Defining Pattern-Specific Outcome Distributions

Why Define a Distribution Within Each Pattern?

With monotone dropout, participants contribute different lengths of an outcome trajectory. A pattern-mixture description organizes the outcome distribution by those patterns. The essential object is not just the distribution of the outcomes recorded before dropout: it is the distribution of the entire trajectory among participants belonging to a given pattern. That definition includes outcomes after dropout, even though those values are not observed for those participants.

This distinction is substantive. The distribution of recorded outcomes within a pattern can be summarized from the observed data, subject to the usual sampling qualifications. The conditional distribution of the full trajectory also includes missing components. Defining that larger object does not, by itself, tell us how to estimate those components or establish that they are identified by the data.

Definition: Consider \(T\) scheduled outcome visits. Let \(Y=(Y_1,\ldots,Y_T)\) denote the full outcome trajectory for a participant, whether or not every component is observed. Use \(D=d\), for \(d=0,\ldots,T\), to denote the monotone dropout pattern in which visits \(1,\ldots,d\) are observed and visits \(d+1,\ldots,T\) are missing. Thus \(D=T\) is the complete pattern and \(D=0\) has no observed outcome visits. The pattern-specific full-trajectory distribution is the conditional distribution of \(Y\) given \(D=d\), defined for patterns with \(P(D=d)>0\).

The pattern is determined by the observation indicators, not by the numerical outcome values. The value \(D=d\) says which visits were observed; it does not say that a participant's outcome was high, low, improving, or worsening. In a probability model, pattern membership and outcomes may nevertheless be associated.

For real-valued outcomes, write the conditional cumulative distribution function as \(F_d(y)=P(Y_1\leq y_1,\ldots,Y_T\leq y_T\mid D=d)\). More generally, for a measurable set \(A\) of possible trajectories, write \(F_d(A)=P(Y\in A\mid D=d)\). This event-based definition also applies to discrete, continuous, or mixed outcome distributions. It does not impose a parametric form, independence across visits, or any particular missingness mechanism.

Conditioning and the Positive-Probability Requirement

The ordinary conditional-probability definition makes the required condition explicit. If \(P(D=d)>0\), then for any trajectory set \(A\),

F_d(A)=P(Y\in A\mid D=d) =\frac{P(Y\in A,\ D=d)}{P(D=d)}.

The denominator is what makes this a valid conditional distribution. For example, \(F_d\) assigns probability one to the whole trajectory space because \(P(Y\in\mathcal{Y},D=d)/P(D=d)=1\), and it is nonnegative and countably additive because the joint probability in the numerator has those properties. A conditional mean, when it exists, is correspondingly \(E(Y_t\mid D=d)\) for each visit \(t\), including visits after dropout.

If \(P(D=d)=0\), this ratio is undefined. A conditional distribution at that pattern is not uniquely determined by ordinary conditioning: values assigned on a probability-zero event do not change the joint distribution. One can stipulate a convention or a model-based extension, but that is an additional choice, not a distribution supplied by conditioning alone. Throughout, \(F_d\) therefore refers to patterns of positive probability unless an extension is explicitly stated.

Key distinction: \(F_d\) is a distribution on the full \(T\)-component trajectory. The distribution of the recorded portion is the conditional distribution of \(Y_{1:d}=(Y_1,\ldots,Y_d)\) given \(D=d\). For \(d<T\), the latter is only a marginal of the former; it does not specify the missing components.

Worked Examples

Worked Example: Conditioning on a Positive-Probability Pattern

Consider a hypothetical population of 10,000 participants measured at three visits. Suppose 4,000 have pattern \(D=2\), so their first two outcomes are observed and their third is missing. Imagine, for illustration, that a simulation retains the full trajectories even after dropout. Among those 4,000 participants, 1,800 have \(Y_3=6\) points, and 900 satisfy all three conditions \(Y_1\leq4\), \(Y_2\leq5\), and \(Y_3\leq6\).

The pattern probability is \(P(D=2)=4{,}000/10{,}000=0.4\), which is positive. Within this pattern, the full-trajectory conditional probability for the specified joint event is

P(Y_1\leq4,\ Y_2\leq5,\ Y_3\leq6\mid D=2) =\frac{900}{4{,}000}=0.225.

Likewise, in this hypothetical complete-data population, \(P(Y_3=6\mid D=2)=1{,}800/4{,}000=0.45\). These are statements about the full conditional distribution: the second concerns a component that is missing for this pattern. In an actual dropout dataset, the third-visit values of these \(D=2\) participants are not observed. The simulation makes the arithmetic possible for illustration; it does not imply those values can be read from ordinary observed records.

Worked Example: The Observed Portion Does Not Determine the Full Distribution

Suppose a hypothetical study has 100 participants with \(D=2\). Their observed outcomes include 30 participants with \(Y_1=4\) points and \(Y_2=5\) points. The observed-pattern empirical probability of this pair is

\widehat P(Y_1=4,\ Y_2=5\mid D=2)=\frac{30}{100}=0.30.

This calculation describes the observed two-component portion. It says nothing by itself about \(Y_3\) for those participants. To see why, consider two hypothetical completions of the same observed records. In completion A, assign \(Y_3=0\) points to every \(D=2\) participant. In completion B, assign \(Y_3=10\) points to every \(D=2\) participant. Both completions leave all recorded data unchanged, but they give different full conditional distributions: in A, \(P(Y_3=0\mid D=2)=1\); in B, \(P(Y_3=10\mid D=2)=1\). Their conditional means are 0 and 10 points, respectively.

Thus the observed distribution of \(Y_{1:2}\mid D=2\) does not uniquely determine \(F_2\). A model or additional assumptions are needed to specify the unobserved part. This is a limitation of the information in the observed portion, not an arithmetic defect in the conditional-distribution definition.

