Why the Assumptions Need to Be Reported Explicitly
A label such as “reference-based imputation” does not by itself tell a reader what values were imputed. The result depends on which arm supplies the reference distribution, how the distribution changes after dropout, what observed information is conditioned on, and how uncertainty is propagated. A report that omits any of these choices may be impossible to reproduce, even if it names a familiar method such as jump to reference (J2R) or copy-reference.
The purpose here is not to select a preferred missing-data assumption. It is to state the selected assumption precisely enough that a reader can identify the distribution assigned to missing outcomes and the procedure used to draw them. The earlier material on J2R and copy-reference establishes the substantive distinction between these rules; the reporting task is to make the exact implementation visible.
What a Complete Specification Contains
Let \(Y_i=(Y_{i1},\ldots,Y_{iK})^\top\) denote the scheduled longitudinal outcomes for participant \(i\), and let \(A_i\) denote randomized arm. Let \(D_i=d\) mean that the last observed outcome is at visit \(d\), so visits \(d+1,\ldots,K\) are missing. Write \(Y_{i,\mathrm{obs}}\) for the observed outcomes and \(Y_{i,\mathrm{mis}}\) for the missing outcomes. A conditioning history \(H_i(d)\) can include baseline covariates and observed outcomes through visit \(d\); the report must say exactly which variables it includes.
A useful mathematical statement of the imputation assumption is the conditional distribution \(Y_{i,\mathrm{mis}}\mid H_i(d), A_i, D_i=d\). For a multivariate normal model, one can report its conditional mean vector and covariance matrix. If the model is not multivariate normal, the report should instead identify the distributional family and the corresponding conditional distribution used to generate imputations. In either case, “use the control data” is not a complete description.
- Reference arm: identify the arm by name and say whether it is the same reference for every randomized group and dropout pattern. In a multiarm trial, identify the mapping explicitly.
- Post-dropout mean: say whether the mean follows a specified reference trajectory, retains a treatment-related component, or follows another stated rule. Clarify the transition visit and how the rule depends on dropout time.
- Covariance: say which covariance model supplies the dependence among missing outcomes and between missing and observed outcomes. State whether the conditional covariance is calculated from that model or replaced by another specified matrix.
- Conditioning history: list the observed outcomes, baseline variables, and other predictors used. Clarify whether treatment history is retained in the conditioning set after dropout.
- Implementation: identify model form, estimation method, handling of parameter uncertainty, number of imputations, random-number or convergence practices when relevant, and the analysis performed on each completed data set.
Conditional Mean and Covariance: The Reporting Target
Suppose the reference-arm outcomes follow a multivariate normal distribution with mean vector \(\mu_R\) and covariance matrix \(\Sigma_R\). Partition the outcomes into the observed components \(Y_{\mathrm{obs}}\) and missing components \(Y_{\mathrm{mis}}\), and partition the mean and covariance accordingly. If \(\Sigma_{\mathrm{obs,obs}}\) is nonsingular, the conditional distribution is normal:
This formula illustrates why a report must address both means and covariance. Even when the reference model supplies the marginal covariance matrix, the distribution used for missing values conditional on observed history has a conditional covariance: the Schur complement in the second argument above. Stating only that the “reference covariance was used” can leave unclear whether the implementation used the full reference covariance, the conditional covariance derived from it, or a separately imposed covariance rule.
Worked Example: Selecting and Naming a Reference Arm
Consider a hypothetical three-arm trial with placebo, Drug A, and Drug B. The prespecified rule is that missing outcomes in Drug A are imputed using the placebo reference distribution, while missing outcomes in Drug B use the Drug A distribution. Dropout can occur after either visit 1 or visit 2.
A complete arm mapping is: for Drug A, reference arm = placebo; for Drug B, reference arm = Drug A. The placebo group itself is not assigned a different treatment reference under this specification. Saying “the control arm is the reference” would be ambiguous here because Drug A serves as the reference for Drug B, while placebo serves as the reference for Drug A.
