Notation for Outcome And Response-Pattern Vectors

Why Use Vectors for Longitudinal Data?

Repeated measurements create two related records: the outcomes a study intends to collect and the visits at which those outcomes are actually available. A compact notation for both records helps prevent a basic but consequential error: treating an unobserved outcome as though it were not part of the outcome vector. The distinction matters when describing the joint distribution, regardless of which missing-data model is eventually used.

Consider a study with a fixed schedule of \(J\) follow-up visits. Each participant has a sequence of outcomes, but some scheduled measurements may not be recorded. The notation below describes the full outcome sequence, the availability of each measurement, the resulting response pattern, and covariates measured at baseline. It does not, by itself, impose a missingness mechanism or a particular factorization of the joint law.

Outcome, Response, and Baseline Vectors

Definition: For \(J\) scheduled follow-up visits, let \(Y_j\) denote the outcome at visit \(j\), for \(j=1,\ldots,J\). The longitudinal outcome vector is \(\mathbf Y=(Y_1,\ldots,Y_J)\). Let \(R_j=1\) if \(Y_j\) is observed and \(R_j=0\) if it is not observed. The response-indicator vector is \(\mathbf R=(R_1,\ldots,R_J)\). A realized value \(\mathbf r=(r_1,\ldots,r_J)\) of \(\mathbf R\) is a response pattern. Let \(\mathbf X\) denote the vector of baseline covariates, including treatment assignment when appropriate.

The visit index refers to the scheduled measurement occasion, not to the order in which records happen to be entered. A value \(R_j=0\) means that the outcome for that scheduled visit is unavailable for analysis; it does not specify why. For example, the visit may have been missed, the participant may have withdrawn, or the measurement may have been unavailable for another reason. If the reason for unavailability is needed, it requires additional notation.

For a given pattern \(\mathbf r\), define the observed and missing visit-index sets by \(O(\mathbf r)=\{j:r_j=1\}\) and \(M(\mathbf r)=\{j:r_j=0\}\). Write \(\mathbf Y_{\mathrm{obs}}(\mathbf r)\) for the subvector of \(\mathbf Y\) whose indices are in \(O(\mathbf r)\), and \(\mathbf Y_{\mathrm{mis}}(\mathbf r)\) for the subvector with indices in \(M(\mathbf r)\). The ordering within each subvector follows the original visit order. These labels depend on the realized pattern: the same visit outcome is observed for one participant and missing for another.

This setup presumes that the scheduled outcomes in \(\mathbf Y\) are meaningful quantities for the participants under study, including at visits where their values are unobserved. It does not require those values to be recorded or known. If an outcome is genuinely undefined after a particular event, rather than merely unobserved, the outcome definition and target population need to be reconsidered before using this notation.

Worked Example: Encoding Four Scheduled Visits

Suppose a hypothetical trial schedules symptom scores at months 1, 2, 3, and 4. For one participant, the full outcome vector is represented as \(\mathbf Y=(y_1,y_2,y_3,y_4)\). The record shows scores of 8 at month 1, 6 at month 2, no score at month 3, and 5 at month 4.

The response indicators are \(R_1=1\), \(R_2=1\), \(R_3=0\), and \(R_4=1\), so \(\mathbf R=(1,1,0,1)\). The observed visit set is \(O(\mathbf r)=\{1,2,4\}\), and the missing visit set is \(M(\mathbf r)=\{3\}\). Thus the observed subvector is \(\mathbf Y_{\mathrm{obs}}(\mathbf r)=(8,6,5)\), while \(\mathbf Y_{\mathrm{mis}}(\mathbf r)=(y_3)\), whose value is not recorded.

The recorded values do not justify replacing the full outcome vector by \((8,6,5)\) and treating it as if it had three scheduled visits. The three observed values retain their original visit labels; the third scheduled outcome remains missing.

Patterns and the Joint Distribution

Each response indicator is binary, so with \(J\) visits there are \(2^J\) possible response patterns before any restrictions are imposed. For example, with three visits the patterns include \((1,1,1)\), in which all outcomes are observed, and \((1,0,1)\), in which the second visit is missing but the third is observed. The latter is a nonmonotone pattern: observation resumes after a missing visit. The notation allows such patterns without asserting that they must occur.

The joint distribution concerns all three components: baseline covariates, outcomes, and response indicators. When probability densities or probability mass functions are available, it can be written as

f_{\mathbf X,\mathbf Y,\mathbf R}(\mathbf x,\mathbf y,\mathbf r) = f_{\mathbf X}(\mathbf x)\, f_{\mathbf Y,\mathbf R\mid\mathbf X}(\mathbf y,\mathbf r\mid\mathbf x).

This is the ordinary conditioning identity: the joint law is the baseline-covariate law multiplied by the conditional joint law of outcomes and response indicators given those covariates. It does not assume that outcomes and response indicators are independent, either conditionally or marginally. If outcomes or covariates are continuous, densities are used for those components; for discrete components, the corresponding probabilities or probability masses are used. The notation \(f\) is a compact way to represent the appropriate joint density/mass, not a claim that every variable is continuous.

A joint law must assign nonnegative probabilities or densities to its possible values and have total probability one. In particular, for each baseline value in a discrete example, summing over all outcome and response values gives the conditional total probability one. The notation keeps separate the baseline information \(\mathbf X\), the full outcome vector \(\mathbf Y\), and the pattern \(\mathbf R\); it does not yet decompose the conditional joint law into smaller pieces.

Worked Example: Listing Patterns for Three Visits

Suppose a hypothetical follow-up study schedules three outcomes. A participant has response indicators \(\mathbf r=(1,0,1)\). Then

O(\mathbf r)=\{1,3\},\qquad M(\mathbf r)=\{2\},\qquad \mathbf Y_{\mathrm{obs}}(\mathbf r)=(Y_1,Y_3),\qquad \mathbf Y_{\mathrm{mis}}(\mathbf r)=(Y_2).

