Why the Reference Arm Must Be Chosen Deliberately
In a multiarm trial, more than one observed group may appear to be a plausible reference for participants who discontinue an experimental treatment. A trial might include placebo, standard care, and a lower-dose regimen alongside a higher-dose regimen. These groups do not necessarily represent the same clinical situation after discontinuation. Choosing among them therefore defines an assumption about the outcomes that would follow dropout; it is not merely a technical choice about which data are easiest to use.
For J2R or copy-reference imputation, the selected reference arm supplies the post-dropout outcome behavior specified by the imputation rule. The choice can affect predicted means, predictive variability, and ultimately the treatment-effect estimate and its uncertainty. It should be justified by the clinical question and trial design, stated before examining which choice produces a preferred result, and reported clearly.
Let \(Y_{ij}\) denote the outcome for participant \(i\) at visit \(j\), and let \(r\) denote a candidate reference arm. For a participant assigned to an experimental arm who drops out after visit \(d\), the observed history through \(d\) is retained. The selected reference model then supplies the distribution used for visits after \(d\), according to the chosen J2R or copy-reference rule. The earlier material on sequential reference-based imputation provides the visit-by-visit framework; the issue here is which arm supplies that reference behavior.
Criteria for a Defensible Reference
The most important question is: Which observed arm best represents the post-discontinuation course assumed by the scientific question? Answer it using the treatment and care participants would receive after stopping, not by searching for the arm whose results make the treatment look strongest or weakest.
- Clinical meaning. Specify what treatment, usual care, or lack of treatment the reference represents after discontinuation. Placebo may be unsuitable if participants would receive effective rescue treatment in practice.
- Trial design and treatment history. Check whether the candidate group represents the same relevant background care, concomitant treatment, and follow-up context. A low dose is not automatically an appropriate reference for a high dose.
- Outcome and visit compatibility. The reference must provide data on the same outcome, measurement scale, and relevant follow-up times, or support a justified model for unequal schedules.
- History overlap. The reference model should support predictions for the observed histories of the participants being imputed. If those histories are unlike the reference arm’s histories, the prediction may rely heavily on extrapolation.
- Prespecification and transparency. Define the reference and the population to which it applies in the analysis plan. If reasonable alternatives exist, explain the primary choice and consider reporting alternatives as sensitivity analyses.
These criteria are connected. For example, a clinically relevant arm may still be a poor statistical reference if its observed histories do not support the conditional predictions needed for the imputation. Conversely, a large arm with good model fit is not automatically clinically appropriate. Both the substantive assumption and its empirical support matter.
How the Choice Enters the Prediction
The effect of selecting a reference can be seen in a simple one-step conditional normal model. Suppose the reference-arm distribution for the outcome at dropout and the next visit is bivariate normal, with means \(\mu_{d,r}\) and \(\mu_{d+1,r}\), standard deviations \(\sigma_{d,r}\) and \(\sigma_{d+1,r}\), and correlation \(\rho_r\). For an observed dropout value \(y_d\), the reference-based conditional mean at the next visit is
This is the conditional mean from the bivariate normal distribution. The selected arm determines the means, standard deviations, and correlation in the expression. Thus, even when the same observed \(y_d\) is used, two plausible reference arms can imply different forecasts. In actual multiple-imputation procedures, uncertainty in model parameters and predictive variation also contribute to imputations; the calculation below isolates the conditional mean to show the role of arm choice.
Worked Example: Choosing a J2R Reference Among Three Arms
Hypothetical setting. A trial compares a high-dose experimental treatment with placebo and standard care. The primary question concerns the high-dose treatment. Investigators specify that, after a participant stops the high dose, the post-dropout course should represent discontinuation to placebo rather than continued standard care. They therefore select placebo as the J2R reference, provided its observed data support the required model.
For illustration, suppose a participant’s observed outcome at dropout is \(y_d=12\) units. The fitted placebo model has \(\mu_{d,r}=10\), \(\mu_{d+1,r}=11\), \(\sigma_{d,r}=2\), \(\sigma_{d+1,r}=2\), and \(\rho_r=0.5\). The one-step conditional mean is
Suppose the standard-care model instead has means \(9\) and \(10\), the same standard deviations, and correlation \(0.4\). Its conditional mean for the same observed history would be
The arithmetic demonstrates that the arm choice changes the conditional forecast: 12 units under placebo versus 11.2 units under standard care. It does not establish which assumption is scientifically correct. In this example, placebo is chosen because it matches the prespecified clinical scenario, not because its forecast is more favorable. The analysis should also check whether placebo histories adequately support prediction at the observed value of 12 units.
Copy-Reference Choice and Composite References
Copy-reference imputations also require a specific reference arm. In a simple location-shift formulation, the participant retains the observed outcome at dropout and subsequent changes follow the selected reference arm’s expected changes. If \(\Delta_{r}\) is the reference arm’s expected change from dropout to the next visit for the relevant history, then the illustrative conditional mean is
This expression is a simple formulation, not a replacement for the fitted conditional model used in a full analysis. Its purpose is to make the reference choice explicit: different arms can imply different post-dropout changes. The exact conditioning and covariance structure must follow the specified imputation model.
Worked Example: Copy-Reference Predictions Under Two Plausible Arms
Hypothetical setting. A trial compares an experimental regimen with placebo and standard care. For participants who discontinue the experimental regimen, investigators have prespecified that placebo is the primary reference because it represents the intended post-discontinuation scenario. A sensitivity analysis will use standard care because some participants could receive it in practice.
