Copy-Reference Analysis in an Active-Controlled Trial

Why Copy the Active-Control Distribution?

In an active-controlled trial, the comparator is an established treatment rather than an untreated or placebo group. If a participant assigned to the experimental treatment discontinues it, their later outcome may be missing. A copy-reference analysis asks what the treatment contrast would look like if those missing outcomes followed the outcome distribution of participants assigned to the active control. This is a modelling assumption about unobserved outcomes, not a claim that discontinuing participants actually received the comparator.

The earlier material on reference-based imputation in a maintenance trial introduced the general idea of using a reference arm to specify missing outcomes. Here the reference is an active control, and the practical question is how that reference distribution affects the contrast between randomized groups.

Definition: Let \(A_i=1\) denote assignment to the experimental treatment and \(A_i=0\) assignment to the active-control reference treatment. Let \(Y_{it}\) be participant \(i\)'s outcome at visit \(t\), with larger values defined here as worse. Under a copy-reference (CR) assumption, post-discontinuation outcomes missing for participants assigned to the experimental treatment are imputed from the specified outcome distribution in the reference arm. In a longitudinal analysis, the reference distribution must specify how its mean and covariance are modelled over time and which baseline covariates are used.

A simple endpoint version makes the mean assumption transparent: within a prespecified baseline-covariate stratum, the missing endpoint values in the experimental arm are represented by draws from the reference arm's endpoint distribution in that stratum. In a repeated-measures analysis, a multivariate reference model can instead supply a joint distribution for several missing visits. The model's conditional structure and covariance matter: imputation should preserve plausible relationships among visits rather than treat repeated outcomes as unrelated.

The CR assumption does not by itself specify every modelling choice. The analysis plan should state the reference population, covariates or strata, outcome model, handling of partial outcome histories, and the treatment of observed outcomes collected after discontinuation. When such post-discontinuation outcomes are observed, they are data, not missing values to replace; the assumption concerns outcomes that remain unobserved.

Assumptions and the Reference Distribution

A copy-reference analysis requires more than the label “CR.” Its interpretation rests on several assumptions and implementation conditions:

Randomized-group definition: participants remain analysed according to their randomized assignment. The estimand must state how discontinuation and subsequent outcomes are handled.
Reference-model adequacy: the model fitted to the active-control arm adequately represents the outcome distribution used for imputation, including relevant covariate relationships and, for repeated outcomes, dependence across visits.
Applicability to experimental-arm discontinuers: after discontinuation, their unobserved outcomes are assumed to follow the chosen reference distribution, conditional on the model's specified predictors. This is an untestable assumption about missing outcomes.
Support: the reference arm must provide enough observations in relevant covariate and outcome-history regions to estimate the required distributions without extreme extrapolation.
Proper uncertainty propagation: imputed values must vary across completed data sets, and inference must include both within-imputation and between-imputation uncertainty.

If the model conditions on baseline variables, it should use those variables consistently when predicting missing outcomes. In longitudinal settings, the precise treatment of observed pre-discontinuation outcomes must also be stated. Different reference-based specifications can treat that history differently and therefore yield different post-discontinuation predictions. The examples below use an endpoint, so the reference distribution is specified within baseline strata and no earlier outcome history is needed.

Worked Example: Copying the Active-Control Endpoint Mean

Suppose a hypothetical trial compares an experimental treatment with an active control on a symptom score, for which lower values are better. There are 80 participants assigned to the experimental treatment. The endpoint is observed for 56, whose mean score is 7.5 points; the endpoint is missing for 24 who discontinued. The reference arm has observed endpoint scores for all 80 participants, with mean 10.0 points. For this illustration, assume the CR model uses the reference-arm endpoint distribution without additional covariate stratification.

To show the mean effect of the assumption, first replace each missing value by the reference mean. (Actual multiple imputation draws individual values from the reference distribution, rather than assigning every missing participant the same value.) The completed experimental-arm mean is

\bar{Y}_{E,\mathrm{CR}} = \frac{56(7.5)+24(10.0)}{80} = \frac{420+240}{80} = 8.25\text{ points}.

The estimated experimental-minus-reference contrast is therefore

\widehat{\Delta}_{\mathrm{CR}} = 8.25-10.0 = -1.75\text{ points}.

Under the stated outcome coding, this estimate favours the experimental assignment by 1.75 points. The observed experimental completers alone had mean 7.5 points, a contrast of \(7.5-10.0=-2.5\) points. Copying the reference mean for the missing outcomes moves the experimental-arm mean upward by 0.75 points and attenuates the apparent benefit. This direction follows from the numbers in this example; CR does not guarantee attenuation in every data configuration.

