Sequential Outcome Distributions Within Patterns

Why factor outcomes within a response pattern?

A pattern-mixture representation describes the outcome distribution separately for each response pattern. Within one such pattern, the outcomes can themselves be described visit by visit: the distribution at a visit may depend on baseline information and on the outcomes at earlier visits. This sequential view makes the pattern-specific distribution easier to interpret and can support models built from familiar conditional distributions.

The key point is that writing a distribution as a sequence of conditional distributions is an exact probability identity. It does not say that outcomes at different visits are independent, and it does not specify which parametric distributions should be used. Those would be additional modeling choices.

The sequential factorization and its conditions

The displayed factorization is the chain rule applied to the conditional joint distribution of \(Y\) after fixing \(R=r\) and \(X=x\). It applies to discrete outcomes through probability mass functions, to continuous outcomes through densities, and to mixed outcome types using the corresponding conditional distributions. For continuous variables, the product is a joint density, not the probability of observing one exact outcome vector.

The factor \(f_{rt}\) is conditional on the whole pattern \(R=r\), as well as on baseline covariates and earlier outcomes. Thus, a later-visit conditional distribution can differ between patterns even when the earlier outcome history and baseline covariates are the same. The response pattern is part of the conditioning information; it is not replaced by the earlier outcome history.

The identity requires a well-defined joint conditional distribution and conditional distributions at the histories being considered. In discrete settings, a factor is needed only for histories with positive conditional probability under \(R=r,X=x\). In continuous settings, conditional distributions are understood in the usual almost-everywhere sense. No assumption of independent outcomes, missing at random, or a particular regression form is needed for the identity itself.

To see the justification, the definition of conditional probability (or conditional density) gives the two-part identity whenever the conditioning event has positive probability. Apply it first to \(Y_1\) and \(Y_2\), then to the resulting history and \(Y_3\), continuing through the visits:

This derivation does not require the successive factors to have the same functional form or parameters. Each can be modeled separately, and a model may include interactions between pattern, baseline covariates, and earlier outcomes. But the factorization alone neither estimates those factors nor establishes that a particular model fits.

Pattern-specific factors need not be the same

The pattern index on each factor is substantive. It allows the outcome distribution among people in one response pattern to differ from that among people in another pattern. A sequential model that omits the pattern index, or constrains its effects to be identical across patterns, imposes additional restrictions; those restrictions are not consequences of the chain rule.

Estimating conditional factors from pattern-specific data

When outcomes and covariates needed for a factor are observed for members of a pattern, empirical conditional distributions can be calculated within that pattern. Such calculations describe the available data; they do not automatically identify factors involving outcomes that are missing for that pattern. In particular, a pattern-specific distribution of an unobserved outcome may require modeling assumptions or other information beyond the observed data.

Diagnostics, alternatives, and limitations

A sequential specification can be useful because each factor has a direct interpretation: it describes one visit’s outcome given the pattern, baseline covariates, and earlier outcomes. It also gives a structured way to inspect model fit by visit and history. For example, compare fitted and observed conditional outcome distributions within patterns where the necessary outcomes are observed; look for sparse histories, implausible fitted probabilities, and residual patterns that remain associated with earlier outcomes or covariates.

These checks have limits. A factor involving a missing outcome for the pattern may not be empirically checkable from that pattern’s observed outcomes alone. Sparse histories can make conditional estimates unstable, and a fitted model may extrapolate where little data are available. A good fit to observed conditional distributions does not, by itself, verify unobserved parts of the pattern-specific outcome distribution.

The main alternative is to specify the pattern-specific joint outcome distribution directly, without expressing it through visit-level conditionals. These representations can describe the same distribution when fully specified consistently; the sequential representation is a chain-rule decomposition, not a competing probability law. A further alternative is to impose restrictions such as independence or shared conditional parameters across patterns. Such restrictions can simplify modeling, but require justification and should not be mistaken for consequences of factorization.

A common mistake is to say, “Because the joint distribution factors into visit-specific terms, the visits are independent.” That conclusion is wrong: the factors after the first are explicitly conditional on earlier outcomes. Independence would require the additional equality \(f_{rt}(y_t\mid y_{1:t-1},x)=f_{rt}(y_t\mid x)\) for every visit and relevant history. Without that restriction, the later outcomes may depend on earlier ones.