Intermittently Missing Marker Measurements

Why Intermittent Missingness Needs Its Own Likelihood

In a multimarker joint model, a subject may have some marker values recorded at a visit and others absent, or may miss one measurement at a visit but provide that marker at later visits. This is intermittent missingness: the observed data are scattered through the planned measurement schedule rather than forming one uninterrupted block. The central likelihood question is whether the recorded values can still inform the marker and survival parameters without filling in every absent value.

The answer is often yes, but not simply because a likelihood can be written down. Likelihood-based estimation can use the available measurements when the marker model is correctly specified and the missingness mechanism is ignorable under appropriate conditions. If the chance a value is missing depends on its unobserved value, the pattern of absences may itself contain information about model parameters and cannot generally be discarded.

A missing value is not a measured zero, and it is not automatically the last value that was observed. It is an unobserved component of the data. The likelihood must account for its absence by using the probability density of the observed components, rather than pretending the full vector was recorded.

The Observed-Data Likelihood

Let \(b_i\) denote subject \(i\)'s vector of marker random effects, with density \(f_b(b_i;\theta)\). Let \(T_i\) and \(\delta_i\) denote the observed event or censoring time and event indicator. Conditional on \(b_i\), the joint model specifies a marker distribution and a survival contribution, denoted \(f_Y(\cdot\mid b_i;\theta)\) and \(L_{Ti}(T_i,\delta_i\mid b_i;\theta)\). Here \(\theta\) collects the parameters of the marker and survival models. The survival contribution is a density for an observed event and a survival probability for a censored observation, as specified by the survival model.

Write \(Y_i^{\mathrm{obs}}\) for the observed marker components and \(Y_i^{\mathrm{mis}}\) for the missing components. If the full marker vector has a density, its contribution after integrating out the missing values is the marginal density of \(Y_i^{\mathrm{obs}}\). Equivalently, for each fixed \(b_i\), \[ \int f_Y(Y_i^{\mathrm{obs}},Y_i^{\mathrm{mis}}\mid b_i;\theta)\,dY_i^{\mathrm{mis}} =f_Y(Y_i^{\mathrm{obs}}\mid b_i;\theta). \] The subject-level joint contribution, when the marker and survival data are conditionally independent given the random effects, is therefore

This expression uses the remaining marker measurements and the subject’s survival information together, while integrating over the unobserved random effects. It does not require estimating a missing marker value first. If the survival model depends on the latent marker trajectories, those trajectories are determined by the model conditional on \(b_i\), and the survival contribution remains inside the same integral.

For a multivariate normal marker model, the rule for partial observations is important. At a visit, take the subvector of observed markers and use its marginal normal distribution, with the corresponding subvector of means and submatrix of covariances. Do not use a complete-vector density with invented values for the absent markers. In particular, when residuals across markers at a visit are correlated, an observed marker’s marginal variance is the relevant variance if the other marker is missing; it is not the conditional variance given an unobserved marker.

When the Missingness Mechanism Can Be Ignored

Let \(R_i\) denote the subject’s full missingness pattern. A missingness mechanism describes the probability of that pattern given the data. For likelihood inference to ignore this mechanism, two conditions are central. First, missingness must be missing at random (MAR) relative to the variables included in the observed-data analysis. Second, the parameters governing missingness must be distinct from the parameters of the marker and survival model. These are conditions for ignorability, not claims that missingness is completely unrelated to the data.

Including \(T_i\) and \(\delta_i\) in this definition matters in a joint model: event and censoring information are observed outcomes in the analysis. For example, a measurement could be more likely to be absent after an observed hospitalization. That dependence need not violate MAR if hospitalization information is included among the observed data and missingness does not additionally depend on the absent marker value.

A short factorization shows why the conditions matter. Let \(g_\psi\) describe the missingness mechanism with parameter \(\psi\), and suppose the full-data density is \(f_\theta\). For a fixed pattern \(R_i\), the observed-data likelihood contribution is

The second equality follows from MAR: the missingness factor is constant with respect to the missing values being integrated out. If \(\psi\) is distinct from \(\theta\), the first factor can be omitted when estimating \(\theta\) by likelihood. The remaining integral is the observed-data likelihood under the joint model. Thus, all observed marker values can contribute even when the marker panel is incomplete.

These conditions do not repair a misspecified marker distribution, random-effects distribution, or survival model. Nor do they establish that the conditional independence assumptions for the marker and survival processes are appropriate. They justify omitting the missingness factor from the likelihood only when the stated MAR and parameter-distinctness conditions hold.

When Ignoring Missingness Is Not Justified

MAR is not the same as “the values are missing at random” in casual speech. If the chance of absence depends on the value that was not recorded, even after conditioning on all relevant observed data, missingness is not MAR. Such a mechanism is often called missing not at random (MNAR). In that case, the observed-data likelihood generally includes the missingness model, or a sensitivity analysis is needed to assess how conclusions change under plausible assumptions about that dependence.

The fact that MAR cannot be verified from observed data alone is an important limitation. Analysts can inspect observed patterns and assess whether the proposed model is plausible, but they cannot use the missing values themselves to prove that missingness does not depend on them. When the scientific setting makes MNAR plausible, a model for the missingness process or a sensitivity analysis should be considered. The assumptions and sensitivity parameters should be reported, because different plausible assumptions may produce different estimates.

Practical Checks and Alternatives

Before relying on the observed-data likelihood, check whether missingness is genuinely intermittent, whether marker and visit identifiers have been aligned correctly, and whether missingness differs across observed groups or observed histories. Such checks can reveal data-processing errors or patterns that challenge a proposed MAR explanation; they cannot establish MAR. Also verify that the fitted likelihood uses the correct marginal covariance for each observed subset. In a multivariate marker model, dropping an entire visit whenever one marker is absent discards usable observations and changes the analysis unnecessarily.

The principal alternative is a complete-case analysis, which uses only subjects with all required measurements. It may lose substantial information and can be biased if inclusion as a complete case is related to outcomes or covariates. Imputation is another strategy, but imputing absent values does not itself solve an MNAR problem; the imputation model must reflect the joint analysis and its missingness assumptions. When MNAR is plausible, selection models explicitly specify missingness conditional on the full data, while pattern-mixture approaches model outcome distributions by missingness pattern. These alternatives require additional assumptions, so comparisons across plausible assumptions are often more informative than presenting one unexamined mechanism as certain.

• Do not substitute zero or the previous measurement. Neither operation follows from the likelihood and both alter the data model.
• Do not confuse missingness with censoring. A missing marker value is an unobserved longitudinal outcome; censoring describes incomplete event-time information.
• Do not assume a missing marker is independent of an observed marker. Use the marginal distribution of the observed subset implied by the fitted covariance structure.
• Do not infer MAR from a nonsignificant comparison. Observed-data checks can flag concerns but cannot test dependence on unobserved values.