There are \(2^3=8\) possible binary patterns, since each of the three indicators has two possible values. The pattern \((1,1,1)\) has no missing outcomes; \((0,0,0)\) has all three outcomes missing; and \((1,0,1)\) has a gap followed by an observed outcome. This count follows by multiplying the two choices at each visit: \(2\times2\times2=8\).

No monotone-dropout restriction is being applied. Such a restriction would rule out patterns where observation resumes after a missing visit; the vector notation itself does not rule them out.

Worked Example: A Joint Probability With Baseline Covariates

Consider hypothetical binary variables: baseline treatment \(X\), two binary outcomes \(Y_1,Y_2\), and two response indicators \(R_1,R_2\). Suppose \(P(X=0)=0.60\), and, conditional on \(X=0\), the joint probability of \(Y_1=1\), \(Y_2=0\), and \(\mathbf R=(1,0)\) is 0.20. The joint probability of all these events is

\begin{aligned} &P(X=0,Y_1=1,Y_2=0,\mathbf R=(1,0))\\ &=P(X=0)\, P(Y_1=1,Y_2=0,\mathbf R=(1,0)\mid X=0)\\ &=0.60\times0.20=0.12. \end{aligned}

Thus the event has probability 0.12, or 12%. Its pattern \((1,0)\) indicates that \(Y_1\) is observed and \(Y_2\) is not; it does not mean \(Y_2=0\) was observed. In the stipulated event, \(Y_2=0\) is a value in the full outcome vector, while the response indicator marks it unavailable. The event probability is valid as stated because the given conditional probability refers to the joint event involving both outcomes and both response indicators—not just to a marginal probability for the pattern.

How to Check the Notation

The notation is useful only if its components agree with the data-collection schedule and the scientific question. Before writing a joint law, check that each outcome coordinate has a clearly defined visit, that the response indicator refers to availability of that coordinate, and that every baseline covariate was measured before the follow-up outcomes it is intended to precede. Treatment assignment can be included in \(\mathbf X\) if it is determined at baseline.

Several mistakes are easy to detect by checking the pattern against the observed record:

Mislabeling an unavailable value as zero. \(R_j=0\) denotes an unobserved outcome, not an outcome value \(Y_j=0\).
Shifting visit indices after a missed visit. A score at visit 4 remains \(Y_4\) even if visits 2 and 3 were unavailable.
Assuming all missingness is dropout. The pattern \((1,0,1)\) is permitted by the definition and represents an intermittent gap.
Assuming a mechanism from notation alone. Writing \(\mathbf Y\) and \(\mathbf R\) together does not say whether response depends on observed outcomes, unobserved outcomes, or both.
Confusing covariates with outcomes. Baseline covariates belong in \(\mathbf X\); repeated outcomes belong in \(\mathbf Y\). A variable's role depends on when and how it is measured.

A practical diagnostic is to compare the response vector with the visit-level data record: every entry \(R_j=1\) should correspond to an available value for \(Y_j\), and every \(R_j=0\) should correspond to an unavailable value. If the data contain different kinds of unavailability, a single binary indicator may be too coarse for the scientific question, and additional reason or status variables may be needed.

What This Representation Does and Does Not Do

The full joint-distribution notation is a neutral starting point. It records the variables whose probabilistic relationships are relevant without deciding how to model them. One may later express a joint law using a selection-model representation or a pattern-mixture representation; those approaches organize the same ingredients differently. At this stage, the key distinction is simply between the full outcome vector and the observed subvector selected by the response pattern.

A long-format data table, with one row per participant per visit, is often convenient for data entry and computation. The vector notation is more compact for describing a participant's repeated outcomes and response pattern. Neither format changes the underlying schedule or the joint distribution. Converting between them requires preserving participant identifiers and visit indices.

The main limitation is that a response pattern records which outcomes are unavailable, not why they are unavailable. It also does not establish that the unobserved outcomes can be recovered from the observed data. Those questions require substantive assumptions and an explicit missing-data model. In particular, the mere fact that the joint law can be written down does not make every part of it identifiable from observed records.

Key takeaway: Use \(\mathbf Y\) for the scheduled outcomes, \(\mathbf R\) for their visit-specific availability indicators, and \(\mathbf X\) for baseline covariates. A response pattern \(\mathbf r\) selects the observed and missing subvectors, while the joint law describes all components without imposing a missingness assumption.

Check Your Understanding

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

For four visits, a participant has \(\mathbf R=(1,0,1,0)\). Give \(O(\mathbf r)\) and \(M(\mathbf r)\).
How many possible response patterns are there for five scheduled visits?
Explain why \(R_2=0\) does not imply \(Y_2=0\).
If treatment was assigned at baseline, in which vector can it be included under this notation?
Does the joint-distribution identity imply that \(\mathbf Y\) and \(\mathbf R\) are conditionally independent given \(\mathbf X\)?

Solutions. (1) \(O(\mathbf r)=\{1,3\}\) and \(M(\mathbf r)=\{2,4\}\), since the first and third indicators are 1. (2) There are \(2^5=32\) patterns, with two choices for each of five indicators. (3) \(R_2=0\) means the second outcome is unavailable; \(Y_2=0\) is a possible outcome value, not the meaning of the response indicator. (4) Include treatment in the baseline-covariate vector \(\mathbf X\). (5) No. The identity \(f_{\mathbf X,\mathbf Y,\mathbf R}=f_{\mathbf X}f_{\mathbf Y,\mathbf R\mid\mathbf X}\) is a conditioning identity and places no conditional-independence restriction on \(\mathbf Y\) and \(\mathbf R\).