Why the Reference Trajectory Matters in a Maintenance Trial
In a maintenance trial, participants who respond to an initial treatment may be randomized to continue the experimental treatment or switch to a reference treatment, such as placebo. If a participant assigned to continued experimental treatment stops it, the later outcome is not automatically observed. A sensitivity analysis must specify what outcomes after that point would look like. The jump-to-reference assumption says that, after experimental-treatment discontinuation, the distribution of missing outcomes follows the reference-arm distribution for participants with comparable observed histories.
This is an assumption about unobserved outcomes, not a claim that every participant’s symptoms change instantly or identically when treatment stops. It is especially consequential when symptom scores are higher for worse health: a reference trajectory that worsens after discontinuation will produce higher imputed scores than an assumption that preserves the experimental-treatment benefit.
The reference distribution must be defined at the relevant visit and conditional on a suitable history. In practice, that history might include baseline severity, the last observed symptom score, and the time of discontinuation. J2R is therefore not generally equivalent to assigning each missing participant the reference arm’s overall average. It is also not a deterministic rule: an imputation model typically draws values with the reference distribution’s residual variability as well as its conditional mean.
The hypothetical example below uses a continuous symptom score in which higher values indicate worse symptoms. Participants who respond to induction are randomized to experimental maintenance treatment \(A\) or reference treatment \(R\). The target comparison is the difference in mean week-24 score between the randomized groups, with negative values favoring experimental maintenance. All trial numbers are hypothetical.
From a Conditional Reference Trajectory to an Imputed Mean
A useful way to see the assumption is to start with a participant’s observed history. Suppose a participant assigned to maintenance treatment has a last observed score of 10 at discontinuation. If comparable reference-arm participants with that history are predicted to worsen by 4 points by week 24, the J2R conditional mean is 14. Under a continued-benefit assumption, the corresponding predicted change might instead be 0, giving a week-24 mean of 10. The difference between these predictions comes from the assumed post-discontinuation trajectory, not from changing the participant’s observed score.
Worked Example: Calculating a Conditional Post-Discontinuation Mean
Suppose a hypothetical reference-arm model predicts an average increase of 4 symptom-score points from discontinuation to week 24 for participants with a last score of 10 and a similar measured history. The same history under a continued-benefit model predicts no average change. Calculate both conditional means.
Jump-to-reference. Add the reference-arm predicted change to the last observed score: \(10+4=14\). The J2R conditional mean at week 24 is 14 points.
Continued benefit. Add the continued-benefit predicted change: \(10+0=10\). The conditional mean under this assumption is 10 points.
The assumptions differ by 4 points for this history. These are conditional mean predictions, not individual guarantees; imputed values would also reflect the modeled outcome variability. The calculation is appropriate only if the reference model supports the stated history and the change is expressed on the same score scale and time interval.
To obtain a treatment-group mean, combine observed endpoint outcomes with imputed missing outcomes. If \(n_{\mathrm{obs}}\) participants have observed week-24 outcomes with mean \(\bar{Y}_{\mathrm{obs}}\), and \(n_{\mathrm{mis}}\) have missing outcomes with imputed mean \(\bar{Y}_{\mathrm{mis}}\), then the completed-data mean is
This identity follows by adding the observed and imputed outcome totals and dividing by the full randomized group size. With multiple imputations, each completed data set supplies its own estimate; uncertainty must account for both within-imputation and between-imputation variation. The calculation below isolates the mean implication of the assumptions.
Worked Example: Combining Observed Outcomes With J2R Imputations
In a hypothetical group of 100 participants assigned to experimental maintenance, 80 have observed week-24 scores with mean 8 points. The other 20 discontinued treatment and have missing week-24 outcomes. Suppose their average J2R imputation is 14 points, as in the conditional calculation above. Calculate the completed-data mean and compare it with continued benefit, which imputes a mean of 10 for those 20 participants.
Under J2R, the observed outcomes contribute \(80\times 8=640\) score-points, and the imputed outcomes contribute \(20\times 14=280\) score-points. Thus
Under continued benefit, the imputed total is \(20\times 10=200\) score-points, so
J2R raises the estimated mean by \(9.20-8.40=0.80\) points relative to continued benefit. The difference is 20% of the 4-point contrast between the missing-outcome predictions, because 20% of the randomized group has missing outcomes: \(0.20\times 4=0.80\). Since higher scores are worse, this J2R scenario is less favorable to experimental maintenance.
Comparing the Randomized Groups
The treatment contrast is formed using the imputed means for both randomized groups. For clarity, suppose the reference group has 90 observed week-24 outcomes with mean 12 points and 10 missing outcomes. Assume the missing reference outcomes are handled so that their imputed mean is also 12 points. The completed reference-group mean is then 12 points. This simplifying assumption is for the example; it does not remove the need to justify the reference group’s missing-data model in an actual analysis.
Worked Example: How J2R Changes the Trial Contrast
Use the group means from the preceding calculation: the experimental group has mean 9.20 under J2R or 8.40 under continued benefit, and the reference group has mean 12.00. Define the contrast as experimental minus reference.
Under J2R, the contrast is \(9.20-12.00=-2.80\) points. Under continued benefit, it is \(8.40-12.00=-3.60\) points. Both estimated contrasts favor experimental maintenance because they are negative and lower symptom scores are better. The J2R estimate shows a 2.80-point lower mean score than reference, compared with a 3.60-point lower mean under continued benefit.
The 0.80-point movement toward zero matches the shift in the experimental-group mean, because the reference-group mean was held fixed. These are point estimates in a hypothetical example, not evidence of statistical significance. In an analysis, uncertainty intervals and any inferential conclusions must be calculated from the imputed data and reported with the estimand and assumptions clearly specified.
