Interpreting Copied Reference Variability

What Is Copied in Copy-Reference Imputation?

A post-dropout mean trajectory does not fully describe an imputation model. Two models can give the same predicted mean at every visit yet generate very different individual trajectories and population distributions because their variances and correlations differ. Copy-reference imputation specifies both aspects: after dropout, outcomes are modeled using the reference arm’s conditional distribution, given the participant’s observed history.

The word conditional matters. A participant’s future outcomes are not generally drawn from the reference arm’s unconditional distribution. They are drawn from a distribution conditioned on the outcomes already observed for that participant. The resulting conditional mean and covariance jointly determine the imputed trajectory.

Definition: In copy-reference imputation, an individual assigned to the treatment arm who drops out has their post-dropout outcomes imputed from the reference arm’s predictive distribution conditional on that individual’s observed outcome history. Under a multivariate normal model, this distribution has a conditional mean and a conditional covariance. Copying reference variability means using the reference model’s conditional covariance for those post-dropout outcomes, as well as its conditional mean.

This is a pattern-mixture assumption about outcomes after dropout, not a claim that an individual’s treatment was actually changed or that the observed treatment effect caused their dropout. The rule describes the assumed distribution for unobserved outcomes. The earlier material on copy-reference imputation establishes the basic imputation rule; here the emphasis is on what its variability means.

Conditional Mean and Conditional Variability

Suppose a participant has an observed history vector \(H\) and a vector of post-dropout outcomes \(F\). In the reference arm, assume the joint distribution of these outcomes is multivariate normal:

\begin{pmatrix}H\\F\end{pmatrix} \sim N\left[ \begin{pmatrix}\mu_H\\\mu_F\end{pmatrix}, \begin{pmatrix} \Sigma_{HH} & \Sigma_{HF}\\ \Sigma_{FH} & \Sigma_{FF} \end{pmatrix} \right].

Here \(\Sigma_{HH}\) is the covariance matrix of the observed-history variables, \(\Sigma_{FF}\) is the covariance matrix of future outcomes, and \(\Sigma_{FH}=\Sigma_{HF}^{\mathsf T}\). Assume \(\Sigma_{HH}\) is invertible. The conditional reference distribution for \(F\), given observed history \(H=h\), is

F\mid H=h \sim N\left( \mu_F+\Sigma_{FH}\Sigma_{HH}^{-1}(h-\mu_H), \ \Sigma_{FF}-\Sigma_{FH}\Sigma_{HH}^{-1}\Sigma_{HF} \right).

The first argument is the conditional mean: it adjusts the reference-arm mean trajectory according to how the participant’s observed history differs from the reference-arm mean history. The second argument is the conditional covariance. It determines the spread of possible future trajectories around that conditional mean and how their deviations move together. It is not generally equal to the unconditional future covariance \(\Sigma_{FF}\), because conditioning on observed history provides information about future outcomes.

Copy-reference imputation uses both quantities. The conditional mean determines where the predictive distribution is centered; the conditional covariance determines the size and dependence of the individual residual draws. Thus an imputed value is not necessarily equal to the conditional mean, and an imputed trajectory is not simply a deterministic copy of the reference mean trajectory.

Why the Conditional Covariance Has This Form

The covariance formula can be justified by removing from \(F\) the part linearly predicted by \(H\). Define the residual vector

U=F-\mu_F-\Sigma_{FH}\Sigma_{HH}^{-1}(H-\mu_H).

Its covariance with \(H\) is zero:

\operatorname{Cov}(U,H) =\Sigma_{FH}-\Sigma_{FH}\Sigma_{HH}^{-1}\Sigma_{HH} =0.

Because \(H\) and \(F\) are jointly normal, \(U\) and \(H\) are jointly normal. For jointly normal vectors, zero covariance implies independence. Therefore, after \(H=h\) is observed, the residual \(U\) still has its unconditional covariance, and the conditional mean of \(F\) is the value obtained by setting \(H=h\) in the linear prediction. Expanding the covariance of \(U\) gives

\begin{aligned} \operatorname{Var}(U) &=\Sigma_{FF} -\Sigma_{FH}\Sigma_{HH}^{-1}\Sigma_{HF} -\Sigma_{FH}\Sigma_{HH}^{-1}\Sigma_{HF} +\Sigma_{FH}\Sigma_{HH}^{-1}\Sigma_{HH} \Sigma_{HH}^{-1}\Sigma_{HF}\\ &=\Sigma_{FF}-\Sigma_{FH}\Sigma_{HH}^{-1}\Sigma_{HF}. \end{aligned}

This argument also shows why the conditional covariance is not an optional embellishment to the mean rule. It is the remaining uncertainty after the information in the observed history has been used.