The report must also say whether each mapping applies at both dropout times. For example: “For each dropout pattern, outcomes after dropout in Drug A were generated from the placebo-based conditional distribution, and outcomes after dropout in Drug B from the Drug A-based conditional distribution.” This identifies the mapping, but is not yet a full specification: the conditional history, mean rule, covariance rule, and model-fitting procedure remain to be stated.
A Short Derivation for a Conditional Distribution
The conditional covariance in the reporting formula is not an arbitrary adjustment. For jointly normal observed and missing outcomes, subtract the linear prediction of the missing outcomes from the observed outcomes. Define \(B=\Sigma_{\mathrm{mis,obs}}\Sigma_{\mathrm{obs,obs}}^{-1}\). Then \(Y_{\mathrm{mis}}-\mu_{\mathrm{mis}}-B(Y_{\mathrm{obs}}-\mu_{\mathrm{obs}})\) has zero covariance with \(Y_{\mathrm{obs}}\), because \(\Sigma_{\mathrm{mis,obs}}-B\Sigma_{\mathrm{obs,obs}}=0\). Joint normality makes zero covariance imply independence. Therefore, after observing \(Y_{\mathrm{obs}}\), the residual distribution is unchanged, and its covariance is:
Adding the observed-history linear prediction gives the conditional mean in the preceding formula. Thus, for a normal reference model, documenting the marginal model and the conditioning history makes the conditional rule derivable; reporting the conditional rule directly makes the implementation even clearer. The derivation relies on joint normality and a valid inverse for the observed-outcome covariance block. If these conditions do not hold, the stated formula cannot simply be assumed to apply.
Worked Example: Reporting a Conditional Mean and Covariance
Suppose a hypothetical placebo reference model for outcomes at three visits has mean \((10,12,13)^\top\) and covariance matrix \(\begin{pmatrix}4&2&1\\2&9&3\\1&3&9\end{pmatrix}\). A Drug A participant has an observed visit-1 outcome of 12 and drops out before visit 2. Under a rule that uses the reference-arm conditional distribution given the observed visit-1 outcome, visits 2 and 3 are missing.
The observed mean is 10, the observed variance is 4, and the covariance of visits 2 and 3 with visit 1 is \((2,1)^\top\). The conditional mean is therefore:
For the conditional covariance, the missing-outcome covariance block is \(\begin{pmatrix}9&3\\3&9\end{pmatrix}\). The covariance adjustment is \(\begin{pmatrix}2\\1\end{pmatrix}\frac{1}{4}\begin{pmatrix}2&1\end{pmatrix} =\begin{pmatrix}1&0.5\\0.5&0.25\end{pmatrix}\). Subtracting gives:
A concise report could state that the post-dropout vector \((Y_2,Y_3)^\top\) was drawn from a bivariate normal distribution with conditional mean \((13,13.5)^\top\) and conditional covariance as displayed, given the observed visit-1 value of 12. The numbers are illustrative hypothetical values. This statement makes the conditioning and the two distributional components explicit.
Describe the Imputation Procedure, Not Just the Assumption
An assumption defines a distribution; an imputation procedure explains how that distribution produced the completed data sets. At a minimum, a reproducible description identifies how reference-model parameters were estimated, how missing values were drawn conditional on each participant’s observed history, and how the completed data sets were analyzed. In multiple imputation, the report should also say how many completed data sets were generated and how the estimates and uncertainty were combined. If parameter uncertainty was included, state how: for example, whether model parameters were drawn before generating outcomes or treated as fixed at estimates.
Specify the data and randomized arm used to estimate each reference distribution, the outcome model, and the estimation method.
Use the participant’s dropout pattern and the declared conditioning history to obtain the post-dropout mean and covariance under the selected rule.
State whether model-parameter uncertainty is incorporated and how draws are generated from the conditional distribution.
Apply the specified analysis to each completed data set and report the method used to combine estimates and uncertainty.
Worked Example: Turning a Method Label Into a Reproducible Report
Suppose a hypothetical two-arm trial measures a continuous outcome at baseline and at weeks 4, 8, and 12. The reference is placebo. For participants randomized to the active treatment who discontinue before a follow-up visit, the chosen rule uses the placebo-based post-dropout conditional distribution. The conditioning history includes baseline outcome and all observed post-baseline outcomes through the last attended visit.
