Purpose and Imputation Rules
When an active-treatment participant discontinues treatment, later outcomes may still be collected, or they may be missing. If those outcomes are missing, a sensitivity analysis must specify how the participant’s post-discontinuation outcomes are related to the reference arm. Jump-to-reference (J2R) assumes that after discontinuation the participant follows the reference-arm outcome process. Copy-reference (CR) assumes that the participant follows the reference arm’s subsequent changes while retaining the active-treatment contrast that had been reached at discontinuation. These are different assumptions about unobserved outcomes, not claims that a participant’s actual outcome necessarily follows either trajectory.
This tutorial gives an implementation for continuous longitudinal outcomes using a multivariate normal model. It focuses on the missing post-discontinuation outcomes of active-arm participants. Outcomes already observed after discontinuation are retained. Missing outcomes outside this situation require their own prespecified imputation rule; do not silently apply J2R or CR to every missing value.
For CR, “carried forward” means an additive constant here: the mean contrast at the discontinuation visit is added to each later reference-based conditional mean. This is a working assumption on the specified scale. If the contrast is defined on a transformed or link scale, the shift must be applied on that scale and interpreted accordingly.
Conditional Distributions From the Full Observed History
Suppose the reference-arm model has mean vector \(\mu^R\) and covariance matrix \(\Sigma\), with the elements ordered by visit. Partition these quantities into observed and missing visits: \(\mu_O^R,\mu_M^R,\Sigma_{OO},\Sigma_{MO},\Sigma_{MM}\). Under a multivariate normal model, the reference-based conditional distribution is
The matrix \(\Sigma_{MO}\Sigma_{OO}^{-1}\) uses all observed visits. It adjusts the reference-arm prediction according to how this participant’s observed history differs from the reference-arm mean history. For J2R, use this conditional mean as written. For CR, add \(\delta_d\) to every element of that conditional mean, where \(\delta_d\) is the active-minus-reference mean contrast at the discontinuation visit \(d\). Under this additive CR rule, the conditional covariance is unchanged.
The reason for the conditional formula can be seen by subtracting the linear predictor from \(Y_M\). The residual \(Y_M-\mu_M^R-\Sigma_{MO}\Sigma_{OO}^{-1}(Y_O-\mu_O^R)\) has zero covariance with \(Y_O\). Under joint normality, zero covariance implies independence, so conditioning on \(Y_O\) leaves this residual’s covariance unchanged. Its covariance is \(\Sigma_{MM}-\Sigma_{MO}\Sigma_{OO}^{-1}\Sigma_{OM}\), giving the stated conditional distribution.
The implementation assumes a correctly specified reference-arm mean and covariance model, a positive-definite observed-history covariance matrix, and the stated multivariate normal distribution (or a suitable model-based analogue). In a practical multiple-imputation analysis, model parameters should also be drawn or otherwise varied to represent their estimation uncertainty; plugging in one fixed set of estimates and drawing only residuals generally omits that source of uncertainty. The examples below use known parameters to make the conditional calculations transparent.
Worked Example: J2R and CR After Discontinuation at Week 4
Consider a hypothetical two-arm trial with lower symptom scores indicating better health. The reference-arm means at weeks 4, 8, and 12 are 12, 13, and 12. The active-arm mean at week 4 is 14, so the active-minus-reference contrast at discontinuation is \(\delta_4=14-12=2\) points: the active mean is 2 points worse. A participant discontinues after week 4 with an observed score of 14. Assume the reference-arm covariance matrix for these visits is
This illustrative model has variance 4 at each visit and correlations 0.5 for visits one interval apart and 0.25 for visits two intervals apart. The observed-history residual relative to the reference mean is \(14-12=2\). The covariance of weeks 8 and 12 with week 4 is \((2,1)^\mathsf{T}\), and the variance at week 4 is 4. Thus the conditional-mean adjustment is \((2/4,1/4)^\mathsf{T}\times2=(1,0.5)^\mathsf{T}\).
For J2R, the missing-outcome conditional means are therefore \((13+1,\ 12+0.5)=(14,\ 12.5)\). For CR, add the 2-point contrast at discontinuation, giving conditional means \((16,\ 14.5)\). In this example the active mean is worse, so CR carries that 2-point disadvantage relative to the reference-based profile; it does not carry an advantage.
