Deriving Copy-Reference Conditional Distributions

Why the Observed History Matters

After a participant assigned to the active arm drops out, the post-dropout outcomes are unobserved. A reference-based imputation rule specifies how to model those outcomes. Under copy-reference (CR), the post-dropout outcomes follow the reference arm’s distribution, conditional on the participant’s observed pre-dropout history. That conditioning is consequential: a participant who entered dropout with an unusually high observed outcome need not be predicted to have the reference arm’s unconditional mean afterward.

This tutorial derives that conditional distribution under a multivariate normal model. The focus is the joint predictive law for the missing outcomes given the observed history, not a full imputation or analysis procedure. The reference arm supplies the conditional regression and residual uncertainty; the active-arm participant supplies the observed values on which the prediction is conditioned.

Definition: Let \(A=1\) denote the active arm and \(A=0\) the reference arm. Let \(D=k\) denote dropout after visit \(k\), and write \(Y_O\) for the vector of outcomes observed through that visit and \(Y_M\) for the post-dropout outcomes that are missing. The CR restriction is \(Y_M\mid(Y_O=y_O,D=k,A=1)\) distributed as \(Y_M\mid(Y_O=y_O,A=0)\). It asserts that, given the observed history, post-dropout outcomes in active-arm dropouts follow the reference arm’s conditional outcome distribution.

The equality is a modeling restriction, not a consequence of randomization. It says that, after conditioning on the observed history, dropout does not provide additional information about the post-dropout outcomes beyond the reference conditional distribution. The restriction does not say that the participant’s observed history was generated by the reference arm: \(y_O\) is the history actually observed in the active arm.

Gaussian Conditional Distribution

Suppose the reference-arm outcome vector over the relevant visits has a multivariate normal distribution with mean vector \(\mu_R\) and covariance matrix \(\Sigma_R\). Partition its components into observed-history and post-dropout parts: \(Y_O\) has mean \(\mu_{R,O}\), \(Y_M\) has mean \(\mu_{R,M}\), and the corresponding covariance blocks are \(\Sigma_{R,OO}\), \(\Sigma_{R,MM}\), and \(\Sigma_{R,MO}\), with \(\Sigma_{R,OM}=\Sigma_{R,MO}^{\mathsf T}\). Assume \(\Sigma_{R,OO}\) is invertible. The reference-arm conditional distribution is then

Y_M\mid(Y_O=y_O,A=0) \sim N\!\left( \mu_{R,M}+\Sigma_{R,MO}\Sigma_{R,OO}^{-1}(y_O-\mu_{R,O}), \ \Sigma_{R,MM}-\Sigma_{R,MO}\Sigma_{R,OO}^{-1}\Sigma_{R,OM} \right).

The first argument of \(N(\cdot,\cdot)\) is the conditional mean; the second is the conditional covariance matrix. By the CR restriction, this is also the distribution used for the missing outcomes of an active-arm dropout with observed history \(y_O\). The conditional mean adjusts the reference mean according to how the observed history differs from the reference mean. The conditional covariance describes remaining uncertainty after that history is known.

Why the Formula Holds

A short derivation shows both how the adjustment arises and why the covariance takes this form. In the reference arm, define the matrix \(B=\Sigma_{R,MO}\Sigma_{R,OO}^{-1}\) and the residual vector \(Z=Y_M-\mu_{R,M}-B(Y_O-\mu_{R,O})\). Its covariance with the observed history is

\operatorname{Cov}(Z,Y_O) =\Sigma_{R,MO}-B\Sigma_{R,OO} =\Sigma_{R,MO}-\Sigma_{R,MO}\Sigma_{R,OO}^{-1}\Sigma_{R,OO} =0.

Because \((Y_O,Y_M)\) is jointly multivariate normal, zero covariance between \(Z\) and \(Y_O\) implies that \(Z\) and \(Y_O\) are independent. Its mean is zero, and its covariance is

\begin{aligned} \operatorname{Var}(Z) &=\Sigma_{R,MM}-B\Sigma_{R,OM}-\Sigma_{R,MO}B^{\mathsf T} +B\Sigma_{R,OO}B^{\mathsf T}\\ &=\Sigma_{R,MM}-\Sigma_{R,MO}\Sigma_{R,OO}^{-1}\Sigma_{R,OM}. \end{aligned}

In the second line, substitute \(B=\Sigma_{R,MO}\Sigma_{R,OO}^{-1}\); the two negative terms are equal, and the final positive term cancels one of them. Since \(Z\) is independent of \(Y_O\), conditioning on \(Y_O=y_O\) leaves the distribution of \(Z\) unchanged. Rearranging its definition gives the displayed conditional mean and covariance. The derivation establishes the Gaussian conditioning result; applying it to active-arm dropouts additionally requires the CR restriction.

