Pattern Labels and Conditional Outcome Distributions

Why Pattern Labels Need Precise Notation

A pattern-mixture analysis organizes people by their missingness patterns and describes outcomes within those groups. The pattern label is not itself a model for the outcomes: it tells us which visits were observed. To state a restriction about missing outcomes precisely, we also need to specify the outcome being described and the information on which its distribution is conditioned.

For a study with scheduled visits, monotone dropout gives a particularly simple set of labels. Each participant has an observed prefix of visits followed by a missing suffix. Once that pattern is named, we can distinguish the distribution of outcomes within a pattern from the distribution conditional on that pattern’s observed history. These distinctions are essential when describing complete-case, available-case, or neighboring-case restrictions; the labels alone do not imply any of those restrictions.

Under this definition, pattern \(K=k\) means that \(Y_1,\ldots,Y_k\) are observed and \(Y_{k+1},\ldots,Y_T\) are missing. The convention for \(k=0\) is that the observed outcome history is empty. If a study has intermittent missingness, such as an unobserved \(Y_2\) followed by an observed \(Y_3\), the monotone pattern labels do not describe it without an additional rule. Do not silently classify such a record as monotone dropout.

Histories and Pattern-Specific Laws

Write the observed history in pattern \(k\) as \(H_k=(X,Y_1,\ldots,Y_k)\). In particular, \(H_0=X\). The history is the information on which a conditional outcome distribution may depend. For \(j>k\), the outcome \(Y_j\) is missing in pattern \(k\), but its conditional distribution can still be defined as a feature of the joint data-generating distribution.

The condition \(K=k\) matters. A conditional distribution among people who stop after visit \(k\) is not automatically the same as one among people who continue beyond that visit, even when their recorded histories match. Also distinguish the conditional law above from the full pattern-specific law \(\mathcal L(Y_{1:T}\mid X=x,K=k)\). The latter describes the complete outcome vector within pattern \(k\) at covariate value \(x\); the former conditions on the pattern’s observed outcomes as well as \(X\).

These are definitions of distributions, not claims that every distribution can be estimated from observed data. For \(j>k\), \(Y_j\) is unobserved in pattern \(k\), so its conditional distribution there is not directly tabulated from that pattern’s records. An assumption or model is needed to connect such an unobserved distribution to observed information. That distinction is especially important when later specifying identifying restrictions.

From Conditional Laws to a Pattern Mixture

Pattern-specific distributions are components of a population distribution. Let \(\pi_k(x)=P(K=k\mid X=x)\) be the probability of pattern \(k\) at covariate value \(x\), and let \(\mathcal F_k(\cdot\mid x)\) denote the distribution of the full outcome vector \(Y_{1:T}\) conditional on \(X=x,K=k\). The pattern-mixture representation combines these components using the pattern probabilities.

Here \(A\) is any set of possible outcome vectors. This identity follows by partitioning the event \(Y_{1:T}\in A\) according to the mutually exclusive and exhaustive values of \(K\), conditional on \(X=x\):

The second equality is the conditional product rule. The identity is a bookkeeping decomposition, not an identifying assumption: it does not tell us how to determine the unobserved parts of \(\mathcal F_k\). Within each pattern, the conditional laws given \(H_k\) provide a more detailed description of those components.

How the Labels Organize Restrictions

The terms used for common pattern-mixture restrictions refer to different groups or comparisons. A complete-case label refers to \(K=T\). For a particular visit \(j\), the available-case group consists of participants with \(K\ge j\), since they have an observed \(Y_j\). A neighboring-case comparison refers to adjacent pattern labels, such as \(K=k\) and \(K=k+1\). These names identify which pattern-specific distributions are involved; they do not state that those distributions are equal or otherwise related.

The conditional distribution notation also makes the conditioning set explicit. For example, a comparison at the boundary between \(K=k\) and \(K=k+1\) may involve the future outcome \(Y_{k+1}\) conditional on a history through \(Y_k\). If the compared distributions are conditioned on different histories, they are not directly the same conditional quantity. Any substantive restriction must state its conditioning variables and the outcome law it relates.

Diagnostics, Comparisons, and Limits

Before interpreting pattern-specific distributions, check that the pattern labels match the recorded data and that the conditioning information is clearly specified. Common warning signs include nonmonotone records being forced into a monotone scheme, sparse patterns, and a conditional comparison being described without saying which history is held fixed. For a conditional comparison, also check whether the relevant history values occur in both patterns. If they do not, a model-based comparison at those values relies on extrapolation.

Pattern-mixture and selection models organize the joint distribution differently. A pattern-mixture description specifies outcome distributions by missingness pattern and combines them with pattern probabilities. A selection-model description instead factors the joint distribution into an outcome distribution and a model for the missingness process conditional on outcomes or covariates. Neither factorization by itself resolves what the observed data leave unobserved. The notation here supports pattern-specific descriptions; it does not provide a method for choosing or verifying a restriction.

A further limitation is that \(K\) records only the last observed visit. It does not encode the reasons for dropout, measurement timing deviations, or unrecorded features of the observation process. If those features matter, a richer pattern definition may be needed, with its assumptions made explicit. Even with a well-defined pattern variable, conditional distributions can be unstable when few participants occupy a pattern or when histories have little overlap.