The calculation checks the imputation rule's mean consequence, but it is not a complete analysis of uncertainty. A proper analysis draws missing values from the reference distribution, repeats the analysis across imputed data sets, and accounts for variation among the resulting estimates.

How the Assumption Changes the Contrast

The effect on the contrast can be derived directly for the simple endpoint case. Suppose \(n_E\) experimental-arm participants are randomized, \(n_o\) have observed endpoints, and \(n_m\) have missing endpoints, where \(n_E=n_o+n_m\). Let \(\bar{Y}_{E,o}\) be the observed experimental-arm mean and let \(\bar{Y}_{R,m}\) be the mean of the reference distribution used for the missing experimental-arm outcomes. Replacing missing outcomes by that mean gives

\bar{Y}_{E,\mathrm{CR}} = \frac{n_o\bar{Y}_{E,o}+n_m\bar{Y}_{R,m}}{n_E}.

Subtracting the observed experimental mean gives the shift in that arm's mean:

\bar{Y}_{E,\mathrm{CR}}-\bar{Y}_{E,o} = \frac{n_m}{n_E} \left(\bar{Y}_{R,m}-\bar{Y}_{E,o}\right).

Thus the shift is determined by both the fraction missing and the gap between the reference prediction and the observed experimental mean. If missing participants are predicted to have worse outcomes than observed experimental participants, the experimental-arm mean rises when higher scores are worse. If the predicted reference mean is lower, it falls. With covariate-specific reference predictions, the same reasoning applies to the weighted sum of the predictions for individual missing participants or strata.

Worked Example: Using Baseline-Stratified Reference Distributions

Consider a second hypothetical endpoint trial with 40 experimental-arm participants and 40 active-control participants. The symptom score is lower when better. Participants fall into two prespecified baseline-severity strata. In the experimental arm, 15 mild-severity participants have observed endpoints with mean 6 points, and 5 have missing endpoints. Among severe participants, 10 have observed endpoints with mean 14 points, and 10 have missing endpoints. The reference-arm means are 8 points for mild participants and 16 points for severe participants; assume all 40 reference endpoints are observed, with 20 participants in each stratum.

Under the specified stratified CR model, the five missing mild-stratum outcomes have predicted mean 8, and the ten missing severe-stratum outcomes have predicted mean 16. The completed experimental-arm mean is

\bar{Y}_{E,\mathrm{CR}} = \frac{15(6)+5(8)+10(14)+10(16)}{40} = \frac{90+40+140+160}{40} = 10.75\text{ points}.

The reference-arm mean, using its two equally sized strata, is

\bar{Y}_R = \frac{20(8)+20(16)}{40} = 12.0\text{ points}.

Consequently, the CR contrast is \(10.75-12.0=-1.25\) points, favouring the experimental assignment by 1.25 points under the stated coding. The experimental completers' pooled mean is \((15(6)+10(14))/25=9.2\) points, so a completer-only contrast would be \(9.2-12.0=-2.8\) points. The difference arises partly because the missing severe-stratum outcomes are assigned a worse reference mean than the observed experimental mean. The example also shows why using one overall reference mean can be inappropriate when baseline severity strongly predicts the outcome.

These calculations use the reference means as a transparent stand-in for the expected imputed values. In a multiple-imputation analysis, the reference model also supplies the spread of possible values, and this variation contributes to uncertainty in the contrast.

Inference Across Imputed Data Sets

For each completed data set \(j\), calculate the treatment contrast \(Q_j\) and its estimated variance \(U_j\), using the same analysis model. With \(m\) imputations, let \(\bar{Q}\) be the mean of the \(Q_j\)'s, \(\bar{U}\) the mean of the \(U_j\)'s, and \(B\) the sample variance of the \(Q_j\)'s. Rubin's total variance combines the average within-imputation variance and the between-imputation variance:

\bar{Q}=\frac{1}{m}\sum_{j=1}^{m}Q_j,\qquad \bar{U}=\frac{1}{m}\sum_{j=1}^{m}U_j,\qquad B=\frac{1}{m-1}\sum_{j=1}^{m}(Q_j-\bar{Q})^2,

T=\bar{U}+\left(1+\frac{1}{m}\right)B.

The standard error is \(\sqrt{T}\). A confidence interval uses a \(t\) reference distribution with degrees of freedom appropriate to the imputation procedure and analysis; the calculation below uses the usual Rubin degrees-of-freedom approximation. A single completed data set, or an analysis that ignores \(B\), understates uncertainty when missing values materially affect the estimate.