Worked Calculations

Worked Example: One Observed Outcome and One Missing Outcome

Suppose, hypothetically, the reference-arm outcomes at visits 1 and 2 have mean vector \((10,12)\) points and covariance matrix \(\begin{pmatrix}4&3\\3&9\end{pmatrix}\) points squared. An active-arm participant drops out after visit 1 with an observed outcome of 14 points. Assume the joint normal model and CR restriction hold for this history, and that the reference conditional distribution is applicable to it.

Here \(Y_O=Y_1\), \(Y_M=Y_2\), and \(y_O-\mu_{R,O}=14-10=4\) points. Thus the conditional mean is \(12+(3/4)(4)=12+3=15\) points. The conditional variance is \(9-(3)(1/4)(3)=9-2.25=6.75\) points squared. Therefore \(Y_2\mid(Y_1=14,D=1,A=1)\sim N(15,6.75)\), with conditional standard deviation \(\sqrt{6.75}=2.598\) points.

The mean is 15 points, not the reference arm’s marginal visit-2 mean of 12 points: the observed visit-1 value is above the reference mean, and the reference outcomes have positive covariance. Under CR, the reference conditional relationship carries that history forward.

Worked Example: Two Post-Dropout Outcomes Together

Consider hypothetical reference-arm outcomes at three visits with mean vector \((5,7,8)\) points and covariance matrix \(\begin{pmatrix}4&2&1\\2&9&3\\1&3&4\end{pmatrix}\) points squared. An active-arm participant drops out after visit 1, having observed \(Y_1=6\) points. Assume the reference model is multivariate normal, the covariance matrix is positive definite, and CR applies to this dropout history.

The observed-history residual is \(6-5=1\) point. The covariance of \((Y_2,Y_3)\) with \(Y_1\) is \((2,1)^{\mathsf T}\), and \(\operatorname{Var}(Y_1)=4\). The conditional mean vector is therefore \((7,8)^{\mathsf T}+(2,1)^{\mathsf T}(1/4)(1)=(7.5,8.25)^{\mathsf T}\) points.

For the conditional covariance, subtract the history contribution from the post-dropout covariance block: \(\begin{pmatrix}9&3\\3&4\end{pmatrix}- \begin{pmatrix}2\\1\end{pmatrix}(1/4)\begin{pmatrix}2&1\end{pmatrix} = \begin{pmatrix}8&2.5\\2.5&3.75\end{pmatrix}\) points squared. Thus the joint distribution of \((Y_2,Y_3)\), conditional on \(Y_1=6\) points, has mean \((7.5,8.25)\) points and the displayed conditional covariance. The two missing outcomes are not conditionally independent: their conditional covariance is \(2.5\) points squared.

Worked Example: Conditioning on the Full Observed History

Use the same hypothetical reference-arm model as in the previous example, but suppose the participant drops out after visit 2 and has observed \(Y_1=2\) points and \(Y_2=5\) points. Only \(Y_3\) is missing. Assume that this history is within the range to which the reference conditional model is being applied and that CR holds.

The observed-history mean is \((5,7)^{\mathsf T}\), so the residual is \((-3,-2)^{\mathsf T}\). The history covariance block and its inverse are \(\Sigma_{R,OO}=\begin{pmatrix}4&2\\2&9\end{pmatrix}\) and \(\Sigma_{R,OO}^{-1}=\frac{1}{32}\begin{pmatrix}9&-2\\-2&4\end{pmatrix}\). The covariance between \(Y_3\) and the history is \((1,3)\). Its regression coefficients are \((1,3)\Sigma_{R,OO}^{-1}=(3/32,10/32)\).

The conditional mean is \(3+(3/32)(-3)+(10/32)(-2)=3-9/32-20/32=2.09375\) points. The conditional variance is \(4-(1,3)\Sigma_{R,OO}^{-1}(1,3)^{\mathsf T} =4-(3/32+30/32)=95/32=2.96875\) points squared. Thus the CR predictive distribution for \(Y_3\) is \(N(2.094,2.969)\), approximately, with outcome measured in points. Conditioning on both observed visits uses information not captured by conditioning on visit 2 alone. This calculation also shows why replacing the observed history by a single last value is generally not justified by the multivariate normal model.

