Avoiding Survivor-Only Claims in a Longitudinal Trial

Why Completer-Only Comparisons Need Care

In a longitudinal randomized trial, participants may stop providing outcome measurements before the planned endpoint. A tempting analysis compares the endpoint averages among participants who completed follow-up in each arm. That calculation is descriptive of those observed completers. It is not automatically an estimate of the randomized treatment contrast for everyone assigned.

The distinction matters because remaining under observation is a post-randomization event. If treatment, response, side effects, or other prognostic factors affect who provides an endpoint measurement, the observed participants in the two arms may no longer be comparable simply because treatment was randomized. A completer-only comparison can then reflect both outcome differences and differences in who remained.

Consider a trial with a continuous symptom-improvement outcome at week 24, measured in points, where larger values indicate greater improvement. The target contrast in this tutorial is the difference in mean week-24 improvement between everyone randomized to treatment and everyone randomized to control. The outcome is conceptually defined for each randomized participant, including participants who discontinue treatment but could still have been assessed. The examples concern dropout and missing measurements, not death or an outcome that is undefined after death.

Randomization supports a causal comparison of assigned groups at baseline, subject to the trial’s design and a clearly defined outcome and treatment strategy. It does not guarantee that the subsets with observed outcomes remain comparable. In particular, the completer contrast is not generally the effect among a common subgroup: treatment-arm completers and control-arm completers need not represent the same kinds of participants.

A Hypothetical Trial With Differential Dropout

Suppose 100 participants are randomized to each arm. At week 24, 60 treatment participants and 80 control participants have observed outcomes. Their mean improvements are 8 and 6 points, respectively. The observed completers’ difference is therefore 2 points. However, 40% of the treatment arm and 20% of the control arm lack the endpoint, so the two observed groups differ substantially in their follow-up patterns.

The concern is not that a difference in dropout automatically proves bias in a particular direction. It is that the observed endpoint means do not reveal the missing outcomes. For example, treatment participants who discontinue because they are not improving may have lower unobserved outcomes than treatment completers. Alternatively, some may discontinue while improving, perhaps because they no longer need treatment. The observed data do not settle which explanation applies.

Pattern-Mixture Sensitivity Analysis

A pattern-mixture analysis describes outcome distributions separately for groups defined by their observation patterns, then combines those groups using their proportions in each randomized arm. For this endpoint, use two patterns: \(R=1\) for observed week-24 outcomes and \(R=0\) for missing week-24 outcomes. Let \(A\) denote randomized arm, with \(A=T\) for treatment and \(A=C\) for control. Write \(\pi_{a,r}=P(R=r\mid A=a)\) and \(\mu_{a,r}=E(Y\mid A=a,R=r)\), where \(Y\) is week-24 improvement.

The observed-data proportions and observed-pattern means can be estimated from the trial. The missing-pattern means cannot be calculated from the missing endpoint values, because those values were not observed. A sensitivity analysis therefore specifies a model-based reference mean for each missing pattern and varies assumptions about how the missing outcomes differ from that reference.

For a transparent illustration, suppose an analysis using observed longitudinal histories supplies reference predictions of 5 points for treatment dropouts and 4 points for control dropouts. These are hypothetical model predictions, not observed dropout outcomes. Let \(\delta_T\) and \(\delta_C\), each measured in improvement points, be offsets from those reference predictions. Then the assumed missing-pattern means are \(5+\delta_T\) and \(4+\delta_C\). A negative offset assumes worse mean improvement than the reference prediction.

Using the trial’s observed means and pattern proportions gives the following estimated contrast under these assumptions:

This is an estimate of the randomized-population contrast under the stated reference predictions and sensitivity offsets. It is not a direct recovery of the missing outcomes. The weights show why assumptions about treatment dropouts have more influence here: 40% of the treatment arm is missing, compared with 20% of the control arm.

Why the Weighted Formula Holds

The pattern-mixture formula follows from conditioning on the observation pattern. For a fixed randomized arm \(a\), the events \(R=r\) divide participants into mutually exclusive patterns. The law of total expectation gives:

For two patterns, this is \(\pi_{a,1}\mu_{a,1}+\pi_{a,0}\mu_{a,0}\). The derivation requires the outcome to be defined for the target population and the patterns to cover that population. It does not identify \(\mu_{a,0}\) from observed outcomes: that conditional mean still requires an assumption or sensitivity parameter. A pattern-mixture analysis makes that dependence visible rather than eliminating it.

The reference prediction also requires care. It may depend on a model fitted to observed histories, and its validity depends on how that model is specified. Calling a value a reference prediction does not make it known or assumption-free. The sensitivity offsets express departures from it; their clinical meaning should be assessed on the outcome scale.

Diagnostics, Alternatives, and Limitations

Diagnostics can show where an analysis deserves scrutiny, but observed data cannot verify assumptions about outcomes that remain missing. Useful checks include:

• Report the number and proportion with missing endpoints by randomized arm, and summarize when dropout occurred.
• Describe recorded reasons for discontinuation and compare observed histories, including prior outcomes, between completers and participants who later drop out.
• Check whether reference-model predictions are plausible on the outcome scale and whether the model behaves sensibly across the histories represented in each pattern.
• Inspect sensitivity results across clinically defensible offsets, including whether conclusions change over a narrow or wide range.

Large differences in observed histories or dropout rates are warning signs, not proof that a particular missing-data assumption is wrong. Conversely, similar dropout rates do not prove that missing outcomes resemble observed outcomes. Sensitivity values that imply impossible scores or implausible trajectories indicate a poor scenario, while plausible values still cannot be selected solely by inspecting observed data.

A completer-only contrast is simple to calculate, but its relevance to a randomized-population question requires justification beyond randomization at baseline. A MAR-based analysis can use observed histories under its assumptions; the earlier material on missing data provides the standard context for that approach. Pattern-mixture sensitivity analysis is useful when conclusions may depend on departures from the reference assumptions, but it does not turn unobserved outcomes into observed data. It supplements, rather than replaces, a clearly defined estimand and a careful account of uncertainty.

The calculated values above are point estimates. A full analysis should also quantify sampling uncertainty using a method appropriate to the model and design, and should report that uncertainty alongside sensitivity results. A point estimate crossing zero does not by itself establish that the treatment contrast is statistically indistinguishable from zero.

A Common Mistake

Wrong reasoning: “Treatment completers improved by 2 points more than control completers, and participants were randomized, so treatment improved outcomes by 2 points for the randomized groups.”

Correction: Randomization applies to assignment, not to the selected subset that completed follow-up. With differential dropout, the completer means describe different observed subsets. To estimate a contrast for all randomized participants, specify how the missing outcomes are handled, state the assumptions, and examine how the result changes under plausible alternatives. The 2-point completer difference remains a valid descriptive statistic, but it is not automatically the randomized-population contrast.