Time-Varying Covariates and Temporal Ordering

Why Temporal Ordering Matters

A covariate can change over visits and still be a legitimate predictor of dropout—but only if its value is available before the dropout decision being modeled. A measurement recorded after a participant has decided to leave cannot be used as though it were known at that decision. The distinction is about the data-generating timeline, not merely about whether a variable is called “current” or “time-varying.”

This matters for both interpretation and estimation. A model that uses information unavailable at the decision time can describe the data retrospectively, but it is not the stated prospective dropout hazard. It can also use values that are missing precisely because dropout has occurred. A clear timeline is therefore part of specifying the selection equation.

Let \(X_{it}^{\mathrm{pre}}\) denote a time-varying covariate measured before decision \(t\). It may be included in \(\mathcal H_{it}\). A value measured only after that decision is not \(X_{it}^{\mathrm{pre}}\) and cannot serve as an observed predictor in this hazard. Prior outcomes such as \(Y_{i,t-1}\) can also be included when they were observed before decision \(t\). Whether a measurement described as “at visit \(t\)” is pre-decision depends on the actual visit and decision schedule; the label alone does not settle the question.

Specifying a Time-Varying Selection Equation

A common specification uses a logistic hazard. Suppose the history before decision \(t\) contains prior outcomes and covariates, as well as the current pre-decision covariate. A simple model is

Here \(\alpha_t\) allows the baseline dropout odds to vary by decision time, \(\mathbf Z_i\) denotes baseline covariates, and the coefficients describe conditional associations with dropout odds among individuals still at risk. Centering or rescaling continuous covariates can make coefficients easier to interpret, but it does not change the temporal requirement. The model also needs an appropriate functional form: a linear term on the log-odds scale is an assumption, not a consequence of a variable being quantitative.

The equation is conditional on \(A_{it}=1\). It does not describe dropout probabilities for people already lost to follow-up. If a covariate changes over time, the risk set and the covariate value must be updated at each decision. A baseline-only equation instead uses \(\mathbf Z_i\) but omits \(X_{it}^{\mathrm{pre}}\); it may miss relevant changes in a participant’s observed status, although adding time-varying terms does not guarantee a better scientific model.

Temporal Ordering in Practice

A useful way to build the history is to write down the sequence of measurements and decisions for one interval. For example, if the symptom score is measured at a scheduled visit and the dropout decision occurs afterward, that score can be a pre-decision predictor. If the decision occurs before the visit or before the score is recorded, it cannot. A variable may be measured before dropout for some participants but not others; in that case, its availability must be represented accurately rather than assumed from the visit number.

This issue is especially important for outcome measurements. A previous observed outcome can be part of the history at a later decision. A current outcome that is measured after the decision cannot be inserted as though it were observed beforehand. If a measurement is genuinely available before the decision, its timing should be documented; its suitability and model are separate questions. Here the key point is only that “current” does not establish “available before dropout.”

Survival Across Several Decisions

With time-varying histories, survival through several decisions needs careful conditioning. The chain rule gives an exact expression conditional on the initial history: each factor is a probability of surviving the next decision, conditional on having survived all earlier decisions and on the initial information. These factors integrate over any future covariate or outcome histories not included in that initial information.

For \(s=t\), the list of preceding \(D\)'s is empty. Later factors are conditional on survival up to that decision, not simply the hazard evaluated at an arbitrary future history. If future histories are random, they must be accounted for. In a model that specifies hazards along a fixed or conditioned-upon covariate/history path, the path-specific survival probability is the product of \(1-q_{is}\) along that path. When conditioning only on the initial history, the survival probability averages the path-specific probabilities over the appropriate distribution of future histories. It is generally incorrect to multiply hazards evaluated at random future values without this conditioning or averaging.

Assumptions, Diagnostics, and Alternatives

A time-varying selection equation is useful only if its timing and statistical assumptions are plausible. Check the measurement schedule and data timestamps, define precisely when each dropout decision occurs, and verify that every proposed predictor is available before that decision. Also check that dropout is represented as an absorbing event and that each hazard is estimated among individuals still at risk. If the covariate itself is unavailable after dropout, that is expected for later measurements; it does not justify replacing an unobserved value with a future value.

The logistic specification assumes that the chosen predictors and functional forms describe conditional dropout odds adequately. Diagnostics should include predicted hazards across decision times and covariate ranges, sparse risk sets, extreme fitted probabilities, and possible separation. Compare observed dropout frequencies with model-based predictions within useful time and covariate groups. For continuous predictors, inspect whether a linear log-odds effect is plausible; for repeated predictors, consider whether a single coefficient across time is defensible. Strong association between current covariates and prior outcomes can also make coefficients unstable, even when predictions appear reasonable.

A baseline-only selection equation is simpler and avoids requiring a new covariate value at every decision, but it cannot represent dropout associations with observed changes over follow-up. A time-varying equation can use updated information, but it adds modeling choices and can be sensitive to measurement timing and missing covariate histories. If covariate values and dropout need to be modeled as jointly evolving processes, a joint model may be more appropriate than treating the covariate path as fixed. These approaches answer different modeling needs; none turns an association in the dropout equation into a causal effect.

A time-varying predictor should therefore be evaluated twice: first as a statistical variable, for example by checking its scale and functional form, and then as an event-time variable, by confirming when its value becomes known. The second check is not a minor data-cleaning step. It determines what conditional probability the selection equation represents and whether the model can be used to describe dropout at the stated decision time.