Worked Example: A Zero-Probability Pattern

Suppose a hypothetical protocol makes pattern \(D=0\) impossible: every enrolled participant must have at least one outcome visit recorded. Then \(P(D=0)=0\). For any trajectory event \(A\), the ordinary formula would require

P(Y\in A\mid D=0) =\frac{P(Y\in A,\ D=0)}{P(D=0)} =\frac{0}{0},

which is undefined. It would be incorrect to report a uniquely determined \(F_0\) from this conditioning event. A researcher could define an extrapolated distribution for that pattern as part of a model, but should label it as an extension and recognize that the joint distribution does not determine it.

Now distinguish this from observing zero \(D=0\) participants in a sample of 100. A sample count of zero does not establish that the population probability is zero. If the population probability is positive, the conditional distribution is mathematically defined, even though the sample provides no direct within-pattern observations and cannot estimate it empirically without further information or assumptions. A zero population probability and a zero sample count are different claims.

What the Definition Does—and Does Not—Assume

To define \(F_d\), we need a specified outcome vector \(Y\), a well-defined pattern variable \(D\), and positive probability for the pattern under consideration. The outcome components must have a meaningful definition at every scheduled visit, including visits that may be unobserved. This is a target-data convention: it says what the full trajectory represents, not that every component is actually recorded.

No assumption that dropout is independent of outcomes is needed to define a conditional distribution. Nor does the definition assert that outcomes cause dropout or that dropout causes outcomes. It describes the distribution of trajectories among members of each pattern. Different patterns can have different full conditional distributions, but whether such differences are present is a property of the joint distribution, not a consequence forced by the notation.

The pattern-mixture viewpoint conditions the outcome distribution on \(D\), as in \(Y\mid D\). A selection-model viewpoint instead describes dropout conditional on outcomes, as in \(D\mid Y\). These are different ways to organize a model for the joint relationship. Simply naming \(F_d\) does not specify a complete joint model, identify missing outcomes, or make the two approaches empirically distinguishable from observed data without further modeling choices.

Diagnostics, Failure Modes, and Limits

Before interpreting a pattern-specific distribution, check that pattern coding matches the observation process. Under the convention here, \(D=d\) means exactly the first \(d\) visits are observed. A pattern with a missing earlier visit and a later observed visit is not one of these monotone patterns; it indicates either a coding error or a setting that does not satisfy the monotone-dropout setup. Also confirm that the outcome vector refers to the same scheduled visits for every participant.

Zero or sparse patterns. A truly zero-probability pattern has no uniquely defined conditional distribution by ordinary conditioning. A rare but positive-probability pattern is defined, but may have little empirical support. Check population or design knowledge separately from observed sample counts.
Confusing observed and full distributions. An empirical distribution of recorded outcomes within pattern \(d\) covers \(Y_{1:d}\), not \(Y\). Do not treat estimates of unobserved components as if they came directly from those records.
Confusing pattern membership with outcome level. \(D=d\) encodes which visits are observed, not a category such as “high outcome.” Any association between \(D\) and \(Y\) must be described or modeled, not built into the definition.
Overinterpreting a stipulated extension. A model can assign a distribution to a zero-probability pattern or to missing components. That assignment is an assumption or convention; the conditional-probability formula does not validate it.

The main practical limitation is therefore clear: the full conditional distribution is a well-defined target for positive-probability patterns, but its missing components are not automatically recoverable from the observed portion. Pattern-specific distributional assumptions can be useful for organizing sensitivity analyses and models, yet conclusions about unobserved outcomes depend on those assumptions. The introductory material on Overlaps Missing Data provides broader context for what observed records can and cannot determine; here, the focus is the precise conditional object used in a pattern-mixture formulation.

Specify the trajectory and pattern.
State the scheduled components of \(Y\) and what each value of \(D\) means.

Check the conditioning event.
For the pattern of interest, establish or assume \(P(D=d)>0\) before defining \(F_d\) by ordinary conditioning.

Name the distribution being discussed.
Distinguish the full \(Y\mid D=d\) distribution from the observed portion \(Y_{1:d}\mid D=d\).

Separate definition from identification.
State which components are observed and which require assumptions or a model for their distribution.

Key takeaway: For \(P(D=d)>0\), a pattern-specific full-outcome distribution is the conditional law of the entire trajectory \(Y\) given pattern \(D=d\). It includes post-dropout outcomes as target variables, but their inclusion in the definition does not make them observed or identified.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

With four scheduled visits, what does \(D=2\) mean under the convention used here, and what is the full trajectory distribution for that pattern?
If \(P(D=1)=0.25\) and \(P(Y\in A,D=1)=0.075\), calculate \(F_1(A)\).
Why can the observed distribution of \(Y_{1:2}\mid D=2\) fail to determine the distribution of \(Y_3\mid D=2\)?
A sample contains no participants with \(D=0\). Does that establish that \(F_0\) is undefined? Explain the distinction needed.

Solutions. (1) \(D=2\) means visits 1 and 2 are observed and visits 3 and 4 are missing. The full distribution is the conditional distribution of \((Y_1,Y_2,Y_3,Y_4)\) given \(D=2\), not merely the distribution of the first two components. (2) Since the pattern has positive probability, \(F_1(A)=0.075/0.25=0.30\). (3) The observed portion contains no recorded values of \(Y_3\) for that pattern; distinct assignments or models for \(Y_3\) can agree on every observed value while implying different conditional distributions. (4) No. A zero sample count does not establish \(P(D=0)=0\). If the population probability is positive, \(F_0\) is defined but unsupported by within-pattern sample observations; if the population probability is truly zero, ordinary conditioning does not uniquely define \(F_0\).