Likelihood Contributions Under Monotone Dropout in Missing-Data Models

Why the Dropout Event Belongs in the Likelihood

With monotone dropout, a subject’s outcomes are observed through some visit and then missing at every later scheduled visit. The observed outcomes alone do not describe the entire observed data: the fact that the subject stopped contributing measurements is also observed. A likelihood contribution must therefore account both for the observed outcome values and for the response pattern that ended in dropout.

The key calculation is at the first missing visit. The subject was retained through the last observed visit, then dropped out at the next one. If dropout at that next visit can depend on its outcome, that outcome is missing precisely when its dropout probability is needed. The event probability must then be averaged over the model for the missing outcome, rather than evaluated at a substituted value.

Notation and Model Factorization

Condition throughout on the subject’s covariates \(X\). Specify the longitudinal outcome model by conditional densities \(f_t(y_t\mid y_1,\ldots,y_{t-1},X)\), with \(f_1(y_1\mid X)\) for the first visit. These densities factor the full outcome distribution; the model need not be Markov, because each density may condition on the full previous outcome history.

For a subject still under observation at visit \(t\), let \(\pi_t(y_1,\ldots,y_t,X)=P(R_t=1\mid R_{t-1}=1,Y_1=y_1,\ldots,Y_t=y_t,X)\) be the probability of being observed at that visit. Set \(R_0=1\). Thus \(1-\pi_t\) is the probability of dropping out at visit \(t\), conditional on having remained through visit \(t-1\) and on the outcome history through \(t\). This is a selection-model specification: dropout may depend on the current outcome, including an outcome that is missing because dropout occurs at that visit.

Contribution for Each Dropout Time

Suppose outcomes \(y_1,\ldots,y_k\) are observed and \(k<T\). At visits \(1,\ldots,k\), the subject both has the observed outcome and remains under observation. At visit \(k+1\), the subject drops out. Write \(u_{k+1},\ldots,u_T\) for the unobserved outcomes. The individual observed-data likelihood contribution is

The product through \(k\) records both the observed outcomes and the retention events. The factor \(1-\pi_{k+1}\) records the dropout event itself. All outcomes after the last observed visit are integrated out. If a dropout probability depends on an unobserved outcome at a later visit, that outcome cannot simply be omitted: it remains inside the integral wherever it enters the response model.

For a completer, every outcome is observed and the contribution has no missing-outcome integral:

For dropout before the first outcome, use the same pattern with \(k=0\): integrate the full outcome distribution multiplied by the probability of dropout at visit 1. For independent subjects, the full likelihood is the product of their individual contributions, or equivalently the sum of their log contributions on the log-likelihood scale.

How the Integration Can Simplify

Sometimes dropout at the first missing visit depends only on the observed history, not on its missing current outcome. Then its event probability can be taken outside the integral. The missing outcome densities integrate to one, provided the response model has no further factors depending on those missing outcomes. This yields a simpler pattern probability while retaining the observed outcome densities and the retention probabilities up to the last observed visit.

A Later Dropout Event and a Complete Pattern

For dropout at a later visit, each earlier observed visit contributes both an outcome density and a retention probability. The next example makes that sequence explicit.

For comparison, a hypothetical subject observed at all three visits contributes the product of three outcome densities and three retention probabilities. The completer’s contribution contains \(\pi_3\), not \(1-\pi_3\); the subject remained observed at visit 3. This distinction is a useful check on the pattern being encoded.

Derivation and Practical Checks

The formula follows directly from the joint model for outcomes and response. For fully specified outcomes, the outcome factor is the product of the \(f_t\)'s. Under the sequential response model, the probability of a pattern observed through \(k\) and then dropping out is the product of retention probabilities through \(k\), followed by the dropout probability at \(k+1\). Multiplying these gives the joint density for the full outcomes and that response pattern. The observed-data likelihood is the marginal density of what was recorded, so integrate that joint density over every missing outcome. This establishes both why the dropout-event factor appears and why it must remain inside the integral when it depends on missing outcomes.

A practical construction check is to read the subject’s pattern from left to right:

Several failures can be detected by checking whether a factor matches the observed pattern. A dropout pattern must not contain a retention factor at its first missing visit; a completer must not contain a dropout factor there. The contribution should also be a probability or density on the appropriate scale, and the integrated missing-outcome model should be normalized. Numerically, an outcome-dependent dropout term can be especially sensitive to the tail behavior of the missing-outcome distribution, so integration limits, quadrature accuracy, and probability calculations should be checked.

Relation to Other Likelihood Representations

The sequential expression is one way to calculate the observed-data likelihood under a specified selection model. A joint-pattern formulation can instead write the probability of each complete response pattern conditional on the outcomes, then integrate missing outcomes. It gives the same contribution when it represents the same joint model. The sequential form is useful for monotone dropout because it makes each retention event and the first dropout event visible.

A pattern-mixture formulation instead models outcomes conditional on the missingness pattern. It organizes the same observed data by pattern, but it is not obtained by simply deleting the dropout factor from the selection-model contribution: the conditional outcome distributions and pattern probabilities must together represent a coherent joint distribution. Complete-case analysis is different again; it discards dropout patterns and does not calculate their likelihood contributions. These alternatives do not remove the need to specify assumptions about unobserved outcomes.

Common Mistake and Limitations

These contributions are valid only relative to the specified outcome and response models, the visit schedule, and the monotone-pattern assumption. They do not establish that the selection model is correct or that its unobserved-outcome assumptions are empirically identified from observed data. If dropout is nonmonotone, the response-pattern factorization must represent the actual return and missingness process rather than treating the first absence as absorbing. In all cases, a likelihood can be calculated precisely while still depending strongly on assumptions about outcomes that were never observed.