Copy-Reference Imputation Across Dropout Patterns

Why dropout time changes a copy-reference prediction

In a longitudinal trial, two participants can both discontinue assigned treatment but have very different amounts of observed outcome history. One may leave after the first follow-up visit; another may leave after the third. A copy-reference imputation must use each participant’s available history to predict what follows. The reference arm supplies the conditional distribution for those future outcomes, while the participant’s observed treatment-arm outcomes remain part of the record.

This distinction matters because conditioning on a longer history can shift a predicted outcome and change its uncertainty. Copy-reference imputation is therefore not one reference-arm mean substituted for every missing value. It is a set of conditional predictive distributions, indexed by the dropout pattern and the history observed in that pattern.

Definition: Suppose $Y_1,\ldots,Y_T$ are repeated outcomes, and a participant in the treatment arm has observed outcomes through visit $k$ before permanent dropout. Under copy-reference (CR), the missing outcomes are imputed from the reference-arm conditional distribution given that participant’s observed history: $$ Y_{k+1:T}\mid Y_{1:k}=y_{1:k} \ \sim\ F_R(Y_{k+1:T}\mid Y_{1:k}=y_{1:k}). $$ Here $F_R$ denotes the reference-arm outcome distribution. The observed treatment-arm outcomes are retained; the reference distribution supplies the conditional law for the missing future outcomes.

A dropout pattern is identified here by the last visit $k$ with an observed outcome. Thus a pattern with $k=1$ uses $Y_1$ as its history, while a pattern with $k=2$ uses both $Y_1$ and $Y_2$. If no post-baseline outcome is observed, $k=0$ and the reference marginal distribution is used, subject to any baseline conditioning in the analysis model. The discussion below assumes monotone dropout after the last observed visit. For intermittent missing visits, the conditioning history must instead be defined from the available outcomes and the specified model for them.

Conditional distributions under a multivariate normal model

A common working model is that the repeated outcomes in the reference arm have a multivariate normal distribution with mean vector $\mu_R$ and covariance matrix $\Sigma_R$. Partition these quantities into the observed history $Y_{1:k}$ and the future outcomes $Y_{k+1:T}$. If $\Sigma_{oo}$ is the covariance matrix of the observed history, $\Sigma_{fo}$ the covariance between future and observed outcomes, and $\Sigma_{ff}$ the covariance matrix of future outcomes, then the conditional distribution is normal:

Y_{k+1:T}\mid Y_{1:k}=y_{1:k} \sim N\left( \mu_{f,R}+\Sigma_{fo}\Sigma_{oo}^{-1}(y_{1:k}-\mu_{o,R}), \ \Sigma_{ff}-\Sigma_{fo}\Sigma_{oo}^{-1}\Sigma_{of} \right).

The first argument gives the conditional mean vector; the second is the conditional covariance matrix. The observed history changes the mean through its deviation from the reference-arm mean history. It also determines how much of the future variation remains after conditioning. The covariance matrix must be invertible for this expression, and in a well-specified multivariate normal model the resulting conditional covariance is positive definite for non-degenerate future outcomes.

In practice, the reference-arm parameters are estimated rather than known. The formula describes the predictive distribution conditional on a particular set of parameters. Multiple imputation should also reflect parameter uncertainty, as well as the remaining outcome variability; imputing only the conditional mean treats predictions as known values and understates uncertainty. Parameter draws and inference after reference-based imputation are treated in the earlier material on parameter draws and Rubin-rule inference.

Worked Example: Predictions After One Versus Two Observed Visits

Consider three follow-up outcomes, measured in points, in a hypothetical trial. Suppose the reference-arm model is multivariate normal with

\mu_R= \begin{pmatrix}0\\1\\2\end{pmatrix}, \qquad \Sigma_R= \begin{pmatrix} 4&2&1\\ 2&4&1\\ 1&1&4 \end{pmatrix}.

Assume a treatment-arm participant has $Y_1=2$ and then drops out. For $k=1$, the reference-arm regression coefficients for $Y_2,Y_3$ on $Y_1$ are $(2/4,1/4)=(0.5,0.25)$. The conditional mean is therefore

\begin{pmatrix}1\\2\end{pmatrix} + \begin{pmatrix}0.5\\0.25\end{pmatrix}(2-0) = \begin{pmatrix}2\\2.5\end{pmatrix}.

