Worked Single-Visit Copy-Reference Imputation

Copy-Reference for One Missing Visit

Suppose a participant in a treatment arm has an observed outcome at one visit and a missing outcome at the next. A single-visit copy-reference rule predicts the missing outcome using the reference arm’s conditional distribution, evaluated at that participant’s observed outcome. The observed value still matters: it is the history on which the prediction is conditioned. What is copied is the reference arm’s conditional relationship and variability, rather than the treatment arm’s post-dropout relationship.

This is a distributional assumption about outcomes after dropout, not a conclusion forced by the observed data. The copy-reference model says what distribution to use for the missing value; it does not establish that the unobserved outcome actually follows that distribution. That distinction is important when interpreting an imputation or assessing how sensitive a treatment comparison is to the chosen assumption.

For the calculations below, use the reference-arm model \( (Y_1,Y_2) \sim N\big((\mu_1,\mu_2)^\mathsf{T},\Sigma\big) \), where \( \Sigma=\begin{pmatrix}\sigma_{11}&\sigma_{12}\\\sigma_{12}&\sigma_{22}\end{pmatrix} \). Here \(\sigma_{11}\) and \(\sigma_{22}\) are variances, while \(\sigma_{12}\) is the covariance. The model assumes a jointly normal pair and a positive-definite covariance matrix, so \(\sigma_{11}>0\) and \(\sigma_{22}-\sigma_{12}^2/\sigma_{11}>0\). In applications, parameters are often estimated from reference-arm data; calculations that substitute estimates treat those estimates as fixed unless parameter uncertainty is handled separately.

Why the Conditional Variance Has This Form

The conditional mean adjusts the reference-arm mean at visit 2 according to how far the observed visit-1 value is from the reference-arm visit-1 mean. The adjustment per unit of that difference is the reference regression coefficient \(\sigma_{12}/\sigma_{11}\). The conditional variance is the part of the visit-2 variance not accounted for by the linear association with visit 1.

A short derivation makes clear why the variance must be reduced by a nonnegative amount. Define the residual \(E=Y_2-\mu_2-\beta(Y_1-\mu_1)\), with \(\beta=\sigma_{12}/\sigma_{11}\). Its covariance with \(Y_1\) is zero:

Because \(E\) and \(Y_1\) are jointly normal, zero covariance implies independence. Thus, once \(Y_1=y_1\) is given, the remaining uncertainty in \(Y_2\) is the variance of \(E\). Expanding that variance gives

The conditional mean follows from \(Y_2=\mu_2+\beta(Y_1-\mu_1)+E\) and \(E\)’s mean of zero. Positive definiteness ensures the resulting conditional variance is positive. If it is negative, the covariance inputs are inconsistent or a calculation has gone wrong.

Worked Calculation With a Conditional Interval

The next example emphasizes checking the conditional variance and interpreting a model-based predictive interval. As in the previous example, the numbers are hypothetical and refer to outcome scores measured in points.

How the Observed History Enters

The conditional mean changes with the observed value \(y_1\), while the conditional variance in this simple bivariate normal model does not. The following hypothetical calculation illustrates both features and shows why a participant’s observed score is not discarded when applying the copy-reference rule.

Assumptions, Diagnostics, and Alternatives

A valid calculation requires more than inserting numbers into a formula. The visits must be defined consistently across arms, the reference group must be appropriate for the trial question, and the reference-arm parameters must describe the relevant outcome scale and population. If the analysis conditions on baseline covariates or randomization strata, the reference distribution should be formed consistently within or conditional on those factors. The observed history should also be reasonably represented in the reference data; predicting far beyond its observed range relies heavily on the assumed model.

Check the estimated covariance matrix for symmetry and positive definiteness, confirm that the conditional variance is positive, and verify that variances and squared units are not confused. Examine whether the bivariate normal model is a plausible summary of the reference-arm outcomes, including the association between visits and the spread of outcomes. Sparse reference data, extreme histories, outliers, or marked departures from the assumed distribution can make the conditional prediction unstable. Diagnostics can reveal poor fit or weak support, but they cannot verify the unobserved post-dropout assumption itself.

Copy-reference differs from the main alternatives in the distribution it specifies. A treatment-arm extrapolation uses the treatment arm’s outcome model to predict after dropout and is often associated with a missing-at-random assumption, given the modeled history. Jump-to-reference (J2R) imposes a different post-dropout mean restriction. Copy-reference instead borrows the reference arm’s conditional regression and conditional variance for the missing visit. These are distinct assumptions; neither the numerical formula nor the observed data alone selects the scientifically appropriate one.

In a single-visit calculation, a conditional mean can be useful for summarizing the prediction, but it is not a substitute for a random imputation when individual outcome variability is required. Conversely, drawing a random value does not resolve uncertainty about the reference-based assumption. The calculation also does not by itself account for estimation uncertainty in the reference parameters. A plug-in distribution treats them as fixed; methods for propagating parameter uncertainty and combining multiple imputations require additional steps.