Informative Dropout In Longitudinal Omics

Why Dropout Can Change the Trajectory We See

Longitudinal omics studies often lose measurements when participants withdraw, become too ill to attend, or stop providing samples. If these losses are related to an unobserved part of a participant’s omics trajectory, the observed measurements may not represent the trajectory the study set out to describe. For example, participants whose latent biomarker levels are rising rapidly might be more likely to withdraw before later samples are collected. An analysis of only the observed measurements can then understate the rise.

The issue is not simply that the number of observations falls over time. The key question is whether the probability of dropout depends on values that are missing, after accounting for the information actually observed. The earlier material on mixed models for repeated omics measurements provides a way to represent participant-specific trajectories; here, the focus is on linking those trajectories to dropout.

MAR does not mean that dropout is unrelated to participants’ data. A participant may withdraw because of a measured prior assay value, provided the dropout mechanism depends only on observed information included in the conditioning set. By contrast, dropout related to a participant’s unobserved current assay value or latent trajectory is generally informative for inference about that trajectory. Whether a particular mechanism is MAR cannot usually be established from observed data alone; the unobserved values needed to distinguish mechanisms are, by definition, unavailable.

Representing Dropout and Its Relationship to a Trajectory

Let \(Y_{ij}\) be an omics measurement for participant \(i\) at scheduled time \(t_{ij}\), and let \(D_i\) be the time of dropout. A simple longitudinal model can be written

Here, \(m_i(t)\) is the participant’s latent mean trajectory, \(x_i(t)\) and \(z_i(t)\) are specified covariate and time functions, \(\beta\) is a vector of fixed effects, and \(b_i\) is a vector of participant-specific random effects. The errors \(\epsilon_{ij}\) describe measurement variation around the latent trajectory. In a shared-parameter model, some or all components of \(b_i\), or the current latent value \(m_i(t)\), also enter the dropout model.

For example, a proportional-hazards dropout model is

where \(h_{D,i}\) is the instantaneous dropout hazard among participants still under follow-up, \(h_{D,0}(t)\) is a baseline hazard, \(w_i(t)\) contains observed predictors, and \(\gamma\) and \(\eta\) are association parameters. A positive coefficient for a random intercept, for example, means that a higher latent intercept is associated with a higher dropout hazard, conditional on the other modeled predictors. It does not establish that the omics level causes withdrawal.

If a clinical event ends the possibility of further measurement, it should not automatically be treated as ordinary withdrawal. Let \(T_i\) be the event time. The event can be modeled separately from dropout, for example with an event hazard that shares \(b_i\) with the longitudinal model. This represents the possibility that latent trajectory features predict both the event and the chance of observing future assays. The event itself and withdrawal are distinct processes, even if both lead to no later measurements. The scientific estimand must also specify whether the target concerns trajectories only while participants are alive, or a hypothetical trajectory beyond an event that makes measurement impossible.

Likelihood and the Role of Shared Parameters

Suppose \(Y_{i,\mathrm{obs}}\) denotes participant \(i\)’s observed assays, and the dropout record includes the observed follow-up time and whether dropout occurred. Write \(f_Y\) for the longitudinal density, \(f_D\) for the dropout contribution, and \(f_b\) for the random-effects density. Under conditional independence given \(b_i\), the participant’s observed-data likelihood contribution is

Here \(\delta_i\) indicates whether dropout occurred rather than follow-up ending for another recorded reason, and \(\theta,\psi,\omega\) are model parameters. The integral matters: the participant’s latent effects are not observed, so the likelihood averages over their possible values rather than treating an estimated random effect as known. If an event process is modeled jointly, its event or censoring contribution is included in the integrand as well.

The logic can be seen by first conditioning on \(b_i\). Given \(b_i\), the model specifies the distribution of observed assays and the distribution of dropout; multiplying these conditional contributions gives the joint contribution at that latent value. Since \(b_i\) is unobserved, averaging over its distribution yields the displayed integral. If dropout depends on \(b_i\), dropout information changes which values of \(b_i\) are plausible for a participant. Consequently, it also changes the implied distribution of that participant’s unobserved assay values.

A mechanism that depends on a latent random effect can be conditionally independent of assay errors given that effect, yet still be informative for the marginal trajectory. This is why “the dropout model shares a random effect with the assay model” is not the same claim as “dropout is MAR given observed data.” Under MAR, with appropriate parameter distinctness, the missingness mechanism can often be omitted for likelihood inference about the outcome model. Under informative dropout, that omission generally loses the dependence needed to describe the observed-data distribution.

Worked Risk Calculation Under a Shared-Parameter Model

A hazard model translates an association with a latent trajectory into dropout probabilities over a specified interval. The following calculation is illustrative, not a fit to real participants. Assume a constant baseline dropout hazard of \(0.10\) per month and a scalar random effect \(b_i\). Let

Assume the model’s hazard applies throughout the six-month interval, there is no additional time-varying predictor, and dropout is the only endpoint considered in this calculation. With a constant hazard \(h\), the probability of dropout by time \(t\) is \(1-\exp(-ht)\).

Using Observed Trajectories to Update Dropout Risk

The random effect is unknown for a participant. A fitted longitudinal model uses observed assays to form a distribution for that participant’s latent effect, rather than assigning the person to a known \(b_i\). The dropout probability conditional on the observed assays is then averaged over this updated distribution. This is a central reason to fit the longitudinal and dropout components together.

Diagnostics, Alternatives, and Limits

No single diagnostic can prove that an informative-dropout model is correct, because the unobserved trajectories that would distinguish MAR from non-MAR are unavailable. Diagnostics instead assess the fit and consequences of a chosen model. Useful checks include comparing predicted and observed dropout frequencies over time and across observed assay-history groups, examining whether the fitted model reproduces the observed distributions of assay values before withdrawal, and checking convergence and sensitivity to the random-effects distribution and baseline dropout hazard.

A model may be going wrong if predicted dropout is systematically too early or too late, if observed dropout varies across observed-history groups in ways the model fails to reproduce, or if association estimates change sharply under modest changes to the trajectory or dropout specification. Sparse late follow-up, few dropouts, and little variation in observed trajectories can leave the association between latent trajectory and dropout weakly identified. A close fit to observed data does not, by itself, validate extrapolations to unobserved assay values.

Shared-parameter models are not the only approach. Selection models specify the outcome distribution and then model missingness conditional on outcomes. Pattern-mixture models specify outcome distributions separately by dropout pattern and require assumptions to relate observed and unobserved patterns. These approaches express assumptions differently; none identifies the unobserved outcomes without assumptions. Shared-parameter models are especially natural when a scientifically meaningful latent trajectory can plausibly link assays and dropout. Selection and pattern-mixture models can be useful alternatives for sensitivity analyses because they make different extrapolation assumptions.

The principal limitation is that informative-dropout conclusions depend on untestable assumptions about unobserved values. A random-effect association parameter may be estimated precisely under one specification while being sensitive to plausible alternatives. Report the dropout model, its link to the trajectory, the time scale, and the assumptions used for participants after dropout. Sensitivity analyses should vary important assumptions and show whether the scientific conclusion changes.