A report that says only “J2R multiple imputation was performed” does not tell the reader how the placebo distribution was fitted, whether the dropout visit is treated as observed treatment history, how covariance is handled, or how parameter uncertainty is treated. A fuller report would specify the outcome model and covariates, state that post-dropout outcomes were drawn jointly conditional on the listed history, and state that the conditional covariance was derived from the fitted placebo multivariate model. It would then state the number of imputations, whether parameter draws were used, the analysis applied to each completed data set, and the pooling method.
For instance, if 100 imputed data sets were generated, parameter draws were taken from the fitted model’s estimated sampling distribution, and the same prespecified treatment comparison was fitted to each completed data set, those are reportable implementation choices—not details that can be inferred merely from the term “J2R.” The specific choices must match the actual analysis; this example does not prescribe them.
Diagnostics, Alternatives, and Limits
A report should provide enough information to assess whether the procedure behaved as intended. Useful checks include the number of participants and missing outcomes by arm and dropout visit; observed and imputed trajectory summaries; the range and distribution of imputed values; and checks for implausible covariance or unstable reference-model estimates. If a reference arm is small, its estimated means and covariance may be unstable. If a required covariance block is singular or nearly singular, the conditional calculation may fail or be numerically unstable; report the remedy rather than silently changing the model. Large differences between observed and imputed values can be consistent with the stated assumption, but should be examined for coding mistakes, mismatched visit definitions, or an incorrectly applied transition rule.
Reference-based imputation is not the only possible approach. A MAR imputation model uses a missing-at-random conditional distribution given the specified observed information; a reference-based rule instead constrains the post-dropout distribution by a selected reference-arm rule. J2R and copy-reference are distinct specifications, and the report should not treat their names as interchangeable. A complete report may compare such assumptions, but it must give the actual rule used for each analysis rather than blending their descriptions.
No amount of reporting makes an unobserved post-dropout distribution empirically identified from the observed data alone. Reference-based assumptions supply a modeling choice for that distribution. Findings therefore depend on the choice of reference, mean and covariance rules, model, and conditioning history. Clear reporting supports scrutiny and reproduction; it does not establish that the chosen assumptions are true.
Common Reporting Pitfalls
- Giving only a method name. “Copy-reference was used” is incomplete unless the reference arm, conditioning information, post-dropout distribution, and covariance treatment are clear.
- Naming an arm without mapping groups to it. In a multiarm trial, list the reference for each randomized group and dropout pattern.
- Reporting means but omitting dependence. Imputed longitudinal outcomes are jointly distributed; state the covariance rule or conditional covariance used.
- Leaving the observed history implicit. State which outcomes and covariates are conditioned on, including how the history at the dropout visit enters.
- Confusing assumptions with implementation. The conditional distribution is the assumption; parameter estimation, random draws, number of imputations, and pooling describe how it was implemented.
Check Your Understanding
Answer each question, then compare with the solutions below.
- Why is “the control arm was used as reference” potentially inadequate in a three-arm trial?
- Under a multivariate normal reference model, what information determines the conditional covariance of missing outcomes given observed outcomes?
- A report states the reference arm and mean rule but says nothing about covariance. What important information is missing?
- Name two implementation details, beyond the distributional assumption, that support reproducibility.
- Does a fully specified reference-based imputation model prove that its post-dropout assumption is correct? Explain.
Solutions. (1) More than one arm may serve as a reference, or different randomized groups may map to different references; the mapping must be listed. (2) It is determined by the missing-outcome and observed-outcome covariance blocks through the Schur-complement formula, under the stated joint normal model. (3) The report has not identified the dependence among imputed outcomes or whether it uses a conditional covariance derived from the reference model or another covariance rule. (4) Examples include the parameter-estimation method, whether parameter uncertainty is incorporated, the number of imputations, and the analysis and pooling method. (5) No. The specification makes the assumption explicit and assessable, but post-dropout outcomes are unobserved, so reporting alone cannot verify the assumption.