at month 6?

At \(t=6\), the recorded status is \(A(6)=1\), because the definition includes \(T\leq t\). But the start of treatment is only learned at month 6, so that value need not have been available just before the event. Its left limit is \(A(6-)=1\{T<6\}=0\).

A predictable treatment history is \(A(t-)=1\{T<t\}\). It is adapted and left-continuous: it remains zero at the start time and takes the value one strictly afterward. At month 8, for example, \(A(8-)=1\); at month 6, \(A(6-)=0\).

Therefore, if the event occurs exactly when treatment starts, the pre-event treatment status is untreated. Using \(A(6)=1\) would attribute treatment to the event at the very time treatment is first observed. This example assumes treatment initiation is not known before it occurs; if a treatment start is scheduled and known in advance, the filtration may contain that information earlier, and the process must be assessed relative to that actual information.

Worked Example: Updating a Biomarker After a Visit

Hypothetical data. A biomarker is 12 units at month 0. A new measurement is taken at month 4, and the result, 15 units, becomes available at month 4. The next result, 11 units, becomes available at month 9. Assume each result is recorded in \(\mathcal{F}_{t_k}\) at its visit time \(t_k\), and no later information is used earlier. Construct a predictable covariate and determine its value at months 4, 6, and 9.

Use the previous result at the visit instant itself, then use the new result strictly after it. The resulting path is left-continuous: it has value 12 through month 4, value 15 on \((4,9]\), and value 11 after month 9. At each time after a measurement becomes available, the updated value is known from the past.

Thus the predictable covariate has values 12 at month 4, 15 at month 6, and 15 at month 9. It changes to 11 strictly after month 9. If an event occurs exactly at month 9, the available pre-event measurement is 15; the value 11 cannot be used for that event under the stated timing convention.

The visit times and availability assumptions are part of the example, not incidental details. If the result from the month-4 sample is not reported until month 5, it cannot be used as the current covariate at month 4; the covariate should remain at 12 until the result is available.

Why the Pre-Event Requirement Is Necessary

The reason can be stated directly in terms of information. An event occurring at time \(t\) is part of the history learned at \(t\), but not generally part of the history just before \(t\). A covariate defined using the event or its consequences may therefore be measurable with respect to \(\mathcal{F}_t\) while failing to be measurable with respect to the pre-event information. Putting that value into a model for the event’s rate would let the outcome help define its own purported predictor.

For example, suppose a covariate records whether a participant has entered a severe disease state, and the transition into that state is itself the event being modelled. The post-transition state cannot be used to describe the rate of entering that state at the transition time. The state just before the transition is the relevant value. This is not merely a preference for lagging variables: it follows from the requirement that the modeled rate use the history available before the event.

The construction can also be seen from the left-limit operation. For a right-continuous, adapted covariate \(Z(t)\), define \(Z(t-)\) as the limit of \(Z(s)\) as \(s\) approaches \(t\) from below. This value is determined by the covariate history strictly before \(t\). Because the left-limit process is predictable, it supplies an admissible time-indexed covariate even when \(Z(t)\) jumps.

Z(t-)=\lim_{s\uparrow t} Z(s) \quad\Longrightarrow\quad Z(t-)\text{ uses the pre-}t\text{ history, not information first observed at }t.

This implication assumes the left limit exists, as it does for a right-continuous process with left limits. The implication concerns information timing; it does not mean that every observation made just before \(t\) is automatically a well-defined or scientifically appropriate covariate. Its meaning, measurement schedule, and availability still need to be specified.

Worked Example: Quantifying the Effect of Event-Time Leakage

Hypothetical model. Suppose a baseline rate is \(0.020\) events per month and the coefficient for a binary treatment indicator is \(\beta=\log(2)\). For an event at month 6, treatment begins at that same month and was not known just beforehand. Compare the rate multiplier using the predictable treatment value with the multiplier using the contemporaneous recorded value.

For the predictable value, \(A(6-)=0\), so the multiplier is \(\exp\{\log(2)\times 0\}=\exp(0)=1\). The model’s rate at that time is \(0.020\times 1=0.020\) events per month.

