Comparing J2R and Copy-Reference in a Trial Example

Why Compare Reference-Based Rules?

When trial participants discontinue the assigned treatment, their later outcomes are unobserved. A reference-based imputation rule specifies how those missing outcomes are predicted relative to outcomes in a reference arm. The rule matters because it encodes an assumption about what happens to the treatment benefit after discontinuation; it does not merely fill in an otherwise neutral blank.

Two common assumptions are jump to reference (J2R) and copy reference (CR). In a simple marginal-mean description, J2R predicts that the outcome mean moves to the reference-arm mean after discontinuation. CR instead retains the mean advantage already achieved on active treatment and applies the reference-arm mean change after discontinuation. They can therefore produce different imputed trajectories and different treatment-effect estimates, even when they use the same observed trial data.

These marginal rules are useful for seeing the assumptions clearly. In an actual reference-based multiple-imputation analysis, predictive distributions may also condition on baseline variables and observed outcome history, and must include appropriate uncertainty. The conditional model determines the precise predictions; the marginal formulas below are a transparent, hypothetical illustration of the contrast between assumptions.

Hypothetical Trial and Assumptions

Consider a randomized trial of 20 participants per arm with a continuous outcome measured at baseline and three follow-up visits. Higher values are better. All values below are hypothetical group means in outcome units. The control-arm means are 0 at baseline, then 2, 4, and 6 at visits 1, 2, and 3. In the active arm, 4 participants complete follow-up and have a visit-3 mean of 10. Eight participants discontinue after visit 1, when their mean is 6. Another 8 discontinue after visit 2, when their mean is 8. For the latter group, their visit-1 mean is also 6.

For this illustration, assume the reference-based marginal predictions are as follows: after dropout, J2R assigns a visit-3 mean of 6 to either dropout pattern; CR applies the control-arm change after each participant group’s last observed visit. The outcome is analyzed as a visit-3 mean difference, active minus control, with each randomized active-arm participant contributing either an observed or imputed visit-3 value. The control mean at visit 3 is 6. These assumptions make the arithmetic explicit; they are not universal predictions for every fitted conditional reference model.

The calculation assumes these pattern counts and means are treated as fixed inputs to the worked example. In a real analysis, imputed values vary across imputations, parameter estimates are uncertain, and inferential uncertainty must be carried through using proper multiple-imputation procedures. The earlier material on proper MI variance and Rubin-rule inference addresses that inferential step.

From Predicted Trajectories to Treatment Effects

A trajectory comparison should be made before summarizing the endpoint effect. The control trajectory is \(0,2,4,6\). The observed active means for the two dropout patterns are \(6\) at visit 1 and \(6,8\) through visit 2; completers have a visit-3 mean of 10. Under J2R, post-dropout predictions move to reference means at the later visits. Under CR, they continue from each pattern’s last active mean by adding the corresponding reference-arm change. In this example, both CR patterns reach a visit-3 mean of 10.

At visit 3, the J2R active-arm mean is the weighted average of the observed completer mean and the imputed means for the two dropout patterns: $$ \widehat{\mu}_{A,3}^{\mathrm{J2R}} = \frac{4(10)+8(6)+8(6)}{20} = \frac{40+48+48}{20} = \frac{136}{20} = 6.8. $$ The treatment-effect estimate is therefore $$ \widehat{\Delta}_{3}^{\mathrm{J2R}} = \widehat{\mu}_{A,3}^{\mathrm{J2R}}-\widehat{\mu}_{C,3} = 6.8-6 = 0.8\text{ outcome units}. $$

Under CR, the observed completers have mean 10 and both dropout patterns have predicted means of 10: $$ \widehat{\mu}_{A,3}^{\mathrm{CR}} = \frac{4(10)+8(10)+8(10)}{20} = \frac{200}{20} = 10. $$ Therefore, $$ \widehat{\Delta}_{3}^{\mathrm{CR}} = 10-6 = 4.0\text{ outcome units}. $$ The CR estimate is \(4.0-0.8=3.2\) outcome units higher than the J2R estimate. This difference is driven by the different post-dropout predictions, not by any change in the observed completer outcomes or the control-arm mean.

Why the Rules Differ: A Short Derivation

Under the CR mean rule, the predicted post-dropout value at visit 3 can be rewritten as the reference endpoint mean plus the active-minus-reference difference at dropout: $$ m_{A,d}+\left(m_{C,3}-m_{C,d}\right) = m_{C,3}+\left(m_{A,d}-m_{C,d}\right). $$

The equality follows by adding and subtracting \(m_{C,d}\): the active mean at dropout is \(m_{C,d}+(m_{A,d}-m_{C,d})\). Adding the reference change \(m_{C,3}-m_{C,d}\) leaves \(m_{C,3}+(m_{A,d}-m_{C,d})\). Thus CR carries the dropout-time mean difference forward on the additive outcome scale. Here the difference at either dropout time is 4 units: at visit 1, \(6-2=4\); at visit 2, \(8-4=4\). The CR endpoint prediction is consequently \(6+4=10\) for each pattern. J2R, as specified for this example, instead predicts 6 at visit 3 without retaining that mean difference.

This derivation also identifies a substantive assumption. CR does not assert that participants remain on active treatment; it assumes that the mean difference accumulated by dropout persists while subsequent mean change follows the reference arm. J2R assumes that the post-dropout mean follows the reference level. Neither rule is established by the observed data alone.

Comparison, Diagnostics, and Limitations

These rules are alternatives to one another, and to other missing-data assumptions such as MAR-based imputation. They should not be treated as two estimators of a single uniquely identified treatment effect: their difference reflects different assumptions about unobserved outcomes. The result is a sensitivity comparison. It does not establish which missing-outcome model is correct.

Several checks can reveal when a calculation or model is going wrong:

• Check the dropout timing. A reference rule applies after dropout, not to observed active-treatment outcomes before it. Each pattern’s last observed visit determines the CR increment and the start of the post-dropout rule.
• Check reference changes by interval. For early dropout here, CR adds \(6-2=4\), not the later-interval change \(6-4=2\). For later dropout it adds 2.
• Plot observed and imputed trajectories. Look for unintended jumps, reversals, or implausible levels at dropout. A large gap between J2R and CR predictions is not automatically an error, but it deserves clinical interpretation.
• Check pattern counts and weights. The three active-arm pattern counts sum to \(4+8+8=20\). An incorrect denominator or duplicated participant changes the endpoint mean.
• Inspect predictive distributions, not just means. Implausible spread, impossible values, or strong dependence on a small reference arm may indicate unstable or unsuitable predictive models. A plausible mean alone does not validate the full imputation distribution.

A common mistake is to average only the observed completers and call that the treatment effect, or to replace every active-arm outcome—including observed pre-dropout outcomes—with a reference value. The first discards the missing patterns and changes the target population; the second applies the post-dropout rule too early. The correction is to retain observed data through each participant’s dropout time, impute only later missing outcomes under the chosen rule, and then combine observed and imputed outcomes with the correct pattern counts.

The simple marginal calculations have important limits. Real longitudinal analyses generally require conditional predictive distributions that respect correlations among repeated measures and incorporate parameter uncertainty. The CR arithmetic shown here is an additive mean-scale illustration; nonlinear outcome scales, changing treatment effects, covariate distributions, or departures from the assumed reference trajectory can alter the predictions. The J2R and CR labels alone do not specify every modeling detail, so an analysis must make its conditioning and distributional assumptions clear.