Deriving Available-Case Mixtures Across Dropout Patterns

Why the Available-Case Distribution Is a Mixture

At a particular visit, the people with observed outcomes may come from several dropout patterns. For example, someone observed at visit 2 might drop out after visit 2 or might remain in the study through visit 3. The distribution among everyone observed at visit 2 therefore combines the visit-2 distributions for each of those patterns. The combination is weighted by how common each pattern is among people observed at that visit, not by equal weights.

This distinction matters because a pattern-specific distribution for an outcome that was not observed cannot be read off from the available-case data. The available-case mixture can be derived and estimated from observed outcomes; using it to represent the missing pattern components requires an additional identifying restriction.

The distribution \(F_{t\mid g}\) is a conceptual pattern-specific distribution even when \(Y_t\) is not observed in pattern \(g\). If \(g\geq t\), it is a distribution of outcomes observed at visit \(t\). If \(g<t\), it concerns outcomes after that pattern's dropout and cannot be directly estimated from the observed visit-\(t\) data.

Deriving the Pattern-Weighted Mixture

Among people observed at visit \(t\), only patterns \(g=t,\ldots,K\) are represented. Conditional on being observed, their pattern weights are \(P(G=g\mid R_t=1)=\pi_g/\sum_{h=t}^K\pi_h\). Applying the law of total probability across these mutually exclusive patterns gives the available-case mixture.

The weights are nonnegative and sum to one. In a sample, if \(n_g\) participants have pattern \(g\), the corresponding empirical weights among the available cases are \(n_g/\sum_{h=t}^K n_h\). The mixture is not generally an unweighted average of the pattern-specific distributions.

The derivation does not require that dropout be independent of outcomes. It is an identity for the distribution among people observed at visit \(t\), given the pattern definition and observed data. Monotone dropout makes the set of contributing patterns especially clear. The identity does not, by itself, determine the outcome distributions in patterns with \(g<t\).

What the Mixture Does Not Identify

The full-population distribution of \(Y_t\), if outcomes at visit \(t\) were defined for everyone, would include all dropout patterns:

For \(g\geq t\), the pattern-specific outcomes at visit \(t\) are observed, subject to ordinary sampling uncertainty. For \(g<t\), they are not observed. Thus the available-case mixture identifies the distribution conditional on being observed at visit \(t\), but does not identify the full-population distribution \(F_t\). The missing components are the pattern-specific distributions \(F_{t\mid g}\) for \(g<t\), and their population weights are \(\pi_g\).

The available-case restriction is an assumption that fills this gap by setting each missing pattern-specific distribution equal to the observed available-case distribution at that visit: \(F_{t\mid g}(y)=F_t^{AC}(y)\) for every \(g<t\). Under this restriction, the full-population distribution equals the available-case distribution. The observed-data mixture does not prove the restriction; its validity cannot be checked directly using outcomes that are missing in those patterns.

This short derivation relies on two facts: the restriction equates every early-dropout component to the available-case distribution, and the pattern probabilities sum to one. Without that restriction, the first line contains unobserved distributions and the full-population distribution is not determined by the available cases alone.

Later Visits and Boundary Patterns

As the visit index increases, fewer dropout patterns remain observable. At the final visit, only \(G=K\) contributes to the available-case distribution. Earlier dropout patterns then all have unobserved final-visit outcomes. This is a boundary case of the same mixture formula: its observed mixture has one component, while the full-population distribution still has components for every pattern.

Assumptions, Checks, and Alternatives

Before using this construction, verify that the pattern labels match the study's observation process. The simple indexing \(G=g\) assumes monotone dropout: after a participant misses a visit, later outcomes are not observed. If the data allow intermittent missingness, the pattern definition must represent the actual observation histories rather than silently treating them as monotone.

Useful checks concern the observable calculation, not the unobserved restriction. Confirm that pattern counts among available cases give nonnegative weights summing to one; that each included pattern actually has outcomes at visit \(t\); and that the weighted probabilities form a valid distribution. If a calculated mixture probability is negative or the probabilities do not sum to one, the weighting or pattern-specific summaries have been mishandled. Sparse patterns can also make empirical pattern distributions unstable.

No diagnostic based only on the observed outcomes can establish that the missing-pattern distributions equal \(F_t^{AC}\). This restriction is not a claim that dropout is unrelated to outcomes in every sense; it is a specific equality of outcome distributions across the observed and unobserved patterns at a given visit. Its plausibility must be assessed using subject-matter knowledge and sensitivity to alternative assumptions.

The main contrast is with a complete-case restriction, which uses only participants observed through all visits as the reference group. At an intermediate visit, the available-case distribution also includes participants who are observed at that visit but drop out later. It can therefore differ from the complete-case distribution. Pattern-specific extrapolation is another alternative: it assigns missing-pattern distributions using a model or sensitivity parameters rather than setting them equal to the available-case distribution. These approaches answer different assumption-based questions; none is made correct by the mixture identity itself.