Observed-Data Contributions by Dropout Pattern

Why the Pattern-Specific Contribution Matters

A longitudinal record with dropout contains two kinds of observed information: which visits were observed, and the outcome values recorded at those visits. A likelihood contribution must account for both. In a pattern-mixture formulation, the contribution for a participant who drops out after visit 2 is not just a density for that participant’s observed outcomes. It also includes the probability of belonging to the dropout pattern, and the outcome density must be the appropriate marginal within that pattern.

This distinction is easy to miss when writing a likelihood from a complete-data model. Future outcomes are not observed for a participant who drops out, so their values cannot be inserted into a complete-data density. They must be integrated out. For monotone dropout over three visits, this gives a simple set of pattern-specific contributions.

Patterns and the Observed-Data Likelihood

Let \(Y=(Y_1,Y_2,Y_3)\) denote the outcomes at three scheduled visits, and let \(D\) be the last visit with an observed outcome. Thus \(D=0\) means no outcome visit was observed, \(D=1\) means only \(Y_1\) was observed, \(D=2\) means \(Y_1,Y_2\) were observed, and \(D=3\) means all three outcomes were observed. This encoding assumes monotone dropout: once a visit is missed because of dropout, later scheduled outcomes are also missing.

Write \(\pi_d=P(D=d)\) for the probability of pattern \(d\). For each pattern, write \(f_d(y_1,y_2,y_3)\) for the joint density of the full outcome vector conditional on \(D=d\). This pattern-specific distribution may describe outcomes that are unobserved for members of that pattern. Define \(f_d(y_{1:d})\) as its marginal density for the outcomes observed through visit \(d\), obtained by integrating out the later outcomes. For \(d=0\), the observed outcome vector is empty and its marginal density is 1.

For \(d=1\) or \(2\), the marginal density is defined by integration over the missing outcomes:

The integral is interpreted as a sum for discrete outcomes and as the corresponding combination of sums and integrals for mixed outcomes. At \(d=3\), there is nothing to integrate; at \(d=0\), integrating the full conditional density over all outcomes gives 1. If covariates \(X=x\) are included, the same expression applies conditionally on \(x\), with \(\pi_d(x)=P(D=d\mid X=x)\) and \(f_d(y_{1:d}\mid x)\).

For independent participants \(i=1,\ldots,n\), with observed patterns \(d_i\) and observed outcomes \(y_{i,1:d_i}\), the observed-data likelihood is

This product is a likelihood for the observed patterns and observed outcomes. It does not evaluate a density at a fabricated value for any missing outcome. The pattern-specific full distributions are part of the model, but the likelihood contribution for an incomplete record uses only their observed-outcome marginals.

Why Marginalization Gives This Contribution

By definition of the pattern-mixture factorization, the joint density of \(D=d\) and the full outcome vector is \(\pi_d f_d(y_1,y_2,y_3)\). The observed record contains \(D=d\) and \(Y_1,\ldots,Y_d\); the remaining outcomes are unobserved. The observed-data density is therefore the joint density integrated over every unobserved outcome:

The pattern probability can be taken outside the integral because it does not depend on the outcome values being integrated. For a participant with \(D=0\), the same reasoning integrates over the entire outcome vector, leaving \(\pi_0\int f_0(y)\,dy=\pi_0\). For a complete participant, the contribution is \(\pi_3 f_3(y_1,y_2,y_3)\).

A useful special case arises when a pattern-specific full density is written as a product of proper sequential conditional densities. Integrating out the post-dropout outcomes removes those future factors one at a time, because each conditional density integrates to 1. That simplification is valid only when the factors are normalized conditional distributions and the relevant marginalization is done correctly; it does not justify evaluating missing outcomes at arbitrary values.

Calculating a Cohort Likelihood

Each record contributes according to its own pattern. To construct a cohort likelihood, identify the pattern and observed outcome vector for every participant, calculate the corresponding \(\pi_d f_d(y_{1:d})\), and multiply the contributions. The following example uses binary outcomes so that the likelihood can be checked directly.

When the Full Density Must Be Integrated

A pattern-specific full density can be written with conditional distributions for all three visits, including visits after dropout. The observed contribution still depends on the marginal distribution of the visits that were actually observed. The next calculation shows why substituting a value for a missing outcome gives the wrong contribution.

Assumptions, Checks, and Common Errors

The derivation relies on a defined pattern variable, a probability model for the pattern and a proper outcome distribution within each pattern. In the monotone three-visit setup, the observed outcomes for pattern \(d\) must be exactly the prefix \(Y_1,\ldots,Y_d\). If records have intermittent missingness rather than monotone dropout, this last-observed-visit encoding does not represent their patterns; the pattern definition and corresponding observed subvectors must be revised.

Before using a likelihood, check that the pattern probabilities are valid: \(\pi_d\geq 0\) for every \(d\), and \(\sum_{d=0}^3\pi_d=1\). Check that each \(f_d\) is a normalized distribution and that its observed-data marginal is taken over exactly the missing components. If a pattern has no or very few participants, its pattern-specific outcome distribution may be weakly informed by the data. A likelihood formula can be correct even when its parameters are difficult to estimate reliably.

• Omitting the pattern probability. Using only \(f_d(y_{1:d})\) ignores the fact that the observed record also includes \(D=d\). Include \(\pi_d\) when the likelihood is for the joint observed record.
• Using the full density without marginalizing. A density for \((Y_1,Y_2,Y_3)\) is not the density for \((Y_1,Y_2)\). Integrate or sum over the unobserved outcome.
• Assigning a value to a missing outcome. Neither zero nor a guessed value represents integration over uncertainty. Such substitution changes the likelihood.
• Misencoding a pattern. Confirm that \(D=d\) means the first \(d\) visits are observed. A different coding convention requires matching changes to the observed subvector and formula.

The observed-data contribution is the pattern-mixture way to express the likelihood. A selection-model likelihood instead factors the joint distribution using an outcome model and a model for missingness conditional on outcomes; it is an alternative factorization, not a reason to omit the pattern term here. A complete-data likelihood would use the full outcome vector, but for incomplete records its missing components must be integrated out to obtain an observed-data likelihood. The pattern-mixture representation also does not, by itself, make unobserved outcomes empirically known: distributions for outcomes after dropout require modeling assumptions beyond the observed record.