Using the post-start value \(A(6)=1\) instead gives \(\exp\{\log(2)\times 1\}=\exp(\log(2))=2\), and a rate of \(0.020\times 2=0.040\) events per month. That doubled rate is not justified for an event occurring at month 6 under the stated information structure: the treatment status used in the calculation was not known before the event. The arithmetic is correct, but the timing makes the second calculation unsuitable as a pre-event rate.

Assumptions, Diagnostics, and Alternatives

A covariate process is predictable only relative to a specified filtration. To assess it, state what information is available at each time, including when a value is measured, when it is reported, and when it is entered into the analysis. The following checks are useful:

Check the event-time convention. For a covariate that changes at an event time, say whether the model uses the value before or after the change. Do not leave simultaneous events and covariate updates ambiguous.
Check availability, not just collection time. A specimen may be collected at a visit but its result may not be known until later. Use the value only from the time it becomes available.
Check the data construction. Plot or tabulate selected covariate histories around events. A value that first appears at the event or is back-filled from a later record may indicate leakage.
Check future information explicitly. If reconstructing a covariate requires knowing whether an event will occur, or what state follows it, that construction cannot serve as a pre-event covariate at that time.

Predictability is different from several common alternatives. A baseline covariate is fixed at the start and is usually predictable, but it cannot represent later changes. A current-value process may reflect changes over time, but if it includes information revealed at the event, it is not suitable as a pre-event predictor at that instant. A lagged covariate can avoid some timing problems, but a fixed lag is not the definition of predictability: a value known immediately beforehand may be predictable without being delayed by a chosen amount. The right construction depends on the real information and measurement process.

Predictability is necessary for the standard event-intensity formulation, but it does not by itself establish a causal interpretation. Nor does it correct confounding, informative measurement, selection, or measurement error. A carefully timed covariate may still be associated with unmeasured causes of the event; those issues require separate assumptions and analyses.

Common pitfall: “The database says the subject was treated at the event time, so treatment status at that time is the covariate.” This confuses a recorded value at \(t\) with information available just before \(t\). If treatment initiation is first observed at \(t\), the event-time pre-treatment value is \(A(t-)\), not \(A(t)\). Define the information set and the timing convention before constructing the covariate.

Practical Construction

A disciplined construction process makes the information condition explicit rather than relying on an informal label such as “time-varying covariate.” For each candidate covariate, specify its source, availability time, and value immediately before any event time at which it may be used.

Specify the filtration.
List what is known by each time, including event records, covariate measurements, and reporting delays.

Define update times.
For each covariate update, record when the new value becomes available, not only when the underlying measurement was taken.

Assign values at update instants.
State explicitly whether the old or new value applies at a time shared by an event and an update. Use the value determined before the event.

Audit event-time histories.
Inspect records immediately before and after events for back-filling, post-event variables, or inconsistent timing conventions.

The key point is that predictability is a mathematical expression of a temporal design requirement: a covariate used to model an event at time \(t\) must be constructed from the history available just before \(t\). Writing that requirement into the data definition makes the model’s meaning clearer and helps prevent outcome information from leaking into its predictors.

Check Your Understanding

Answer each question using the stated event-time information convention. Brief solutions follow the questions.

A biomarker result is reported at month 3. Under a convention that uses the old value at the reporting instant and the new value strictly afterward, which value applies to an event at month 3?
Treatment begins at a random observed time \(T\), and the start is not known beforehand. Write a predictable binary treatment process and give its value at \(t=T\).
Explain why a disease stage first recorded after an event cannot be used as a pre-event covariate for that event.
Does predictability alone justify interpreting a covariate coefficient causally? Give one reason.

Solutions. (1) The old value applies at month 3; the new result is used strictly afterward. (2) Use \(1\{T<t\}\), which is adapted and left-continuous under the stated setup; at \(t=T\), its value is 0. (3) The post-event stage uses information unavailable just before the event and may be a consequence of it, so it cannot define the pre-event history. (4) No. Predictability addresses timing, not confounding; for example, an unmeasured factor may affect both the covariate and the event.