For a participant with an observed dropout value of 12 units, suppose the fitted conditional expected next-visit change is \(0.8\) units under placebo and \(0.2\) units under standard care. Under the simple copy-reference location-shift formulation, the placebo-based mean is
Using standard care instead gives
These are conditional means, not individual imputed values: predictive variation and parameter uncertainty remain part of multiple imputation. The primary choice is justified by the prespecified clinical scenario, while the alternative shows how conclusions may depend on a different plausible reference. The calculation assumes the models estimate changes on the same outcome scale and for a comparable history; those assumptions must be checked before interpreting the contrast.
Sometimes a trial has several arms that together represent the intended reference condition. A composite reference can be prespecified, but pooling should not be automatic. The component arms must have a defensible shared interpretation, and the analysis must specify how their data contribute to the reference distribution. Simply combining outcomes because the arms are available can obscure meaningful differences in care or response.
Worked Example: A Prespecified Pooled Reference
Hypothetical setting. A three-arm trial includes two randomized arms that differ only in a delivery schedule but provide the same background standard care. The protocol defines their combination as the reference for participants discontinuing the experimental treatment. Suppose the next-visit mean changes are 1.0 units in the first reference component, based on 40 participants, and 0.5 units in the second, based on 60 participants. For this simple illustration, use the participant-count-weighted mean change:
For a participant whose observed outcome at dropout is 12 units, the corresponding location-shift conditional mean is \(12+0.70=12.70\) units. This calculation assumes the components are intentionally treated as one reference population and that the weighted mean is appropriate for the target composition. In a full model, pooling must also address covariates and covariance, not just the mean. If the component arms represent substantively different post-dropout courses, a pooled mean could hide that heterogeneity and may not represent either course well.
Diagnostics, Alternatives, and Limitations
A reference-arm justification should be accompanied by checks that the model can use the selected arm responsibly. Examine observed trajectories by arm and visit, compare outcome distributions and relevant covariates, and assess whether the observed histories of participants needing imputation fall within the range supported by the reference data. Check that the fitted conditional predictions are clinically plausible and that the model converges and produces valid predictive distributions. Large extrapolations, implausible jumps at dropout, or marked differences among candidate-arm trajectories deserve investigation rather than automatic acceptance.
These checks assess support and model behavior; they cannot prove that the assumed post-dropout course is true. The missing outcomes are not observed, so the reference-based rule remains an assumption. A well-fitting model in the reference arm does not resolve whether that arm represents the clinical scenario of interest.
| Approach | What it assumes | When it may be useful |
|---|---|---|
| One prespecified reference arm | One observed arm best represents the intended post-dropout course. | When the clinical scenario is clearly represented by a single comparator. |
| Prespecified composite reference | Several arms can defensibly be treated as one reference population under a stated pooling rule. | When the components represent a common care condition and the target mixture is defined. |
| Alternative reference sensitivity analyses | More than one reference is plausible; each analysis states a different assumption. | When clinical practice or uncertainty about post-discontinuation care supports credible alternatives. |
The alternatives answer different questions; they are not interchangeable repairs for an inconvenient result. A pooled reference can increase the amount of information contributing to a model, but it may conceal meaningful differences. Separate reference analyses preserve those differences but require a clear interpretation for each. Sensitivity analyses can show the dependence of conclusions on the assumption, but they do not identify which unobserved outcomes actually occurred.
A second common error is pooling arms solely because their sample sizes are convenient. Similar labels do not establish that groups share the same post-dropout meaning or outcome behavior. Before pooling, state the composite reference population, why its components belong together, and how the model handles their contributions. If that justification is weak, use a clinically coherent single arm or present separate alternatives.
The central limitation is that reference selection concerns outcomes after dropout, which are unobserved. Diagnostics can reveal poor overlap, instability, or implausible implications, but cannot validate the assumed missing outcomes directly. The analysis should therefore name the selected arm, the reason for choosing it, the rule applied after dropout, and any important alternative assumptions examined.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A trial includes placebo and standard care. What should determine which is the primary J2R reference?
- For a reference model, let \(\mu_{d,r}=8\), \(\mu_{d+1,r}=9\), \(\sigma_{d,r}=2\), \(\sigma_{d+1,r}=3\), \(\rho_r=0.4\), and \(y_d=10\). Find the one-step conditional mean.
- In a simple copy-reference formulation, the observed dropout value is 15 units and the reference expected change is \(-0.6\) units. Find the conditional mean and state what it represents.
- Two comparator arms have similar sample sizes but represent different post-discontinuation care. Is pooling justified by sample size alone?
- Why should the primary reference not be selected because it yields a preferred treatment estimate?
Solutions. (1) Choose the arm whose post-discontinuation clinical scenario matches the scientific question and trial context, then verify that its data support the required predictions. (2) \(9+0.4(3/2)(10-8)=9+1.2=10.2\) units. (3) \(15+(-0.6)=14.4\) units; this is the illustrative conditional mean under the selected reference change, not a realized outcome or a complete predictive distribution. (4) No. Pooling requires a defensible shared reference interpretation and an explicit pooling rule; sample size alone is not enough. (5) That makes the assumption outcome-driven and risks favoring a desired conclusion. Prespecify the clinically justified primary reference and use credible alternatives as transparent sensitivity analyses.