Checking the J2R Transition at Dropout

Why Check the Transition at Dropout?

Jump to reference (J2R) changes the model used to generate outcomes after a participant discontinues the treatment strategy. Before dropout, the participant’s observed outcomes are retained. After dropout, the conditional distribution for future outcomes is based on the reference arm, with the participant’s observed history carried into that conditional prediction. A useful check is therefore to inspect what this switch implies for the change from the last observed outcome to the first imputed outcome.

That change is not required to be zero, and it is not generally equal to the difference between the treatment-arm and reference-arm marginal means at that visit. It depends on the participant’s conditioning history and the reference model’s conditional mean. A check should compare the observed transition with the transition implied by that model—not impose continuity or a marginal-mean target that J2R does not specify.

What the Reference-Model Transition Means

For clarity, consider a continuous outcome measured at scheduled visits. Suppose a participant assigned to treatment has observed outcomes through visit \(g\), then drops out. Write \(H_{ig}\) for the history used by the sequential imputation model. It may contain the observed outcome at visit \(g\), earlier outcomes, and any other conditioning variables specified by that model. Under J2R, the first missing outcome is generated from the reference-arm conditional distribution given this history. In notation,

Here \(F_0\) is the reference conditional distribution and \(m_{i,g+1}^{J2R}\) is its conditional mean. The expected transition for this participant is \(m_{i,g+1}^{J2R}-Y_{ig}\). If the imputation model is normal, for example, it also specifies a conditional variance; the realized imputed value can differ from its conditional mean because it includes a random residual and, in multiple imputation, may also reflect parameter uncertainty.

The key assumption is about the post-dropout distribution: conditional on the history included in the model, the participant’s unobserved outcomes follow the reference-arm conditional distribution. The transition check does not verify that this untestable assumption is true. It checks whether the implementation and resulting predictions are consistent with the stated assumption. The imputation model, including its conditioning history and sequential use across visits, must be correctly specified for this interpretation to apply.

From Individual Predictions to a Pattern-Level Check

Participants who drop out after the same visit form a dropout pattern. For a pattern \(g\) containing \(n_g\) participants, the model-based mean transition at the first missing visit is the average of their conditional expected transitions:

For one completed imputation, the corresponding realized average is \(\overline{\Delta}_{g,\mathrm{draw}}^{J2R} =n_g^{-1}\sum_i(Y_{i,g+1}^{J2R}-Y_{ig})\). These two quantities need not match exactly. The realized average includes random outcome variation, and may additionally include parameter-draw variation. Across many imputations, their averages should be reasonably aligned, allowing for Monte Carlo error. A small pattern can show substantial variation even when the model is implemented correctly.

A useful short derivation makes the distinction concrete. If the reference conditional mean has the linear form \(m_{i,g+1}^{J2R}=\alpha+\beta^\mathsf{T}H_{ig}\), then

Thus, the pattern-level expected change depends on the pattern’s histories and last observed outcomes. It is not obtained by subtracting two marginal means unless the model and data make that calculation equivalent. This is why a plot of arm-level means alone cannot fully diagnose the transition.

Numerical and Graphical Checks

A practical review combines calculations with plots. Start by separating participants by their last observed visit. Within each dropout pattern, calculate the mean last observed outcome, mean conditional prediction for the first missing visit, and their difference. Also calculate individual expected transitions and inspect their spread. This prevents an overall average from concealing patterns with different dropout times or histories.

For plots, mark each participant’s dropout visit and distinguish observed outcomes from imputed outcomes. A spaghetti plot can reveal whether a few extreme participant-level transitions dominate the group average. A pattern-specific mean trajectory can show the average at the final observed visit and the first missing visit, with separate symbols or line styles for observed and imputed values. Displaying the model-based conditional means separately from one set of stochastic imputed values helps distinguish a mean-model feature from random draw variation.

A transition at the boundary may look abrupt and still be consistent with J2R. Conversely, a smooth plotted line does not demonstrate that the J2R model was applied correctly: plotting choices can connect points across dropout, and averages can look smooth while participant-level predictions are misaligned. Checks should therefore include the numerical values and the data-processing rules, not just visual appearance.

Common Pitfalls, Alternatives, and Limits

The main alternatives answer different questions. A treatment-arm missing-at-random model continues to use the treatment-arm conditional distribution after dropout, whereas J2R uses the reference-arm conditional distribution after dropout. Copy-reference approaches make a different assumption about what treatment history is carried forward. These approaches can therefore imply different transition patterns. The transition check is not a method for choosing among them: it shows the implications of the chosen model and can reveal whether those implications match its intended specification.

• Wrong reasoning: J2R means the first missing mean must equal the reference-arm marginal mean. The correction is to use the reference conditional mean given the history specified by the model. Conditional and marginal means need not coincide.
• Wrong reasoning: the trajectory must be continuous at dropout. The correction is that J2R changes the post-dropout conditional model. A nonzero expected transition can follow from that change and the participant’s last observed outcome.
• Wrong reasoning: one realized imputation must equal the conditional mean. The correction is that a draw includes random outcome variation, and possibly parameter uncertainty. Compare expected predictions with suitable averages across draws, not an individual draw with an exact mean requirement.
• Wrong reasoning: an overall mean transition is enough. The correction is to stratify by dropout visit and inspect participant-level transitions. Opposing or heterogeneous changes can cancel in an overall average.

These checks are diagnostics, not tests that establish the missing-data assumption. The post-dropout outcomes are unobserved, so their distribution cannot be confirmed from the observed data alone. A plausible transition plot can coexist with a misspecified reference model, and an unusual-looking transition may reflect genuine model implications rather than a coding error. The check is most informative when paired with careful review of visit indexing, conditioning variables, reference-arm fit, and the handling of parameter and outcome draws.