Monotone Dropout Patterns Under Reference-Based Rules

Why the Dropout Visit Matters

When a participant stops attending scheduled visits, the missing outcomes do not all have the same history. Someone who drops out after visit 1 has less observed treatment experience than someone who drops out after visit 3. Reference-based rules therefore assign predictive distributions according to the participant’s dropout pattern, conditioning on the outcomes that were actually observed. This tutorial focuses on that assignment for monotone dropout: once a visit is missed, all later scheduled visits are also missing.

The assignment is a modeling choice, not a direct consequence of the observed data. J2R (jump to reference) and copy-reference (CR) encode different assumptions about outcomes after treatment-arm dropout. For either rule, a valid prediction must use the participant’s observed history and a joint distribution compatible with the chosen rule.

Here \(T\) denotes the treatment arm and \(R\) the reference arm. The target is the conditional distribution of the missing outcomes for a treatment-arm participant in pattern \(k\): \(Y_{k+1:J}\mid \bar Y_k,D=k,A=T\). The rule and its model parameters determine this distribution; the observed history supplies the conditioning values. Participants with different \(D\) therefore need not receive the same predictive distribution, even if they are assigned to the same treatment arm.

Rule-Specific Distributions for Each Pattern

For a Gaussian specification, a pattern-specific joint model can be summarized by a mean vector \(m^{(r,k)}\) and covariance matrix \(V^{(r,k)}\), where \(r\) is J2R or CR. These cover all scheduled outcomes for the pattern, including its observed and missing visits. The mean restrictions express the chosen reference-based assumption. The covariance restrictions must also be specified consistently with that rule; the mean rule alone does not determine a predictive distribution.

Under J2R, the post-dropout marginal means are set to the corresponding reference-arm means. The treatment-arm trajectory before dropout is retained. Under CR, the post-dropout mean change follows the reference-arm change, so the treatment-arm mean at dropout is carried forward as an offset. In symbols, for a pattern with \(k\geq1\), a standard mean specification is \(m^{(J2R,k)}_j=\mu_{R,j}\) for \(j>k\), while \(m^{(CR,k)}_j=\mu_{T,k}+(\mu_{R,j}-\mu_{R,k})\) for \(j>k\). For \(j\leq k\), the mean trajectory is the treatment-arm trajectory. These expressions describe marginal means; conditional predictions also depend on covariance and the observed history.

For either rule, partition the rule-specific mean and covariance into observed and missing components: \(m=(m_o,m_m)\) and \(V=\begin{pmatrix}V_{oo}&V_{om}\\V_{mo}&V_{mm}\end{pmatrix}\), where \(o\) indexes visits \(1,\ldots,k\) and \(m\) indexes visits \(k+1,\ldots,J\). If \(V_{oo}\) is nonsingular, the conditional Gaussian distribution is

The first term adjusts the missing-outcome mean for how the participant’s observed history differs from its model mean. The second is the conditional covariance after accounting for that history. This is a predictive distribution, not merely a list of post-dropout marginal means.

Worked Predictions Under Two Rules

Why the Conditional Formula Holds

The Gaussian conditional formula follows by subtracting from the missing vector its linear prediction from the observed vector. Set \(B=V_{mo}V_{oo}^{-1}\) and \(Z=Y_m-m_m-B(Y_o-m_o)\). Then \(\operatorname{Cov}(Z,Y_o)=V_{mo}-BV_{oo}=0\). Because \((Z,Y_o)\) is jointly Gaussian, zero covariance implies independence. Its mean is zero, and its covariance is

Independence means the distribution of \(Z\) is unchanged after conditioning on \(Y_o=y_o\). Rearranging its definition gives \(Y_m=m_m+B(y_o-m_o)+Z\) under that conditioning, which yields the stated conditional mean and covariance. This also explains why the observed-history adjustment and the reduction in uncertainty arise together from the same covariance structure.

Assigning Patterns, Checking Assumptions, and Comparing Rules

For monotone dropout, the assignment is organized by the last observed visit. A participant with \(D=1\) receives a prediction for visits \(2,\ldots,J\); one with \(D=2\) receives a prediction for visits \(3,\ldots,J\). A participant with \(D=J\) has no missing follow-up outcome to predict. For \(D=0\), there is no postbaseline history to condition on, so the prediction is the specified pattern-specific marginal distribution for all follow-up visits.

J2R and CR can be compared by the assumptions they make after dropout. J2R places post-dropout marginal means at reference-arm levels. CR carries the treatment mean at dropout forward while applying reference-arm mean changes. Under both rules, predictions still depend on observed history and covariance. Neither rule is a data-driven finding that the participant’s unobserved outcomes would follow that trajectory.

The pattern-specific setup also clarifies how this analysis differs from a single common missing-outcome model. A common model may impose one conditional relationship across participants, whereas reference-based pattern-mixture assumptions explicitly assign distributions according to dropout pattern. This distinction matters when dropout timing is associated with observed history or when the scientific sensitivity question concerns what happens after treatment discontinuation.

Several checks help identify implementation errors or model strain:

• Verify the pattern: Confirm that every visit up to \(D\) is observed and every later visit is missing. A missed visit followed by a recorded visit is not monotone dropout.
• Check the mean assignment: Under J2R, post-dropout marginal means should be reference means. Under CR, the mean difference at dropout is carried forward while subsequent mean changes follow the reference trajectory.
• Check the covariance: The joint covariance must be symmetric and positive definite (or positive semidefinite where appropriate), and the conditional covariance must not have negative variances. Large or impossible conditional values can signal incorrect indexing or inconsistent covariance parameters.
• Check observed support: If a participant’s history is far outside the model’s plausible range, a Gaussian conditional prediction may be sensitive to the assumed linear relationship and normal tails.
• Check pattern size: Sparse patterns can make pattern-specific parameters unstable or heavily dependent on borrowing and modeling choices. A numerically available prediction is not evidence that the assumption is well supported.

A common mistake is to assign reference-arm marginal means to all missing visits and call those values the predictive distribution. That ignores the conditioning on observed outcomes and omits uncertainty and dependence among the missing visits. The correction is to construct the rule-specific joint model for the participant’s pattern, partition its mean and covariance, and calculate the conditional distribution. Another mistake is to treat CR as identical to J2R because both use reference behavior after dropout. Their post-dropout mean restrictions differ: CR retains the treatment advantage at dropout, whereas J2R moves to reference mean levels.

The approach is limited by its assumptions. The unobserved post-dropout outcomes cannot confirm whether J2R or CR is appropriate, and Gaussian predictions can be unsuitable for outcomes with strong bounds, skewness, or discrete support unless the model accounts for those features. Estimates may also be sensitive to covariance choices and to how reference-arm information is used. These are reasons to report the chosen assumptions and examine sensitivity, not reasons to interpret one rule as automatically correct.