The conditional covariance of $(Y_2,Y_3)$ is

\begin{pmatrix}4&1\\1&4\end{pmatrix} - \begin{pmatrix}2\\1\end{pmatrix} \frac{1}{4} \begin{pmatrix}2&1\end{pmatrix} = \begin{pmatrix}3&0.5\\0.5&3.75\end{pmatrix}.

Now suppose a different participant has observed $Y_1=2,Y_2=3$ and drops out after visit 2. The covariance matrix for the observed history is $\begin{pmatrix}4&2\\2&4\end{pmatrix}$, whose inverse is $\frac{1}{12}\begin{pmatrix}4&-2\\-2&4\end{pmatrix}$. The regression coefficients for $Y_3$ on this history are

\begin{pmatrix}1&1\end{pmatrix} \frac{1}{12} \begin{pmatrix}4&-2\\-2&4\end{pmatrix} = \begin{pmatrix}1/6&1/6\end{pmatrix}.

Thus the conditional mean and variance of the missing $Y_3$ are

E_R(Y_3\mid Y_1=2,Y_2=3) =2+\frac{1}{6}(2-0)+\frac{1}{6}(3-1) =2.667, \qquad \operatorname{Var}_R(Y_3\mid Y_1,Y_2)=4-\frac{1}{3}=3.667.

The later dropout pattern uses the additional observed value $Y_2=3$, so its prediction for $Y_3$ differs from the prediction based only on $Y_1=2$, which was 2.5. These are conditional predictions under the stated reference model, not claims that either participant’s missing outcome will equal the conditional mean.

Why the pattern-specific predictions have this form

The normal conditional formula follows by subtracting the linear prediction from the future vector. Let $H=Y_{1:k}$, $F=Y_{k+1:T}$, and $B=\Sigma_{fo}\Sigma_{oo}^{-1}$. Define the residual $U=F-\mu_{f,R}-B(H-\mu_{o,R})$. Its covariance with $H$ is

\operatorname{Cov}(U,H) =\Sigma_{fo}-B\Sigma_{oo} =\Sigma_{fo}-\Sigma_{fo}\Sigma_{oo}^{-1}\Sigma_{oo} =0.

For a jointly multivariate normal vector, zero covariance implies independence. Consequently $U$ is independent of $H$, has mean zero, and has covariance

\operatorname{Var}(U) =\Sigma_{ff}-\Sigma_{fo}\Sigma_{oo}^{-1}\Sigma_{of}.

Conditioning on $H=y_{1:k}$ therefore changes the mean of $F$ by $B(y_{1:k}-\mu_{o,R})$, while leaving this residual covariance. This derivation also clarifies why the same conditional rule must be recalculated when $k$ changes: both the observed-history block and its covariance with future outcomes change.

Worked Example: Generate a Coherent Post-Dropout Draw

Use the reference model from the previous example for a participant with $Y_1=2$ who drops out after visit 1. A multiple-imputation draw must represent uncertainty, not just set the missing values to their conditional means. One way to draw jointly is to proceed sequentially using the conditional distributions implied by the same multivariate normal model.

First, $Y_2\mid Y_1=2\sim N(2,3)$. If a standard normal draw is $Z_2=0$, then the imputed value is $Y_2=2+\sqrt{3}(0)=2$. Given $Y_1=2,Y_2=2$, the conditional mean of $Y_3$ is

2+\frac{1}{6}(2-0)+\frac{1}{6}(2-1)=2.5,

and its conditional variance is $11/3$, as calculated from the two-visit history formula. If the next standard normal draw is $Z_3=-0.5$, then

Y_3=2.5+\sqrt{11/3}(-0.5) =2.5-0.9574 =1.543.

The draw $(Y_2,Y_3)=(2,1.543)$ is one possible imputed pair; it is not the conditional mean pair $(2,2.5)$. Sequential sampling is valid here because each step uses the conditional distribution from the same joint reference model. In a full analysis, parameter uncertainty would also be incorporated in the imputation procedure.

Applying the rules across patterns

For a treatment-arm participant with dropout after visit $k$, the observed history is kept as observed and only the future vector is drawn from the corresponding reference conditional distribution. Across a trial, this operation is repeated for every dropout pattern. In particular, predictions for the pattern dropping after visit 1 should not be reused for participants who have observed outcomes through visit 2.

Worked Example: Combining Participants From Several Patterns

Suppose a hypothetical treatment arm has four participants with three scheduled follow-up outcomes. One participant completes all visits and has observed $Y_3=4$. Two drop out after visit 1, with observed $Y_1=2$ and $Y_1=0$. A fourth has $Y_1=1,Y_2=2.5$ and drops out after visit 2. Use the reference model in the first example and calculate the conditional-mean completed $Y_3$ average.

