Likelihoods With Predictable Time-Varying Covariates

Why the Covariate’s Timing Matters

When an event process has a time-varying covariate, its intensity at an event time depends on the covariate information available immediately before that event. The likelihood also accounts for every interval during which the subject was at risk, including intervals with no event. These are two distinct uses of the covariate history: its value at each observed jump and its path throughout the accumulated intensity.

This distinction is essential when covariates change during follow-up. A treatment may begin partway through observation, a laboratory measurement may be updated, or a subject’s state may change. Using a value recorded after an event to explain that event reverses the temporal order. Conversely, using only the covariate value at the event omits its contribution to the event-free periods.

Let \(N(t)\) count events observed from entry time \(a\) through an end time \(\tau\), and let \(t_1,\ldots,t_K\) be its event times. Let \(Y(t)\) indicate whether the subject is at risk immediately before \(t\). For a simple event process with a continuous-time intensity, the event-process likelihood contribution has the form

When the intensity is zero outside the risk period, the integral automatically accrues only while \(Y(t)=1\). In the common specification \(\lambda(t;\theta)=Y(t)q(t;\theta,X(t))\), the event-time factor uses the predictable covariate value at that jump, while the integral uses the covariate path at every time in the risk period. For an individual with no observed events, the empty product equals 1, leaving only the exponential term.

This expression is the event-process likelihood contribution under the stated intensity model. If covariates are stochastic, it is not automatically the full likelihood for all observed data: a parameter-dependent model for their process may also need to be included. Likewise, censoring and entry mechanisms must be handled according to the observation design; the intensity integral begins at the actual start of observation and risk.

Why the Likelihood Has These Two Parts

The product records the event intensities at observed jumps. The exponential term records the likelihood contribution of not observing an event during the rest of the risk time. The following short derivation explains both parts. Over a small interval of width \(\Delta t\), conditional on the history just before the interval, the probability of one event is approximately \(\lambda(t;\theta)\Delta t\), and the probability of no event is approximately \(1-\lambda(t;\theta)\Delta t\). The log contribution for no event is therefore approximately \(-\lambda(t;\theta)\Delta t\), because \(\log(1-u)\approx-u\) for small \(u\). Adding these contributions across intervals gives the integrated negative intensity; at observed events, the corresponding event factors contribute \(\log\lambda(t_j;\theta)\).

This argument assumes that the event process is simple, with no simultaneous jumps, and has a well-defined intensity with respect to the stated history. At each observed jump, the intensity must be positive; otherwise the likelihood is zero. The covariate values used in the intensity must be available predictably, and the intensity must be integrable over the observed risk period. These conditions make the event factors and cumulative-intensity term coherent parts of one path likelihood.

Using the Whole Covariate Path

For a piecewise-constant covariate, split the cumulative-intensity integral at each covariate change, risk-set change, or other point where the intensity formula changes. On each resulting interval, multiply its duration by its constant intensity. This is an exact calculation when the intensity really is constant within each interval; it is not an approximation that permits ignoring intervals without events.

The same principle applies when a predictable covariate varies continuously, though the integral may require calculus or numerical integration. For example, suppose a hypothetical subject is at risk from time 0 until an event at time 1.5, with deterministic predictable covariate \(X(t)=t\) and intensity \(\lambda(t)=0.05\exp\{0.3X(t)\}\) per day. At the event, the intensity is \(0.05\exp(0.45)\approx0.07842\) per day, and the cumulative intensity is

The event-time likelihood density is \(0.07842\exp(-0.09472)\approx0.07133\) per day. The integration ends at the event, not at an arbitrary later time, because the subject is no longer at risk after this event in the stated single-event setting.

Checks, Alternatives, and Limitations

Before evaluating a likelihood, check that each event-time covariate value was available just before that event; that risk intervals begin at entry and end at the correct censoring or event time; and that the integral covers every at-risk interval. For piecewise-constant covariates, verify that the interval durations sum to the observed risk time. For continuously varying covariates, verify that the integration uses the stated time scale and the same covariate process as the event factors.

A useful implementation strategy is to split records at covariate changes and calculate event and exposure contributions separately. Splitting records does not itself alter the likelihood if the resulting intervals preserve the original covariate and risk histories. If the covariate changes continuously, arbitrarily coarse intervals instead approximate the integral and may introduce numerical error; use a sufficiently accurate integration method and check its stability.

A common mistake is to insert the covariate value recorded after an event into that event’s intensity, reasoning that it is the latest available measurement in the data file. That value may reflect information arising after the event and is not the predictable value for the event-time factor. Use information available immediately before the jump; if the recorded process is right-continuous, use its left limit. A second mistake is to multiply event intensities but omit the integrated-intensity term. That ignores the information in event-free risk time and is not the event-history likelihood.

The likelihood also depends on the adequacy of the intensity model and the recorded event, covariate, entry, and censoring histories. It does not, by itself, establish a causal effect of a time-varying covariate. If covariate evolution is informative or its distribution depends on model parameters, a joint modeling strategy may be needed. The displayed product assumes no simultaneous jumps; tied or simultaneous events require additional treatment rather than an arbitrary ordering of jumps.