Available-Case Restrictions With Unequal Pattern Weights

Why Pattern Weights Matter

When dropout patterns have different sizes, the available cases at a visit are not an equally weighted collection of patterns. A pattern representing half the study contributes more individuals to the available-case distribution than a pattern representing one tenth. That distinction matters when the available-case distribution is used to specify the distribution of outcomes that were not observed.

The available-case restriction makes this borrowing explicit: for a particular visit, the distribution among patterns missing that visit is set equal to the distribution among all patterns that observed it. The latter is a mixture, and its weights come from the proportions of the observed patterns. This tutorial derives that mixture and follows its implications for a three-visit monotone-dropout setting.

Patterns and the Available-Case Mixture

Suppose each participant is scheduled for visits 1, 2, and 3, with monotone dropout: after a participant misses a visit, they do not return at a later visit. Write \(Y_j\) for the outcome at visit \(j\). Use pattern labels \(111\), \(110\), and \(100\): the \(j\)th digit is 1 if the outcome at visit \(j\) is observed and 0 otherwise. Let \(\pi_g=P(G=g)\) be the proportion of the study population in pattern \(g\), where \(G\) is the dropout pattern.

The weights are conditional proportions among people who observed the visit. They are nonnegative and sum to 1. In the three-visit setting, \(\mathcal{A}_2=\{111,110\}\) and \(\mathcal{A}_3=\{111\}\). Thus visit 2's available cases mix the \(111\) and \(110\) distributions, while visit 3's available cases come only from pattern \(111\).

The available-case restriction is an identifying assumption for patterns missing a visit. At visit 2, it sets \(F_{2,100}=F_2^{AC}\). At visit 3, it sets \(F_{3,110}=F_3^{AC}\) and \(F_{3,100}=F_3^{AC}\). These equalities concern distributions, not merely means. They are assumptions about unobserved outcomes; the observed data alone do not verify them.

Deriving the Implied Marginal Distribution

For any fixed visit \(j\), the overall population distribution is a mixture across all dropout patterns. Write \(\mathcal{M}_j\) for patterns missing \(Y_j\), and let \(q_j=\sum_{g\in\mathcal{A}_j}\pi_g\) be the population proportion with \(Y_j\) observed. The available-case restriction assigns \(F_{jg}=F_j^{AC}\) for every \(g\in\mathcal{M}_j\). Substitution into the full mixture gives:

The first equality is the law of total probability over dropout patterns. In the second, the observed-pattern contribution is \(q_jF_j^{AC}\) by the definition of the available-case mixture, while the missing-pattern contribution is \((1-q_j)F_j^{AC}\) by the restriction. Their coefficients sum to 1, proving the result. Thus the restriction implies that the full-population marginal distribution at that visit equals the available-case distribution.

This result does not mean that every pattern has the same outcome distribution. At visit 2, for example, \(F_{2,111}\) and \(F_{2,110}\) may differ; the restriction assigns their weighted mixture to pattern \(100\). The overall distribution equals that mixture because the missing-pattern contribution has been defined to match it.

Means and Pattern-Weighted Reconstruction

If the outcome has a finite mean, the distributional restriction implies the corresponding mean restriction. Let \(\mu_{jg}=E(Y_j\mid G=g)\) when this expectation exists. The available-case mean is \(\mu_j^{AC}=\sum_{g\in\mathcal{A}_j}w_{jg}\mu_{jg}\). For any pattern missing the visit, the restriction gives \(E(Y_j\mid G=g)=\mu_j^{AC}\). The overall mean is then \(\sum_g\pi_g\mu_{jg}=\mu_j^{AC}\). A mean-only restriction can also be imposed directly, but it is weaker: matching means does not establish matching distributions.

Conditioning Changes the Weights

The mixture must use the conditioning set specified by the restriction. If a restriction is imposed within baseline-covariate strata, compute the observed-pattern weights separately within each stratum. A marginal mixture generally is not the same as a mixture within strata: both pattern proportions and outcome means can vary with the covariate.

Interpretation, Diagnostics, and Alternatives

The pattern weights determine which observed distributions contribute to the borrowed distribution. A small observed pattern receives a small mixture weight, even if its mean or distribution differs sharply from that of a larger pattern. At a later visit in this three-visit monotone setting, only pattern \(111\) observes \(Y_3\); the available-case distribution there is simply \(F_{3,111}\). The restriction assigns that distribution to both patterns missing visit 3. At visit 2, by contrast, the borrowed distribution is a mixture of two observed patterns.

The restriction is not a consequence of dropout being monotone, and it is not established by the observed outcomes. It is an identifying choice. Its implications can be checked for coherence with observed pattern summaries and the restrictions being imposed, but agreement of observed summaries cannot prove the assumption about outcomes that were not observed. A later analysis may reveal conflicts among restrictions or other incompatibilities.

The complete-case restriction is a different borrowing choice: it uses the complete-pattern distribution rather than the available-case pool. For visit 2, that replaces the mixture of patterns \(111\) and \(110\) with pattern \(111\) alone. An unweighted average of observed-pattern distributions is another possible calculation, but it is not the available-case mixture unless the observed pattern proportions are equal. Neighboring-case restrictions make a different assignment based on selected adjacent patterns; they should not be mistaken for the pooled available-case rule.

A common mistake is to say, “There are two observed patterns, so average their distributions 50–50.” That reasoning counts pattern labels rather than participants. The correction is to normalize each observed pattern's population proportion by the total proportion observed at that visit. Another mistake is to conclude that the restriction says all pattern-specific distributions are equal. It says that each missing pattern is assigned the available-case mixture; observed patterns may retain different distributions.