For the first visit-1 dropout, the conditional mean is $2+0.25(2-0)=2.5$. For the second, it is $2+0.25(0-0)=2.0$. For the visit-2 dropout, it is

2+\frac{1}{6}(1-0)+\frac{1}{6}(2.5-1) =2+\frac{2.5}{6} =2.417.

Using these conditional means as replacements solely to calculate this illustrative conditional-mean summary gives

\bar{Y}_3^{\,\text{completed}} =\frac{4+2.5+2.0+2.417}{4} =2.729\text{ points}.

This arithmetic shows how different patterns contribute predictions based on their own histories. It is not a complete multiple-imputation analysis: repeated completed data sets should use predictive draws, and inferential uncertainty must reflect both within- and between-imputation variation. The computed 2.729 points is only the average after inserting conditional means for this small hypothetical set.

Assumptions, diagnostics, and alternatives

The calculations rely on a reference-arm model that adequately describes the conditional outcome distribution over the histories to which it is applied. The multivariate normal assumption supplies a particular form for the conditional distribution; if it is inappropriate, the formula may produce misleading predictions even when the covariance arithmetic is correct. The reference-arm covariance must be estimable and nonsingular for the histories used. With baseline covariates or stratification, the model and conditional predictions must respect those features consistently.

Copy-reference also makes a substantive assumption: after dropout, the future outcome distribution is taken to be that of the reference arm conditional on the observed treatment-arm history. It does not claim that the dropout mechanism itself is random or ignorable. Rather, it specifies a distribution for unobserved outcomes under a chosen post-dropout rule. Whether that rule is appropriate for the clinical question is a separate judgment.

Useful checks include examining the size and composition of each dropout pattern, verifying that the reference model has adequate data for the histories being conditioned on, and checking that conditional covariance matrices are positive and numerically stable. Compare observed histories in the treatment dropout patterns with histories represented in the reference arm. Predictions for histories far outside reference-arm support are extrapolations; a mathematically computable conditional mean does not make such extrapolations reliable. Also check that the predicted trajectories and variability are clinically plausible and that visit ordering and outcome units are consistent.

A frequent error is to use the reference-arm marginal mean at every missing visit, on the reasoning that copy-reference means “replace missing treatment outcomes with reference outcomes.” That discards observed history. Under CR, the reference distribution is conditional on the retained treatment history, so the prediction generally differs from the reference marginal mean. Another error is to use a longer history for one dropout pattern but accidentally apply it to a pattern that dropped out earlier; each pattern has only the outcomes actually observed before dropout.

Copy-reference differs from jump-to-reference (J2R), which specifies a transition in post-dropout behavior rather than simply using the reference-arm conditional distribution given the observed treatment history. It also differs from a missing-at-random continuation based on the treatment-arm outcome model. These approaches encode different assumptions about unobserved outcomes, so their results need not agree. CR’s strengths are that it preserves observed treatment outcomes and uses a clearly defined reference conditional law. Its limitations include dependence on the chosen reference model, possible extrapolation beyond observed reference histories, and sensitivity to the assumption that reference-arm conditional behavior is appropriate after treatment dropout.

Key takeaway: For each dropout pattern, retain the treatment-arm history actually observed and condition the reference-arm distribution on that history. More observed visits can update predictions and uncertainty, so the missing outcomes must be imputed pattern by pattern.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

Under copy-reference, what distribution is used for missing outcomes for a participant who drops out after visit 2?
In the worked normal model, what is the conditional mean of $Y_3$ for history $Y_1=0,Y_2=1$?
Why is substituting the reference-arm marginal mean for every missing outcome not generally copy-reference imputation?
What does a positive conditional variance represent in a predictive draw?

Solutions. (1) Use the reference-arm distribution of the future outcomes conditional on the participant’s observed treatment-arm history $Y_1,Y_2$. (2) The reference mean for $Y_3$ is 2, and both history deviations are zero, so $2+\frac{1}{6}(0-0)+\frac{1}{6}(1-1)=2$ points. (3) Copy-reference conditions on the observed history, which can shift the predictive mean and alter the future covariance; a marginal mean ignores that information. (4) It represents remaining outcome variation after conditioning on the observed history, so predictive draws vary around the conditional mean rather than all equaling it.