Notation for Reference-Based Imputation

Why Notation Matters in Reference-Based Imputation

Reference-based imputation for a longitudinal clinical trial needs a precise account of who was assigned to which arm, which visits were scheduled, and which outcomes were observed before follow-up ended. Small ambiguities in those labels can change the meaning of a restriction: for example, the visit at which dropout occurs is not necessarily the same as the last visit with an observed outcome. This tutorial sets out notation for describing the trial data and dropout patterns. It does not impose a rule for filling in missing outcomes.

The notation is designed for a trial with a treatment arm and a reference arm, such as an experimental treatment and a control condition. The reference arm is the designated comparator for the analysis; it need not mean that participants received no care. Assignment to an arm is also distinct from adherence or treatment actually received.

Participants, Arms, and Visits

This convention assumes the same visit index has the same scheduled meaning across arms. If visits are scheduled at weeks 4, 8, and 12, for example, then \(j=1,2,3\) refer to those weeks. The index is a label for the scheduled visit, not necessarily the number of weeks since a participant's last visit. When the schedule differs by arm or participant, the index must be supplemented with the actual visit time or a schedule-specific label.

Let \(\mathcal I_T=\{i:A_i=T\}\) and \(\mathcal I_R=\{i:A_i=R\}\) be the sets of participants assigned to the two arms. Their sizes are \(n_T=|\mathcal I_T|\) and \(n_R=|\mathcal I_R|\), with \(n_T+n_R=n\). These definitions refer to randomized assignment. A participant assigned to \(T\) remains in \(\mathcal I_T\) even if they stop the assigned treatment or receive another treatment.

Dropout Time and Observed History

For monotone dropout, a participant's observed post-baseline outcomes form an uninterrupted sequence: after the participant stops providing outcomes, all later scheduled outcomes are missing. This is a convenient representation, not a claim that all trials have this missingness pattern.

The \(D_i\) convention avoids an off-by-one ambiguity. If the last observed visit is week 8, then \(D_i=2\) under the schedule in the example above, and the first missing scheduled outcome is at week 12, so \(C_i=3\). Some reports use a dropout index to mean the first missed visit instead; that is \(C_i\), not \(D_i\). State which convention is in use.

In this notation, a post-dropout outcome is one of \(Y_{ij}\) for \(j>D_i\). Such an outcome is missing from the observed data; its value is not known merely because the notation names it. If baseline covariates are denoted by \(X_i\), one may write the observed history as \(\mathcal H_i(D_i)=(X_i,Y_{i0},\ldots,Y_{iD_i})\). Include \(X_i\) only when those covariates are part of the information being recorded or conditioned on.

Dropout Patterns and Missing-Cell Counts

A dropout pattern is the set of participants sharing a particular last-observed visit, sometimes considered separately within each randomized arm. For arm \(a\in\{T,R\}\), define \(\mathcal I_{a,d}=\{i:A_i=a,\ D_i=d\}\) and \(n_{a,d}=|\mathcal I_{a,d}|\). Under monotone dropout, these groups partition the participants in arm \(a\): each participant has exactly one value of \(D_i\), so \(\sum_{d=0}^{J}n_{a,d}=n_a\).

For a participant with \(D_i=d\), exactly \(J-d\) scheduled post-baseline outcomes are missing. Consequently, among participants in arm \(a\), the number of missing post-baseline outcome cells is \(\sum_{d=0}^{J}(J-d)n_{a,d}\). This counts cells, not participants: one participant who drops out early can contribute several missing outcomes.

What the Notation Assumes—and What It Does Not

The simple \(D_i\) representation assumes outcomes are observed at every scheduled visit up to \(D_i\) and missing at every later scheduled visit. It is appropriate when a participant's outcomes stop after a permanent departure from follow-up. It does not represent every reason that a visit outcome might be absent. For example, a participant may miss week 4, return at week 8, and then drop out before week 12. That is intermittent missingness before permanent dropout, and a single last-observed-visit index would incorrectly imply that week 8 was missing.

A record may also end because the study ends or a participant reaches the administrative end of follow-up. If all scheduled outcomes through \(J\) are observed, record \(D_i=J\); do not label a participant a dropout merely because the trial has ended. Conversely, stopping assigned treatment does not by itself establish that outcome follow-up stopped. If outcomes continue to be collected, their observed values remain part of the history.

For intermittent missingness, it is safer to record observedness visit by visit. For example, define \(O_{ij}=1\) when \(Y_{ij}\) is observed and \(O_{ij}=0\) when it is missing. Then a sequence such as \((O_{i0},O_{i1},O_{i2},O_{i3})=(1,0,1,0)\) is expressible without pretending that all visits after the first missed one are also missing. The simple \(D_i\) notation is a concise special case for monotone patterns.

Common Pitfalls and Limits

• Using “dropout visit” without defining it. A last-observed index \(D_i=2\) and a first-missed index \(C_i=3\) describe the same boundary, but they are different indices.
• Treating assigned arm as treatment received. \(A_i=T\) records randomized assignment. It does not assert that the participant adhered to treatment throughout follow-up.
• Equating a missing outcome with a dropout. One missed visit followed by a later observed visit is intermittent missingness, not monotone dropout.
• Counting participants instead of missing cells. A participant with \(D_i=0\) contributes \(J\) missing follow-up outcomes, not one.
• Reading an unobserved value as known. Naming \(Y_{ij}\) identifies which outcome is missing; it does not supply its value or establish a distribution for it.

The main alternative to the last-observed-visit summary is a full visit-by-visit observedness record. The summary is compact and useful when dropout is monotone. The indicator representation is more flexible but requires more entries. Neither notation, by itself, explains why outcomes are missing or specifies how missing values should be completed. Reference-based analyses add assumptions about missing outcomes separately; those assumptions must not be confused with the bookkeeping used to describe the observed data.