J2R for Dropout at Different Follow-Up Visits

Why the Dropout Visit Matters

Under jump-to-reference (J2R), the post-dropout distribution is based on reference-arm behavior, conditional on the participant’s observed history. That history changes with the dropout visit: someone dropping out early contributes fewer observed outcomes to the conditional prediction than someone dropping out later. Consequently, dropout time affects both the predicted post-dropout means and their uncertainty.

The calculations below concern a participant in the treatment arm who has observed outcomes through visit \(d\) and then permanently drops out. The observed history is the sequence of outcomes through that last observed visit. For clarity, the examples use four scheduled visits and monotone dropout. The same conditional-Gaussian calculation applies to other visit counts when the observed and missing sets are specified.

Definition: Let \(Y=(Y_1,\ldots,Y_K)^\mathsf{T}\) denote a participant’s outcomes at \(K\) visits. If dropout occurs after visit \(d\), the observed history is \(Y_O=(Y_1,\ldots,Y_d)^\mathsf{T}\), and the missing post-dropout outcomes are \(Y_M=(Y_{d+1},\ldots,Y_K)^\mathsf{T}\). Under J2R, the conditional distribution for \(Y_M\) is taken from the reference-arm model, conditional on the participant’s observed history \(Y_O=y_O\).

This definition describes a conditional prediction, not a replacement of the participant’s observed outcomes by reference-arm outcomes. Outcomes before dropout remain observed treatment-arm outcomes. The J2R assumption applies after dropout, while the observed history still informs the predictions through its relationship with future outcomes in the reference model.

Conditional Prediction Under a Reference Model

Suppose the reference-arm outcome vector has a multivariate normal distribution with mean vector \(\mu_R\) and covariance matrix \(\Sigma\). Partition the mean vector and covariance matrix according to the observed visits \(O\) and missing visits \(M\). If \(\Sigma_{OO}\) is invertible, the conditional distribution used for J2R is

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

The first argument of \(N\) is the conditional mean vector; the second is the conditional covariance matrix. The conditional mean starts at the reference-arm mean for the missing visits and adjusts it according to how the participant’s observed history differs from the reference-arm mean history. The conditional covariance describes uncertainty remaining after conditioning on that history.

Assumptions for this calculation: The reference-arm outcomes follow the specified multivariate normal model; its mean and covariance adequately describe the relevant reference distribution; and \(\Sigma_{OO}\) is nonsingular. The J2R identifying assumption is that, after dropout, the participant’s outcomes follow the reference-arm conditional distribution given the observed history. These are modeling and sensitivity assumptions, not consequences established by the observed data alone.

A short derivation explains the formula. Define the residual \(U=Y_M-\mu_{R,M}-\Sigma_{MO}\Sigma_{OO}^{-1}(Y_O-\mu_{R,O})\). Its covariance with \(Y_O\) is zero:

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

Because the joint distribution is multivariate normal, zero covariance here implies that \(U\) and \(Y_O\) are independent. The mean of \(U\) is zero, and its covariance is \(\Sigma_{MM}-\Sigma_{MO}\Sigma_{OO}^{-1}\Sigma_{OM}\). Thus conditioning on \(Y_O=y_O\) gives the stated conditional mean and covariance. This also shows why the history enters through the reference model’s covariance, rather than through an arbitrary adjustment.

Worked Predictions at Three Dropout Visits

Use the same hypothetical reference model for the following examples. Outcomes are measured in points. The reference-arm mean vector is \(\mu_R=(5,6,7,6.5)^\mathsf{T}\), and the covariance matrix is \(\Sigma=2I+2J\), where \(I\) is the identity matrix and \(J\) is the all-ones matrix of the appropriate size. Thus each visit has variance 4 points squared, and every pair of distinct visits has covariance 2 points squared. These values are hypothetical and are used only to show the calculations.

Worked Example: Dropout After the First Visit

A hypothetical treatment-arm participant has an observed score of 8 points at visit 1 and then drops out. The observed history has one value, \(y_O=8\), while the reference mean at visit 1 is 5. The covariance between each later visit and visit 1 is 2. Therefore, the coefficient on the observed residual is \(2/4=0.5\).

The observed residual is \(8-5=3\) points, so the history adjustment for each missing visit is \(0.5(3)=1.5\) points. The predicted conditional means are \(6+1.5=7.5\) points at visit 2, \(7+1.5=8.5\) points at visit 3, and \(6.5+1.5=8.0\) points at visit 4.

For each missing visit, the conditional variance is \(4-2^2/4=3\) points squared. For any two missing visits, the conditional covariance is \(2-(2)(2)/4=1\) point squared. Thus the participant’s early dropout leaves substantial uncertainty about the later trajectory, even though the conditional means use the observed score.

Worked Example: Dropout After the Second Visit

A second hypothetical participant has observed scores \(7\) and \(9\) points at visits 1 and 2, then drops out. The observed residual vector relative to the reference means \((5,6)^\mathsf{T}\) is \((2,3)^\mathsf{T}\). The observed covariance block is \(\Sigma_{OO}=2I+2J\) for two visits, with inverse \(\frac12 I-\frac16J\).

The covariance row between either missing visit and the two observed visits is \((2,2)\). Multiplying by the inverse gives \((1/3,1/3)\). Therefore, the history adjustment is \((1/3)(2+3)=5/3\), or about 1.667 points. The conditional means are \(7+5/3=8.667\) points at visit 3 and \(6.5+5/3=8.167\) points at visit 4.

For either missing visit, the conditional variance is \(4-(2,2)\Sigma_{OO}^{-1}(2,2)^\mathsf{T}=4-4/3=8/3\), or about 2.667 points squared. The conditional covariance between visits 3 and 4 is \(2-4/3=2/3\) point squared. Relative to the early-dropout example, the additional observed outcome changes the residual adjustment and reduces the conditional variance.

