Depression Trial With Intermittent Visits and Later Dropout

Why the Observation Pattern Matters

In a repeated-measures depression trial, a participant may miss one assessment and return for the next, then later stop providing outcome data. Those events produce a different observation pattern from uninterrupted follow-up or dropout after a particular visit. If a trajectory is calculated from everyone observed at each visit, the set of participants contributing to it can change over time. A late-visit mean can therefore reflect both symptom change and a change in who remains observed.

This case study uses hypothetical Hamilton Depression Rating Scale scores (HAM-D), for which lower scores indicate fewer depressive symptoms. The goal is to compare available-case trajectories with trajectories completed under an explicit neighboring-pattern restriction. A restriction makes assumptions about outcomes that were not observed; it does not reveal those outcomes or establish that the assumption is true.

Define the Visits and Patterns

Let \(Y_{it}\) denote participant \(i\)'s HAM-D score at visit \(t\), where \(t=0\) is baseline and \(t=1,2,3,4\) are follow-up visits. Let \(R_{it}=1\) when that score is observed and \(R_{it}=0\) otherwise. The observation pattern for a participant is the vector \((R_{i0},R_{i1},R_{i2},R_{i3},R_{i4})\). Assume baseline is observed for everyone in this illustration.

Three follow-up patterns will be used. Pattern A is complete follow-up: \((1,1,1,1,1)\). Pattern B misses visit 2 but returns at visits 3 and 4: \((1,1,0,1,1)\). Pattern C is observed through visit 2 and then drops out: \((1,1,1,0,0)\). Thus B is intermittent, while C represents later dropout. These labels describe observation, not symptom severity or treatment response.

The basic pattern-mixture representation describes the outcome distribution separately by observation pattern. In particular, the mean at visit \(t\) can differ across patterns. Outcomes observed within a pattern inform its observed means; means at visits missing for that pattern are not identified by those data alone. Any restriction used to fill such a gap must be stated as an assumption.

Hypothetical Trial Data

Suppose a randomized trial has 40 participants in each arm, with 20 in pattern A, 10 in pattern B, and 10 in pattern C per arm. The entries below are pattern-specific observed mean HAM-D scores in points. A dash means no score was observed for that pattern at that visit; it is not a zero.

These pattern means are hypothetical summaries, not real trial findings. They are sufficient to illustrate means and restrictions, but not to calculate a valid standard error for an individual-level treatment effect: within-pattern variation and the full analysis specification would also be needed.

Available-Case Trajectories

An available-case mean at visit \(t\) is the mean among participants with an observed score at that visit, calculated separately by randomized arm. It describes the observed participants at that visit. It equals the full-arm mean only under additional conditions, such as the observed participants being representative for that visit's outcome or a valid adjustment restoring that representation.

Here \(A_i=1\) denotes treatment and \(A_i=0\) control; \(I(\cdot)\) is 1 when its condition holds and 0 otherwise. The observed contrast is in HAM-D points. Since lower scores are better, a negative treatment-minus-control contrast favors treatment descriptively among those observed at that visit.

A Neighboring-Pattern Restriction

To illustrate how an identifying restriction changes the trajectory, define the following neighboring-pattern rule for this example. For a missing visit-2 mean in pattern B, borrow the observed visit-2 mean averaged across the two other patterns, A and C, weighting by their pattern sizes. For missing visits 3 and 4 in pattern C, set its missing pattern mean equal to the corresponding mean in pattern A, the complete-follow-up pattern. Apply the rule separately within each arm.

This is an explicit restriction on pattern-specific means, not a universal or uniquely defined “neighboring-case” method. It assumes that the chosen neighboring observed patterns provide an appropriate value for the missing pattern mean. In particular, assigning C the complete-pattern means at late visits assumes away any additional worsening or improvement associated with C's dropout pattern.

Justification, Diagnostics, and Alternatives

The reconstruction is a weighted average over patterns. If \(n_{a,g}\) is the number of participants in arm \(a\) and pattern \(g\), and \(m_{a,g,t}\) is the observed or restricted pattern mean, the reconstructed mean is \(\sum_g n_{a,g}m_{a,g,t}/\sum_g n_{a,g}\). Its justification is simply partitioning the arm into disjoint patterns: each participant belongs to one pattern, so the arm mean is the pattern-size-weighted mean of the pattern means. For missing pattern means, however, this identity yields a population mean only if the assigned values correctly represent those unobserved pattern means.

In the example, the visit-3 restricted treatment mean is \( (20\times10+10\times12+10\times10)/40=10.500 \); the control mean is \( (20\times16+10\times18+10\times16)/40=16.500 \). The restriction therefore changes the arm means from the available-case values 10.667 and 16.667 to 10.500 and 16.500. At visit 4, it changes 7.667 and 14.667 to 7.500 and 14.500. These shifts are consequences of including pattern C with borrowed means, not evidence supplied by its unobserved outcomes.

The assumptions and conditions for interpreting these calculations should be kept distinct:

• The visits, outcome scale, randomized arms, and pattern definitions are consistently recorded; pattern counts refer to the same participants throughout.
• Available-case means are descriptive means for those observed at each visit. Generalizing them to all randomized participants requires appropriate missingness assumptions or a suitable adjustment.
• The neighboring restriction must specify which patterns supply each missing mean and how they are weighted. It is an identifying assumption, not a consequence of randomization.
• These calculations concern means. Inference requires uncertainty estimates that account for sampling variability and, for imputation-based analyses, uncertainty due to missing outcomes.

Useful diagnostics include tabulating pattern counts by arm, plotting observed pattern-specific means over visits, and checking whether the proposed neighbors have comparable observed histories. A pattern with few participants can make its observed mean unstable. If the complete pattern has unusually mild symptoms, borrowing its late means for a dropout pattern may make the reconstructed trajectory look too favorable. Strongly different observed means among neighboring patterns are a warning that the restriction deserves scrutiny, though agreement among observed means cannot verify equality at missing visits.

Available-case summaries avoid assigning values to missing outcomes, but their visit-specific participant composition can change. A neighboring-pattern restriction yields a trajectory for all pattern counts, but depends on untestable equalities for missing means. Other pattern-mixture restrictions, including the delta-adjustment approaches considered in earlier material on pattern-mixture models, can encode different departures for missing outcomes. No one approach is automatically correct; the substantive assumptions and the resulting sensitivity of conclusions should be made visible.

Common Pitfall and Limits

Wrong reasoning: “The visit-4 available-case contrast is \(-7\) points, so treatment reduced symptoms by 7 points for all randomized participants.” This ignores that visit-4 means use only patterns A and B, and that the outcomes for pattern C are absent. It also turns a descriptive contrast into a causal claim without an appropriate estimand and analysis.

Correction: State which participants contribute to each mean and describe the contrast as available-case unless the assumptions and analysis justify broader interpretation. If a neighboring restriction is used, report it as an assumption and show how it changes the trajectory. In this example, the visit-4 restricted contrast is also \(-7\) points, but that numerical agreement occurs because the rule assigns pattern C the same arm-specific mean as pattern A; it does not validate the rule.

A further limitation is that pattern-level means conceal individual variability and correlations across visits. They cannot, on their own, support a complete repeated-measures analysis or quantify uncertainty. The case study is intended to expose how observation patterns and identifying restrictions affect estimated trajectories, not to prescribe a single definitive analysis for a trial.