Worked Example: One Participant’s Correlated Post-Dropout Draw

Suppose the reference-based conditional mean for two post-dropout visits is 12 and 14 outcome units, and the conditional covariance is

\Sigma_{F\mid H}= \begin{pmatrix} 9 & 4.5\\ 4.5 & 16 \end{pmatrix}.

The conditional standard deviations are 3 and 4 outcome units. The conditional correlation is \(4.5/(3\cdot4)=0.375\). To generate a predictive draw, let \(Z_1\) and \(Z_2\) be independent standard normal variables and use the factorization

\begin{pmatrix}3&0\\1.5&\sqrt{13.75}\end{pmatrix} \begin{pmatrix}3&1.5\\0&\sqrt{13.75}\end{pmatrix} = \begin{pmatrix}9&4.5\\4.5&16\end{pmatrix}.

For a hypothetical draw \(Z_1=-2\) and \(Z_2=0.75\), the residual vector is

\begin{pmatrix} 3Z_1\\ 1.5Z_1+\sqrt{13.75}Z_2 \end{pmatrix} = \begin{pmatrix} 3(-2)\\ 1.5(-2)+\sqrt{13.75}(0.75) \end{pmatrix} \approx \begin{pmatrix}-6\\-0.219\end{pmatrix}.

In the second component, \(\sqrt{13.75}\approx3.7081\), so \(1.5(-2)+3.7081(0.75)=-3+2.7811\approx-0.2189\). Adding these residuals to the conditional means gives the imputed trajectory \((12-6,\ 14-0.2189)\approx(6.00,\ 13.78)\) outcome units. The draw is one possible realization, not a new estimate of the conditional mean. Its deviations at the two visits are correlated because the imputation uses the reference conditional covariance.

Consequences for a Population of Participants

At the individual level, copying the conditional covariance determines how widely imputed trajectories vary around their conditional means and which combinations of outcomes are more or less likely. At the population level, the consequences depend on more than that covariance alone. Participants may have different observed histories, different dropout patterns, or both. Their conditional means and covariances can therefore differ.

A useful population calculation separates variation within a dropout group from variation between groups. If \(M\) denotes dropout pattern and \(Y\) is an outcome, then

\operatorname{Var}(Y) =E\{\operatorname{Var}(Y\mid M)\} +\operatorname{Var}\{E(Y\mid M)\}.

The first term averages the within-pattern variances; the second reflects differences between pattern-specific means. Consequently, copying a reference covariance does not mean that the variance of the entire treatment-arm outcome distribution must equal the reference-arm variance. The pooled distribution also depends on the prevalence of each pattern and on the means and variances for other patterns.

Worked Example: Pooled Variability Across Dropout Patterns

Consider a hypothetical trial population at one follow-up visit. Suppose 60% of treatment-arm participants complete follow-up, with outcome mean 15 and variance 16 outcome-units squared. The remaining 40% drop out earlier; copy-reference imputation gives this pattern a conditional mean of 12 and variance 9 outcome-units squared. Assume these pattern proportions and distributions describe the population for this calculation.

The pooled mean is

$$ E(Y)=0.60(15)+0.40(12)=9+4.8=13.8 $$

outcome units. To calculate the pooled variance, first average the within-pattern variances:

$$ 0.60(16)+0.40(9)=9.6+3.6=13.2 $$

outcome-units squared. The between-pattern contribution is

$$ 0.60(15-13.8)^2+0.40(12-13.8)^2 =0.60(1.44)+0.40(3.24) =0.864+1.296=2.16. $$

The pooled variance is therefore \(13.2+2.16=15.36\) outcome-units squared. It is not just the dropout-group variance of 9, nor is it the simple unweighted average of 16 and 9. The imputed reference variability contributes within the dropout pattern, while the difference between pattern-specific means contributes additional population variability.

Correlations Shape Summaries of the Trajectory

Copied variability includes covariances, not just separate visit-by-visit standard deviations. A correlation between future outcomes affects summaries such as an average across visits or a change from one visit to another. Ignoring that dependence can misstate the variability of these summaries even when every visit-specific variance is correct.