The conditional covariance for weeks 8 and 12 is
For a concrete joint residual draw, a Cholesky factor of this matrix is approximately \(\begin{pmatrix}1.732&0\\0.866&1.732\end{pmatrix}\). If the two simulated standard-normal values are \(z_1=0.4\) and \(z_2=-0.2\), the residual draw is \((1.732(0.4),\ 0.866(0.4)+1.732(-0.2))=(0.693,\ 0)\). The resulting J2R imputation is approximately \((14.693,\ 12.500)\); the corresponding CR imputation is \((16.693,\ 14.500)\). These are illustrative random draws, not the conditional means. The assumptions used are explicit: the specified normal reference model, the stated covariance, and an additive CR shift of 2 points.
Use Every Observed Visit
A sequential implementation can be useful, but a later draw must preserve the joint predictive distribution conditional on the participant’s entire observed history. Conditioning week 12 only on an imputed week-8 value is valid only under an explicitly specified model in which week 4 is conditionally irrelevant once week 8 is known. The general multivariate formula above does not require that shortcut: it conditions directly on all observed visits.
Worked Example: Conditioning on Two Observed Visits
In a second hypothetical calculation, the participant has observed scores 10.5 at week 0 and 14 at week 4, then discontinues. Reference means at weeks 0, 4, 8, and 12 are \((10,12,13,12)\); the active-minus-reference contrast at week 4 is again 2 points. Use a variance of 4 and correlation 0.5 between adjacent visits, with correlation 0.25 between visits two intervals apart. The observed residual vector is \((0.5,2)^\mathsf{T}\).
For observed visits weeks 0 and 4, \(\Sigma_{OO}=\begin{pmatrix}4&2\\2&4\end{pmatrix}\), whose inverse is \(\frac{1}{12}\begin{pmatrix}4&-2\\-2&4\end{pmatrix}\). The covariance between missing weeks 8 and 12 and the two observed visits is \(\Sigma_{MO}=\begin{pmatrix}2&4? \end{pmatrix}\) only if entries are placed incorrectly; using the specified time-lag correlations, the correct matrix is \(\Sigma_{MO}=\begin{pmatrix}1&2\\0.5&1\end{pmatrix}\). The entries are 1 and 2 for week 8’s covariances with weeks 0 and 4, and 0.5 and 1 for week 12’s covariances with those visits.
Multiplying gives \(\Sigma_{MO}\Sigma_{OO}^{-1}=\begin{pmatrix}0&0.5\\0&0.25\end{pmatrix}\). Applying this to the full observed residual vector yields \((1,0.5)^\mathsf{T}\). Hence the J2R conditional means for weeks 8 and 12 are \((14,12.5)\), and the CR conditional means are \((16,14.5)\). Week 0’s residual has coefficient zero under this particular covariance structure, but that result follows from the specified matrix calculation; it was not assumed away. The conditional covariance is the same \(\begin{pmatrix}3&1.5\\1.5&3.75\end{pmatrix}\) as in the preceding example.
This calculation checks the conditioning history rather than presuming that a recent observation makes older observations irrelevant. A different covariance structure could give week 0 a nonzero coefficient and change both the conditional means and covariance.
Implementation Workflow and Checks
A reproducible implementation separates the identification of eligible missing values from the model-based draws. For each imputation, retain observed outcomes, apply the chosen rule only to eligible missing post-discontinuation outcomes in the active arm, and record the parameters and random-number state needed to reproduce the draw.
Use treatment assignment, discontinuation time, visit, and outcome-observation status. Do not overwrite observed outcomes or apply the active-arm reference rule to reference-arm participants.
For the participant’s visits and covariates, obtain the reference-arm mean vector and covariance matrix. Record the outcome scale, visit order, and model used.
Construct \(y_O,\mu_O^R,\mu_M^R,\Sigma_{OO},\Sigma_{MO},\Sigma_{MM}\) using the same visit order. Calculate the J2R conditional mean and covariance from the full observed history.
For J2R, use the conditional reference mean. For CR, add the prespecified discontinuation contrast to that mean. Draw the missing vector jointly, and include parameter uncertainty in the multiple-imputation procedure.
Check that observed data are unchanged, missing target values are filled, draws are on the intended scale, and the covariance and imputation labels are saved with the completed data.
