Deriving J2R Conditional Means from Gaussian Models

Why a J2R Conditional Mean Needs More Than a Mean Restriction

Jump-to-reference (J2R) specifies that, after an active-arm participant drops out, the post-dropout mean trajectory follows the reference-arm mean trajectory. But a participant’s missing outcomes are predicted using their observed history as well. The prediction therefore depends on how observed and missing outcomes covary, not just on the post-dropout means.

This tutorial derives that prediction for a Gaussian pattern-mixture model. The key distinction is between the mean restriction, which sets the post-dropout mean, and the covariance model, which determines how the participant’s observed history shifts their conditional prediction. J2R’s mean restriction alone does not specify every covariance needed for that calculation.

Model and Assumptions

Let \(Y=(Y_1,\ldots,Y_K)^\mathsf{T}\) be a participant’s outcomes at \(K\) ordered visits. Consider a participant assigned to the active arm who drops out after visit \(s\), so their observed history is \(Y_o=(Y_1,\ldots,Y_s)^\mathsf{T}\) and their post-dropout outcomes are \(Y_m=(Y_{s+1},\ldots,Y_K)^\mathsf{T}\). We condition on this dropout pattern. Write \(y_o\) for the participant’s realized observed history.

For this pattern, suppose the joint distribution of observed and missing outcomes is multivariate normal with mean vector \(\mu^{(s)}\) and covariance matrix \(\Omega^{(s)}\). Partition them according to the observed and missing visits:

Assume \(\Omega_{oo}^{(s)}\) is nonsingular. For the standard J2R mean restriction, the observed-history means for this active-arm pattern are the active-arm means, while the post-dropout means are the corresponding reference-arm means:

Here \(A\) denotes the active arm and \(R\) the reference arm. The covariance blocks must also be specified by the model. For example, an analysis might use a common covariance matrix across arms or choose covariance blocks from a reference-arm model. The J2R mean restriction does not, by itself, choose between these options or imply that their covariance blocks are equal.

Deriving the Conditional Predictive Mean

For a jointly Gaussian vector partitioned into observed and missing components, the conditional expectation is

Substitute the J2R pattern means. The result is

The multiplication is dimensionally consistent: \(\Omega_{mo}^{(s)}\) has one row per missing visit and one column per observed visit, while \(\left(\Omega_{oo}^{(s)}\right)^{-1}(y_o-\mu_{A,o})\) is a vector with one entry per observed visit. If there is only one missing visit, the result is a scalar. With several missing visits, it is a vector of conditional means.

A short derivation explains the covariance adjustment. Set \(B=\Omega_{mo}^{(s)}(\Omega_{oo}^{(s)})^{-1}\) and define \(Z=Y_m-\mu_m^{(s)}-B(Y_o-\mu_o^{(s)})\). Its covariance with \(Y_o\) is

Because \(Z\) and \(Y_o\) are jointly Gaussian, zero covariance implies independence. Thus the conditional expectation of \(Z\) given \(Y_o\) is its mean, zero. Rearranging the definition of \(Z\) gives the conditional expectation formula above. In J2R, substituting \(\mu_m^{(s)}=\mu_{R,m}\) and \(\mu_o^{(s)}=\mu_{A,o}\) yields the stated expression.

What the Formula Does—and Does Not—Say

The centering term is \(y_o-\mu_{A,o}\), not automatically \(y_o-\mu_{R,o}\). This follows from the stated J2R pattern mean: the observed visits are modeled with their active-arm means, while the missing visits use reference-arm means. If instead one conditions entirely under the reference-arm joint model, the resulting conditional mean is

That expression uses reference-arm centering and reference-arm covariance blocks. It is a distinct modeling choice unless the relevant means and covariance blocks make the two expressions equal. Similarly, continuing the active-arm mean trajectory after dropout would replace the reference post-dropout mean and would not implement the J2R mean restriction.

These distinctions matter because a mean restriction alone is not a full conditional model. Two analyses may agree on \(\mu_{R,m}\) but give different participant-specific predictions if they use different \(\Omega_{mo}\) or \(\Omega_{oo}\). If the covariance blocks are assumed common across arms, use those common blocks in the J2R formula; do not silently substitute reference-arm covariance blocks unless that is the specified model.

Diagnostics, Limitations, and Common Pitfalls

Before using the formula, check that the Gaussian pattern model and the covariance blocks are appropriate for the intended analysis. In particular, \(\Omega_{oo}^{(s)}\) must be invertible. A nearly singular block can produce unstable weights, so small changes in estimated covariances may cause large changes in predicted means. Check the covariance estimates for symmetry and positive definiteness, and examine whether the resulting conditional predictions are plausible in the outcome’s units and range. An implausible result is a reason to investigate the model and data, not to silently alter the formula.

The calculation is conditional on the specified dropout pattern. It does not establish that dropout is unrelated to unobserved outcomes, nor does it make the J2R assumption a fact verified by the observed data. The reference-arm post-dropout trajectory is a modeling restriction. As in other pattern-mixture analyses, conclusions can depend on that untestable choice.

• Common mistake: “J2R sets the missing conditional mean equal to the reference-arm mean, so the prediction is just \(\mu_{R,m}\).” Correction: J2R sets the post-dropout component of the pattern mean vector to \(\mu_{R,m}\). Conditioning on the participant’s observed history adds the covariance adjustment unless that adjustment is zero.
• Common mistake: Centering the observed history at \(\mu_{R,o}\) while claiming to use the hybrid J2R pattern mean. Correction: With active-arm observed-history means and reference-arm post-dropout means, the adjustment is centered at \(\mu_{A,o}\). Reference-arm centering corresponds to a different conditional-model specification.
• Common mistake: Treating the J2R mean rule as if it also specified every covariance. Correction: State which covariance blocks are used. The conditional mean depends on them.
• Common mistake: Reporting the conditional mean as a realized missing outcome. Correction: It is a model-based expected value given the observed history and pattern, not a guarantee about an individual outcome or a complete predictive distribution.

Compared with continuing the active-arm mean after dropout, J2R substitutes a reference-arm post-dropout mean. Compared with conditioning wholly under a reference-arm model, the hybrid J2R expression retains active-arm centering for the observed history. The choice is substantive: neither the reference trajectory nor the covariance specification is determined by the observed participant history alone.