Assumptions, Diagnostics, and Alternatives

The formula is exact for the stated Gaussian model, but its use for active-arm dropouts depends on substantive and statistical assumptions. In practice, the analysis needs a suitably estimated reference-arm mean vector and covariance matrix, a nonsingular covariance matrix for the observed history, and a conditional model that is meaningful at the histories seen among active-arm dropouts. Most importantly, CR must be considered a defensible assumption: conditional on the observed history, the post-dropout distribution for these active-arm participants is taken to be the reference distribution.

Observed data cannot confirm how the missing outcomes of active-arm dropouts would have behaved. Diagnostics can reveal problems with the model or its application, but cannot establish the unobserved CR restriction. Useful checks include:

Inspect whether the dropout histories fall within the range of reference-arm histories. A prediction for a history far outside that range relies on extrapolation.
Check the estimated reference covariance matrix for inadmissible or unstable estimates. A non-positive-definite covariance estimate cannot define the assumed multivariate normal distribution as stated.
Assess whether the Gaussian model gives plausible conditional distributions over the observed-history range. Severe skewness, outliers, or changing relationships across histories can make its conditional predictions misleading.
As a model diagnostic, compare predictions with held-out reference-arm outcomes or use reference-arm participants’ observed histories to assess prediction of later observed outcomes. Such checks assess predictive fit within the reference arm, not the CR assumption for active-arm dropouts.

The main alternatives answer different questions. Under a missing-at-random model, the active arm’s conditional outcome model supplies the prediction, given the included observed information and the missingness assumptions. Under CR, the reference arm’s conditional model supplies it. Jump-to-reference (J2R) places a restriction on post-dropout means relative to the reference arm; it should not be confused with copying the entire reference conditional distribution given the observed history. In particular, CR’s conditional mean can differ from the reference marginal mean when the observed history differs from its reference mean.

Common Pitfall and Limitations

A common mistake is to set every post-dropout conditional mean equal to the reference arm’s marginal mean and call the result CR. In the first worked example, that would set the visit-2 mean to 12 points despite an observed visit-1 value of 14 points. The CR calculation instead gives 15 points, because it conditions on the observed visit-1 value using the reference covariance. The correction is to partition the reference mean and covariance by observed and missing outcomes, then use the conditional-normal formula.

CR is not automatically preferable because it uses reference-arm data, nor does Gaussian conditioning remove uncertainty about the identifying restriction. Predictions may be unreliable when the reference sample is small, the covariance structure is poorly estimated, or dropout histories differ substantially between arms. The normal model also imposes a particular shape and linear conditional mean. A careful analysis should state the restriction, show which observed history is conditioned on, and consider whether the resulting predictions are plausible. Sensitivity to alternative assumptions is important because the post-dropout outcomes are not observed to adjudicate between them.

Key takeaway: Under copy-reference, the missing outcomes follow the reference arm’s conditional distribution given the participant’s observed pre-dropout history. In a multivariate normal model, that distribution has a history-adjusted mean and a reduced conditional covariance; it is not generally the reference arm’s marginal distribution.

Check Your Understanding

Answer the questions, then compare your reasoning with the solutions.

In the first worked example, why is the conditional mean above the reference marginal visit-2 mean?
Suppose the reference observed-history covariance block is singular. Which part of the stated formula fails, and what should not be done without further modeling?
For a reference model with \(\mu_{R,O}=4\), \(\mu_{R,M}=10\), \(\Sigma_{R,OO}=4\), \(\Sigma_{R,MO}=2\), and \(\Sigma_{R,MM}=9\), calculate the conditional mean and variance when the observed history is 6.
Does good prediction of later outcomes for held-out reference-arm participants prove that CR is correct for active-arm dropouts? Explain.

Solutions. (1) The observed history is 4 points above its reference mean and has positive covariance 3 with the later outcome, so the reference conditional adjustment is \((3/4)(4)=3\) points; the conditional mean is 15 rather than 12. (2) The inverse \(\Sigma_{R,OO}^{-1}\) does not exist, so this formula cannot be applied as written. One should not silently substitute an ordinary inverse or claim the conditional law is identified by this expression; a suitable reduced history or other justified model is needed. (3) The history residual is \(6-4=2\). The mean is \(10+(2/4)(2)=11\), and the variance is \(9-(2)(1/4)(2)=8\). The conditional distribution is \(N(11,8)\) in the outcome’s units. (4) No. That check assesses predictive fit within the reference arm. It cannot verify the unobserved conditional distribution for active-arm dropouts, which is imposed by the CR restriction.