Worked Example: Pooling Copy-Reference Contrasts

Suppose five hypothetical imputations produce experimental-minus-reference estimates, in symptom-score points, of \(-1.6,-1.8,-1.4,-2.0,\) and \(-1.7\). Suppose each completed-data analysis has estimated variance \(U_j=0.25\text{ points}^2\). The pooled estimate is

\bar{Q} = \frac{-1.6-1.8-1.4-2.0-1.7}{5} = -1.70\text{ points}.

Here \(\bar{U}=0.25\text{ points}^2\). The deviations from \(-1.70\) are \(0.10,-0.10,0.30,-0.30,0.00\) points, so

B = \frac{0.10^2+(-0.10)^2+0.30^2+(-0.30)^2+0^2}{4} = \frac{0.20}{4} = 0.0500\text{ points}^2.

The total variance and standard error are

T = 0.25+\left(1+\frac{1}{5}\right)(0.0500) = 0.310\text{ points}^2, \qquad SE=\sqrt{0.310}=0.557\text{ points}.

Using the Rubin degrees-of-freedom approximation gives \(df\approx4(1+0.25/0.06)^2=106.8\), and a 95% critical value of about 1.983. The interval is

-1.70\ \pm\ 1.983(0.557) = (-2.80,-0.596)\text{ points}.

Under the specified analysis and CR assumption, this interval describes uncertainty in the experimental-minus-reference mean contrast; its values favour the experimental assignment because lower scores are better. It is not a probability statement about the fixed contrast. The calculation is illustrative: an actual analysis should use an appropriate degrees-of-freedom method for its design and model.

Diagnostics, Alternatives, and Limitations

Diagnostics can assess whether the fitted reference model is credible as a model for the observed reference data, but cannot establish that missing experimental-arm outcomes truly follow that distribution. Check whether covariate strata have adequate reference observations, whether predicted values lie in plausible ranges, and whether the reference model captures important outcome patterns and repeated-measure correlations. Very small strata, extreme predictions, unstable model fits, and large sensitivity of results to reasonable model changes are warnings that the assumed reference distribution is poorly supported. Earlier material on failed fits and unstable imputations is relevant when these problems arise.

CR is one assumption-based approach, not a consequence of randomization alone. Under a missing-at-random analysis, predictions typically use the observed data model and its stated predictors; that can give different missing-outcome predictions. Other reference-based assumptions can retain or discard aspects of the experimental-arm trajectory differently after discontinuation. These alternatives should be named and specified precisely rather than treated as interchangeable versions of CR. In an active-controlled trial, the choice of reference distribution is particularly consequential: the comparator's outcomes reflect an active therapy, not absence of treatment.

The central limitation is that the unobserved post-discontinuation outcomes cannot verify the CR assumption. The estimated contrast is therefore conditional on the assumption, the reference model, and the chosen estimand. A precise confidence interval does not remove uncertainty about whether the assumption is scientifically appropriate. In particular, CR should not be described as evidence that participants who discontinued the experimental treatment would actually have experienced the active-control outcomes.

Common mistake: “The active-control mean is the right value for every missing experimental outcome, so I can fill all missing values with that constant and analyse the completed data once.”
Correction: The reference arm supplies a distribution, not generally a fixed value for each person. Use the prespecified reference model to generate appropriate imputations, retain relevant covariate structure, and combine estimates so that both within- and between-imputation uncertainty are represented.

Key takeaway: Copy-reference analysis replaces missing post-discontinuation experimental outcomes with draws from a specified active-control distribution. The resulting treatment contrast depends on how many outcomes are missing and on how that reference distribution differs from the experimental-arm outcomes that were observed.

Check Your Understanding

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

If higher outcome values are worse, when does copying the reference mean raise the experimental-arm mean?
Why might a baseline-stratified reference model be preferable to one overall reference mean?
In the first worked example, what fraction of the experimental arm had missing endpoints, and why does that fraction affect the mean shift?
What uncertainty does the between-imputation variance \(B\) represent?
Can good fit of the reference model to observed control outcomes prove that CR is true for missing experimental outcomes?

Solutions. (1) It raises the mean when the reference prediction for the missing outcomes exceeds the observed experimental mean, since the shift is \((n_m/n_E)(\bar{Y}_{R,m}-\bar{Y}_{E,o})\). (2) Stratification can reflect prognostic differences, preventing missing participants in different baseline groups from all receiving an unsuitable pooled prediction. (3) The fraction was \(24/80=0.30\); the mean shift is scaled by this fraction, so a larger missing fraction gives the reference prediction more weight. (4) \(B\) represents variation among estimates across imputations, reflecting uncertainty from filling in missing values. (5) No. Fit to observed reference outcomes can be assessed, but the assumed distribution of unobserved experimental-arm outcomes remains untestable from those observed data alone.