Sequential Copy-Reference Imputation Across Follow-Up Visits

Why Generate the Outcomes Sequentially?

When a participant discontinues study treatment, several scheduled outcomes may remain unobserved. Imputing each missing visit separately can give plausible marginal distributions but fail to represent the fact that outcomes from the same participant are related. A sequential procedure instead generates one outcome conditional on the participant’s available history, then generates the next conditional on that history and the value just drawn.

This tutorial develops that procedure for copy-reference (CR) imputation. After dropout, the conditional distribution is based on the reference arm. The CR assumption retains the treatment-versus-reference mean difference at dropout as a location shift in the post-dropout trajectory. Dependence between future visits comes from the reference arm’s conditional covariance structure.

The formulas below assume a specified dropout visit and a multivariate normal model. They describe how to generate an individual trajectory conditional on the observed history; they do not address uncertainty in the fitted model parameters.

Model and Notation

Here \(\mu_{A,s}\) and \(\mu_{R,s}\) are the treatment- and reference-arm means at dropout visit \(s\). The shift \(\delta_s\) is in the outcome’s units. A positive value shifts each reference-based post-dropout conditional mean upward by that amount; a negative value shifts it downward. This formulation copies the reference conditional distribution’s dependence and variability while retaining a constant mean offset anchored at dropout.

For a joint multivariate normal reference model, partition the reference mean and covariance into observed-history and future blocks. The conditional reference moments are

The matrix \(\Sigma_{R,HH}\) must be invertible for these expressions as written. In practice, the reference model must also provide a defensible estimate of the means and covariance blocks, and the covariance matrix must be valid. These are substantive modeling requirements: the equations do not make an unsuitable reference arm or an unsupported normal approximation appropriate.

From a Joint Conditional Distribution to Sequential Draws

The conditional distribution above concerns the entire missing vector. A sequential implementation factors it into one-visit conditional distributions. First compute the CR conditional mean vector \(m^*_{F\mid H}=m^R_{F\mid H}+\delta_s\mathbf{1}\) and covariance \(V^*_{F\mid H}=V^R_{F\mid H}\). For each successive visit, condition on the observed history and all previously generated future outcomes.

For the first future visit, \(P\) is empty, so the draw has mean \(m^*_k\) and variance \(V^*_{kk}\). At later visits, the generated values update the conditional mean. The conditional variance also changes because uncertainty about a later outcome is reduced by knowing earlier outcomes. This is the key to retaining dependence: the procedure uses the joint reference covariance, not a separate independent draw at every visit.

Why the Sequential Rule Gives the Intended Joint Distribution

For a multivariate normal vector, conditioning one block on another gives a normal distribution with the conditional mean and covariance shown above. Apply that result first to \(Z_k\) given \(Z_P\), then factor the joint conditional density as a product of successive conditional densities:

Each factor is precisely the one-visit normal distribution in the sequential rule. Therefore, drawing in visit order from those conditionals produces the specified joint normal distribution, rather than only matching its marginal means and variances. This also explains why the earlier generated draws must be retained and used in subsequent conditional means.

Worked Sequential Calculation

What the Anchor Means—and What It Does Not

The CR shift distinguishes this construction from jump-to-reference (J2R). Under the formulation here, the reference conditional mean is shifted by the treatment-versus-reference difference at dropout. J2R instead specifies a transition to the reference mean behavior without retaining that same constant offset. The two assumptions can therefore yield different post-dropout means even when they use a reference-based covariance structure.

The shift is an assumption about the post-dropout location, not a claim that an individual’s treatment effect is known. In applications, \(\delta_s\) is estimated from arm-level model quantities. It is also not a command to add \(\delta_s\) to a realized outcome at dropout and copy that exact value forward. Future outcomes remain random, and their conditional means may change as the observed history and previously drawn values enter the reference conditional model.

The formulas require a coherent reference distribution for the relevant visits and a suitable conditional model for the participant’s observed history. In particular, the normal conditional formulas assume a joint normal model (or use of that model as an approximation), a nonsingular history covariance block, and a positive-definite conditional covariance. They also presume that the reference-arm relationship between histories and future outcomes is a reasonable basis for prediction in the treatment-arm participants after dropout.

Diagnostics and Failure Modes

A numerical algorithm can run successfully while the assumed trajectory is inappropriate. Useful checks focus on both the distribution and the resulting paths:

• Check the covariance structure. Verify symmetry, positive definiteness, and the expected conditional variances. Negative conditional variances indicate an invalid covariance estimate or a calculation error.
• Check the anchoring. Confirm that the same \(\delta_s\) is added to every future conditional reference mean under this specified CR formulation, and that the reference covariance is not inadvertently shifted or replaced by independent variances.
• Inspect trajectories in outcome units. Compare generated paths with plausible ranges and with observed reference trajectories. A constant location offset can produce implausible late-visit values if outcomes have strong ceilings, floors, or nonlinear time trends.
• Check visit ordering and conditioning history. Each draw after the first must condition on the observed pre-dropout history and all previously generated post-dropout outcomes. Omitting a prior draw changes the conditional distribution.
• Look for weak reference information. Sparse data at later visits or an unstable reference covariance can make conditional predictions unreliable, even when the formulas are evaluated correctly.

A particularly common mistake is to generate each missing outcome independently from its CR marginal distribution. That preserves individual visit means and variances but removes the specified cross-visit covariance. The correction is to draw from the sequential conditional distributions, using each generated value in the next step. Another mistake is to apply the dropout-time shift to the covariance; in this formulation, \(\delta_s\) changes location only.

Scope and Practical Interpretation

Sequential CR imputation is useful when the analysis requires complete post-dropout trajectories and the scientific assumption is that the reference arm’s conditional evolution applies after discontinuation, while the treatment-versus-reference mean difference at dropout is retained as a constant offset. It makes the dependence structure explicit and ensures that imputed repeated outcomes are not treated as unrelated.

It does not establish that this post-dropout behavior is true, nor does it remove the need to assess sensitivity to alternative assumptions. Results can depend on the reference arm, the mean and covariance models, the dropout visit, and the chosen CR location shift. A model may be algebraically valid yet extrapolate poorly. Interpret the imputation as a consequence of the stated assumption, not as observed evidence about the missing outcomes.