Worked Example: Variability of an Average and a Change

Use the same two-visit conditional covariance matrix from the first example. For the average of the two future outcomes, \(A=(F_1+F_2)/2\), the conditional variance is

\operatorname{Var}(A\mid H) =\frac{1}{4}\{9+16+2(4.5)\} =\frac{34}{4}=8.5

outcome-units squared. If the covariance were incorrectly ignored, the calculation would instead give \((9+16)/4=6.25\), understating the variance by 2.25 outcome-units squared. For the change \(D=F_2-F_1\),

\operatorname{Var}(D\mid H) =16+9-2(4.5)=16

outcome-units squared. Assuming independence would give 25 outcome-units squared, an overstatement in this example. The direction of the error depends on the summary: positive covariance increases the variability of a sum but reduces the variability of a difference relative to an independence calculation.

Assumptions, Comparisons, and Limitations

Interpreting copied variability requires keeping the model assumptions explicit. The formulas above rely on a multivariate normal reference model with a valid, estimable covariance matrix and an appropriate model for the observed history. Applying copy-reference imputation also assumes that, after dropout, the reference arm’s conditional distribution is a suitable distribution for the missing outcomes of the treatment-arm participants with that history. This is a substantive assumption about unobserved outcomes; the observed data alone do not verify it.

Copy-reference imputation should not be confused with imputing every missing value at the conditional mean. A mean-only replacement removes predictive residual variation and can create an artificially concentrated completed-data distribution. Nor does the word “copy” mean that each missing value is copied from an actual reference participant. The method draws from a modeled conditional distribution.

The main alternatives differ in what they assume about the missing outcomes. Under a missing-at-random analysis, the prediction is based on the specified treatment-arm model conditional on observed information. Copy-reference instead uses the reference-arm conditional distribution after dropout. Jump-to-reference (J2R) is defined by a post-dropout mean rule; its covariance specification should also be stated, since the mean rule alone does not determine the variability of draws. These approaches answer different assumption-based questions, not questions that the observed data can settle by themselves.

Several limitations follow. A poorly estimated reference covariance can produce unreliable predictive draws, particularly when the reference arm is small or the model has many outcomes. A normal model can also give a poor description of skewed, bounded, or otherwise non-normal outcomes. Finally, conditional reference variability can be implausible for a particular treatment-arm population even if the arithmetic is correct. Results should therefore be interpreted as consequences of the chosen model and assumptions, not as direct measurements of what happened after dropout.

Key takeaway: Copy-reference imputation imports a conditional reference distribution, not only a mean. Its conditional covariance governs individual residual draws and their dependence; population variability additionally reflects the mix of dropout patterns and their different means.

Common Mistakes

Treating the conditional mean as the imputed value. The conditional mean is the center of the predictive distribution. An individual draw generally differs from it because a residual is added.
Copying only marginal variances. A trajectory also depends on covariances. Omitting them changes the distribution of averages, changes, and other multi-visit summaries.
Calling the pooled variance the reference variance. The pooled treatment-arm variance includes the variability within patterns and the differences between their means; it need not match any one arm’s variance.
Interpreting the assumption as a factual claim. The imputed distribution is an assumption about unobserved outcomes, not evidence that each participant would actually have followed that trajectory.

Check Your Understanding

Answer each question, then compare with the solutions below.

If the conditional mean is 20 units and the conditional standard deviation is 5 units, is every imputed value equal to 20? What does the standard deviation describe?
Suppose two future outcomes have variances 4 and 9 and covariance 3. Calculate the variance of their average.
In a population with two dropout patterns, why can the pooled variance exceed both the within-pattern contribution averaged across patterns and the variance in either one pattern?
State one assumption needed to interpret copied reference variability as appropriate for treatment-arm dropouts.

Solutions. (1) No. A predictive draw is the conditional mean plus a residual draw; 5 units is the conditional standard deviation, describing the spread of that predictive distribution. (2) The variance is \(\{4+9+2(3)\}/4=19/4=4.75\) outcome-units squared. (3) Differences between pattern-specific means add the nonnegative between-pattern term to the average within-pattern variance. (4) One required assumption is that, conditional on the relevant observed history, the reference arm’s post-dropout outcome distribution is suitable for the treatment-arm participants who drop out.