More generally, if both scenarios use the same observed outcomes and the same reference-group estimate, the difference in the experimental-group mean is
This follows because the observed-outcome term is identical under both scenarios and cancels when the two weighted means are subtracted. The relation makes two practical points: a larger difference between the assumptions matters more when more outcomes are missing, and even a substantial difference in imputed means may have a modest effect on the group mean when few outcomes are missing.
Assumptions, Diagnostics, and Failure Modes
J2R is a sensitivity assumption for outcomes that are missing after discontinuation; the observed data do not identify the post-discontinuation distribution in the experimental group. Its interpretation depends on several conditions and modeling choices:
- Comparable observed histories. The reference data must include participants with histories sufficiently similar to those of experimental-group discontinuers. Conditional predictions outside that support rely heavily on model extrapolation.
- Appropriate conditional model. The reference outcome model must represent the relation between prior history and later outcomes, and should reflect the visit schedule and relevant covariates. Incorrect functional form or omitted predictors can distort the imputed mean.
- Reference-arm missingness is addressed. If the reference outcomes used to estimate \(F_R\) are themselves missing, their handling also requires assumptions. Calling a model J2R does not automatically resolve missing reference-arm outcomes.
- Meaningful discontinuation and missingness rules. Discontinuation is not the same as loss to follow-up. If a participant stops experimental treatment but still provides a week-24 outcome, that measured outcome remains observed; it is not replaced by a J2R imputation. The analysis must specify which missing post-discontinuation outcomes receive the assumption.
- Compatible outcome definitions. The reference trajectory must concern the same outcome, scale, and time point. A predicted change cannot be added to a score unless both use compatible units and the stated time interval.
A first diagnostic is to examine histories in experimental discontinuers and the reference participants used to predict their outcomes. Sparse or absent overlap in baseline severity, prior scores, or discontinuation time warns that the conditional reference mean may be unstable. Also check predicted values for implausible ranges, residual patterns, and whether the model reproduces observed reference-arm means across relevant history strata. Large changes from small model adjustments or a few extreme imputations are signs to investigate, not reasons to select a preferred answer silently.
A derivation clarifies what a useful comparison can and cannot show. Suppose the J2R and continued-benefit scenarios share the same observed scores, group size \(n\), and reference-group estimate. Write each experimental-group mean as the observed outcome total plus its scenario-specific imputed total, divided by \(n\). Subtract the continued-benefit mean from the J2R mean. The observed total cancels; the remaining imputed total difference is \(n_{\mathrm{mis}}\) times the difference in average imputed outcomes. Dividing by \(n\) gives the formula above. Therefore the movement in the overall contrast is driven by both the assumed post-dropout difference and the missing fraction. It is not a test of which assumption is true.
How J2R Fits Among Alternative Assumptions
| Approach | Post-discontinuation implication | Key interpretation |
|---|---|---|
| Jump-to-reference | Missing outcomes follow the conditional reference-arm distribution after discontinuation. | Experimental benefit is not carried forward for these missing outcomes according to the reference trajectory. |
| Continued benefit | Missing outcomes follow a conditional trajectory that preserves the experimental-treatment benefit. | Represents a different assumption about outcomes after discontinuation; it is not established by observing participants who remain on treatment. |
| MAR-based imputation | Uses observed information in a specified missing-at-random model, without imposing the J2R post-discontinuation restriction. | Its implications depend on the variables and model included; it need not equal either reference or continued benefit. |
These are not interchangeable computational settings with a single empirically determined answer. They encode different assumptions about unobserved outcomes. A credible report states the J2R restriction, explains how the conditional reference distribution was estimated, and presents its result alongside relevant alternatives rather than describing J2R as a fact about what participants would have experienced.
A common mistake is to say, “Everyone who discontinues is assigned the overall reference-arm mean.” That ignores history and may incorrectly treat a person with severe symptoms at discontinuation like a reference participant with mild symptoms. The correction is to use a conditional reference distribution at the relevant visit, supported by comparable observed histories, and to preserve actually observed post-discontinuation outcomes. Another mistake is to interpret a less favorable J2R contrast as proof that discontinuation caused worsening. The contrast is conditional on the assumed missing-outcome model; it does not by itself establish a causal mechanism for discontinuation or symptom change.
Check Your Understanding
Answer each question using the definitions and calculations in this tutorial. Solutions follow the questions.
- A participant’s last symptom score is 7. The conditional reference model predicts an increase of 3 points by the endpoint. What is the J2R conditional mean?
- In a group of 50, 40 observed outcomes have mean 6 and 10 missing outcomes have J2R mean 11. What is the completed-data mean?
- In the same group, continued benefit assigns the 10 missing outcomes a mean of 8. By how much does J2R change the group mean relative to continued benefit?
- A participant stops experimental treatment but has a measured endpoint score. Should that score be replaced by a J2R value? Explain.
- What is a warning sign that a conditional reference-arm prediction is relying on extrapolation?
Solutions. (1) \(7+3=10\) points. (2) The observed total is \(40\times6=240\), and the imputed total is \(10\times11=110\), so the mean is \((240+110)/50=7.00\) points. (3) The missing-outcome mean differs by \(11-8=3\) points, and the missing fraction is \(10/50=0.20\); the group mean therefore rises by \(0.20\times3=0.60\) points under J2R. (4) No. The outcome is observed, so it remains in the analysis; J2R specifies a distribution for missing post-discontinuation outcomes. (5) Little or no overlap between discontinuers’ histories and reference-arm histories, such as no comparable reference participants at the relevant severity or discontinuation-time profile, signals extrapolation.