Worked Example: A Week-8 Discontinuation and One Missing Visit
Suppose a hypothetical participant has observed scores 12.5 at week 4 and 15 at week 8, discontinues after week 8, and has a missing week-12 score. Reference means at these visits are 12, 13, and 12. Under the same variance-4, adjacent-correlation-0.5 model, the observed residuals are \((0.5,2)\). The covariance of week 12 with weeks 4 and 8 is \((1,2)\), and \(\Sigma_{OO}^{-1}=\frac{1}{12}\begin{pmatrix}4&-2\\-2&4\end{pmatrix}\). Therefore the conditional regression coefficients are
The week-12 conditional adjustment is \(0(0.5)+0.5(2)=1\), so the J2R conditional mean is \(12+1=13\). The week-12 conditional variance is \(4-(0,0.5)(1,2)^\mathsf{T}=3\). The active mean at week 8 is 15 and the reference mean is 13, so \(\delta_8=2\); the CR conditional mean is \(13+2=15\). If the simulated standard-normal residual is \(-0.3\), then the residual draw is \(\sqrt{3}(-0.3)\approx-0.520\). The imputed values are approximately 12.480 under J2R and 14.480 under CR.
Both observed visits were included in the calculation. The week-4 coefficient happens to be zero for this specified covariance, but that is a consequence of the matrix calculation, not a general license to condition only on week 8. The example also assumes the stated Gaussian model and the additive CR contrast at week 8.
Diagnostics, Alternatives, and Limitations
Check the implementation at both the record and distribution levels. At the record level, verify treatment arm, discontinuation visit, missingness status, and visit ordering. At the distribution level, inspect conditional means and variances for plausible values, confirm that conditional covariance matrices are symmetric and positive semidefinite, and look for failed matrix inversions or extreme draws. Compare generated summaries across imputations with the reference model and observed histories. Unexpectedly identical imputations can indicate that residual draws or parameter draws were omitted; implausible discontinuities can indicate a wrong visit index, scale, or contrast sign.
J2R and CR differ from MAR because they impose reference-based assumptions on the post-discontinuation distribution rather than relying only on the outcome model for observed data. They also differ from delta adjustments: a delta adjustment shifts imputations according to a specified sensitivity parameter, whereas J2R and CR define the shift through a reference-arm trajectory and, for CR, the treatment contrast at discontinuation. These alternatives encode different assumptions; none is selected by a diagnostic test that reveals the unobserved outcomes.
The normal formulas are not universal. Binary, ordinal, count, or strongly non-normal outcomes need a suitable predictive model and an appropriately defined reference-based rule; a Gaussian draw should not be applied mechanically. Results can also depend on the reference-arm model, covariance structure, contrast scale, and whether model-parameter uncertainty is represented. J2R and CR are sensitivity assumptions, not guaranteed descriptions of post-discontinuation outcomes. Save the completed data sets and the rule, parameters, and diagnostics used to create them; subsequent analysis and pooling are separate steps.
Common Mistakes
- Using only the most recent observed value. That is not generally the conditional distribution given the full observed history. Use the full covariance formula, or state and justify a model that makes older history conditionally irrelevant.
- Reversing the contrast sign. Define \(\delta_d\) as active mean minus reference mean. If lower scores are better and the active mean is higher, a positive CR shift carries forward a disadvantage.
- Confusing target means with imputed draws. A conditional mean is not an individual imputation; add a draw from the conditional covariance distribution.
- Overwriting observed post-discontinuation outcomes. Reference-based rules fill the specified missing outcomes; they do not erase observed data.
- Ignoring parameter uncertainty or covariance failures. Residual-only draws with fixed fitted parameters can underrepresent uncertainty, while a non-positive-definite covariance can make the intended draw impossible.
Check Your Understanding
Answer each question, then compare with the solutions below.
- At discontinuation, the active mean is 8 points and the reference mean is 10 points. On an additive outcome scale, what CR shift is carried forward?
- Why is conditioning a week-12 draw only on an imputed week-8 value not generally sufficient when week 4 was also observed?
- Under the week-8-discontinuation example above, what are the J2R conditional mean and variance at week 12?
- Name two checks that can identify an implementation problem before the imputed data are analyzed.
Solutions. (1) \(\delta_d=8-10=-2\), so CR carries a 2-point advantage (lower outcome) relative to the reference profile. (2) The joint conditional distribution given week 4 and week 8 depends on the full observed-history covariance unless the model establishes conditional irrelevance of week 4; the general calculation uses both. (3) The J2R mean is 13 and the variance is 3, from the stated full-history calculation. (4) For example, verify that observed values were not overwritten and that visit indexing and treatment arm are correct; also check covariance symmetry and positive semidefiniteness, and confirm that missing target records were filled.