Missing Marker Measurements in the Observed-Data Likelihood

Why Missing Marker Values Need Careful Treatment

Longitudinal follow-up rarely produces a complete row of marker measurements for every participant. A scheduled visit may be missed, a specimen may be unusable, or a measurement may not be recorded. In a joint model, the available marker values still inform the subject-specific trajectory and, through shared random effects, the event-time contribution. The central task is therefore not to discard every subject with an incomplete marker record, but to write the likelihood for what was actually observed and state when the process that produced the missing values can be left out.

A missing scheduled measurement is different from a measurement that was never due, or one scheduled after the subject’s event or censoring time. Only a scheduled measurement within the subject’s observation period is a candidate for being missing. The observation-schedule and risk-time conventions established earlier determine which occasions belong in the record; missingness concerns an eligible occasion whose marker value was not observed.

Observed Marker Densities in the Joint Likelihood

For subject \(i\), let \(b_i\) denote the shared random effects, with density \(p(b_i;\theta_b)\). At scheduled time \(t_{ij}\), let \(Y_{ij}\) be the marker and let \(R_{ij}=1\) indicate that its value was observed; \(R_{ij}=0\) indicates that the scheduled value is missing. Define \(\mathcal O_i=\{j:R_{ij}=1\}\), the set of observed scheduled marker occasions. Write \(T_i\) for the observed event or censoring time and \(\delta_i\) for the event indicator. Conditional on \(b_i\), the longitudinal model supplies marker densities \(f_Y(y_{ij}\mid b_i;\theta_y)\), and the survival model supplies the event or censoring contribution \(f_T(T_i,\delta_i\mid b_i;\theta_T)\).

The expression is conditional on a set of assumptions. The marker and event models must be correctly specified, including their conditional structure given the shared random effects. The model must use the actual observed values and the applicable risk period. In particular, a marker value after the event is not a scheduled pre-event value with a missing observation. The formula also presumes that the missingness process is ignorable under the conditions stated below. If those conditions fail, the displayed likelihood alone may not represent the likelihood of the observed data.

For a right-censored subject, \(f_T\) is the survival-model contribution appropriate to the observed time and event indicator; for an observed event it includes the event density. Thus missingness changes the longitudinal product, not the rule for constructing the event contribution. The integral over \(b_i\) is still necessary: the observed markers and event record jointly contribute information about the shared latent effects.

Why an Unobserved Marker Density Disappears

The omission of a missing marker density is not an assumption that its value is zero, nor is it a substitution of a guessed value. It follows from integrating over the unobserved value under the longitudinal model. Conditional on \(b_i\), a properly normalized density integrates to one. If the missingness mechanism is ignorable, integrating over a missing marker contributes no additional factor to the likelihood for \(\theta\).

When the Missingness Process Can Be Ignored

Let \(Y_{\mathrm{obs}}\) and \(Y_{\mathrm{mis}}\) denote all scheduled marker values that are observed and missing, respectively. Let \(R\) denote the full observation pattern, and let \(D\) denote other observed data used in the joint likelihood, including covariates and the event or censoring record. A standard sufficient condition for ignoring the missingness process is missing at random (MAR): conditional on the observed data, the probability of the observation pattern does not depend on the unobserved marker values. Formally, for missingness parameters \(\psi\), \(p(R\mid Y_{\mathrm{obs}},Y_{\mathrm{mis}},D;\psi)=p(R\mid Y_{\mathrm{obs}},D;\psi)\).

MAR alone is not enough. The parameters governing the data model, \(\theta\), and those governing missingness, \(\psi\), must be distinct, so the parameter space factors into separate components rather than imposing a shared constraint. Under MAR and parameter distinctness, the observed-data likelihood factors into a data-model likelihood and a missingness factor that depends on observed quantities and \(\psi\), not on \(\theta\). Inference about \(\theta\) can then use the observed-data likelihood above without modeling the missingness process.

The observed information relevant to MAR must be defined carefully. In this setting it includes the observed event or censoring history, not only the observed marker values. For example, missingness may depend on an already observed event status or on a previously recorded marker value and still satisfy MAR, provided it does not additionally depend on the unobserved marker value after conditioning on the observed data. A dependence on an unobserved current marker, or on a latent random effect that remains associated with that marker after conditioning on observed data, is not automatically ignorable.

Practical Checks, Alternatives, and Limits

Before fitting the likelihood, verify which visits were scheduled and eligible, distinguish missing values from visits after the terminal time, and check that every observed marker density corresponds to an actual recorded value. Inspect missingness rates by visit and compare them across observed covariates, prior marker values, and event or censoring histories. Such checks can reveal patterns inconsistent with a proposed observation model, but they cannot establish MAR: the dependence on the unobserved values is not directly testable from the observed data alone.

Likelihood-based use of incomplete records differs from complete-case analysis. Complete-case analysis discards subjects with any missing marker value; it can lose substantial information and may induce selection bias. The observed-data likelihood uses each subject’s available marker measurements and event record, while integrating over shared random effects. It is not a general cure for informative missingness: if the mechanism is nonignorable, omitting it can bias estimation even when the longitudinal and survival submodels are otherwise correctly specified.

Other approaches include explicitly modeling the missingness probability, or conducting sensitivity analyses using selection or pattern-mixture models. Multiple imputation is another possible strategy, but imputations must respect the longitudinal structure, event information, and dependence represented by the joint model. These approaches require additional modeling choices; none makes the missingness assumptions disappear. The appropriate analysis depends on the scientific context and on how plausible the proposed assumptions are.

Common Pitfall

A frequent error is to say, “The marker is missing, so its likelihood contribution is zero,” or to substitute a predicted marker value and treat it as observed. Both are incorrect. A missing value contributes no observed marker density under an ignorable mechanism because its density integrates to one, not because its value is zero. The subject’s other observed markers and event contribution remain in the likelihood, and uncertainty about the shared random effects is handled by integration. If missingness depends on the unobserved value, the missingness process may itself carry information and cannot simply be discarded.