Worked Example: Dropout After the Third Visit

A third hypothetical participant has scores \(7,7,8\) points at visits 1, 2, and 3, then drops out. The observed residuals relative to the reference means \((5,6,7)^\mathsf{T}\) are \((2,1,1)^\mathsf{T}\), with sum 4 points. The observed covariance block is \(2I+2J\) for three visits, so its inverse is \(\frac12 I-\frac18J\). Multiplying the cross-covariance row \((2,2,2)\) by this inverse gives \((0.25,0.25,0.25)\). The history adjustment is therefore \(0.25(2+1+1)=1\) point.

The conditional mean at visit 4 is \(6.5+1=7.5\) points. The conditional variance is \(4-(2,2,2)\Sigma_{OO}^{-1}(2,2,2)^\mathsf{T}\). The row product just calculated gives the quadratic form \(0.25(2+2+2)=1.5\), so the conditional variance is \(4-1.5=2.5\) points squared.

This calculation makes the role of the history explicit: the reference mean at visit 4 is adjusted by the weighted residuals across all three observed visits. The late-dropout prediction is not simply the reference mean, nor is it obtained by carrying the last observed score forward.

How to Compare Dropout Times

In the examples, the covariance structure makes the pattern especially transparent. For \(d\) observed visits, the observed covariance block is \(2I_d+2J_d\), whose inverse is \(\frac12 I_d-\frac{1}{2(d+1)}J_d\). The covariance between any future visit and the observed visits is a row of \(2\)s. Their product with the inverse is a row with each entry \(1/(d+1)\). Hence the conditional mean for a future visit \(j\) is

E(Y_j\mid Y_O=y_O) =\mu_{R,j}+\frac{1}{d+1} \sum_{k=1}^{d}(y_k-\mu_{R,k}).

The history adjustment uses the sum of observed residuals, scaled by a factor that depends on the number of observed visits. In this particular model, the conditional variance of any one future outcome is \(4-2d/(d+1)\). It is 3, \(8/3\), and 2.5 points squared when dropout follows visits 1, 2, and 3, respectively. More history reduces the conditional variance here, but it does not guarantee that the conditional mean moves in a particular direction: that depends on the participant’s residuals and the reference trajectory.

These comparisons are model-dependent. With a different covariance structure, the weights on earlier outcomes need not be equal, and outcomes close in time may be more strongly correlated than distant ones. The general partitioned conditional formula, rather than the simplified equal-weight expression, is the applicable result in that case.

Assumptions, Diagnostics, and Common Pitfalls

The numerical predictions rely on the specified reference means and covariance. In an analysis, these quantities are estimated, and reference-based multiple imputation also represents parameter and outcome uncertainty through draws, as covered in the earlier material on parameter draws and Rubin-rule inference. The conditional means shown here are not a complete uncertainty analysis by themselves.

Useful checks include confirming that the observed history is aligned with the correct visit order, that the reference means and covariance use the same visit definitions, and that the observed covariance block is invertible and numerically stable. Inspect predicted values against the outcome’s possible range and clinical scale. Also compare predictions across dropout patterns: abrupt changes may reflect real changes in conditioning information, but can also reveal a visit-indexing error, an incorrectly partitioned covariance matrix, or a model that fits poorly.

Wrong reasoning: “J2R means replace every post-dropout value with the reference-arm mean.” Correction: J2R uses the reference conditional distribution given the participant’s observed history; the conditional mean generally includes a history adjustment.
Wrong reasoning: “More observed visits must make the predicted mean closer to the reference mean.” Correction: More visits change the adjustment weights and available residuals. The mean can move either way; only under particular covariance models does a simple shrinkage description apply.
Wrong reasoning: “The conditional variance is the marginal variance at the future visit.” Correction: Conditioning reduces variance by the quadratic term \(\Sigma_{MO}\Sigma_{OO}^{-1}\Sigma_{OM}\), which must be calculated using the correct observed block.

J2R is an assumption about outcomes after dropout, not a fact that can generally be verified from observed post-dropout outcomes. It is useful because it states a clear reference-based scenario, but conclusions can depend on that scenario. It differs from complete-case analysis, which does not provide these post-dropout predictions, and from last observation carried forward, which substitutes a fixed observed value rather than a conditional reference distribution. Copy-reference imputation is another reference-based rule, but its assumptions differ; the earlier material on copy-reference imputation introduced that alternative.

Key takeaway: For each dropout pattern, identify the observed history, partition the reference mean and covariance accordingly, and calculate the conditional distribution for the missing visits. The dropout visit matters because it determines which outcomes condition the J2R prediction.

Check Your Understanding

Answer each question using the definitions and model above. Solutions follow the questions.

Under the four-visit covariance model, what is the history adjustment for dropout after visit 1 if the observed outcome is 6 points and the reference mean at that visit is 5 points?
For dropout after two visits, suppose the observed residuals are 1 and \(-1\) points. What is the conditional mean adjustment for a future visit?
Under the stated covariance model, what is the conditional variance of a future outcome after dropout following visit 3?
Why is a J2R prediction generally not equal to the reference-arm marginal mean at the future visit?

Solutions. 1. The residual is \(6-5=1\), and the one-visit coefficient is \(1/2\), so the adjustment is \(0.5\) point. 2. The adjustment is \((1/3)(1+(-1))=0\) points. 3. It is \(4-2(3)/(3+1)=2.5\) points squared. 4. J2R conditions on the participant’s observed history, so the conditional mean includes a covariance-weighted adjustment for how that history differs from the reference mean history.