Conditional Carryover of Observed Treatment History

Why Observed History Can Matter After Dropout

A participant who leaves an active treatment may have several outcomes already recorded. Those measurements are not missing, and reference-based imputation does not replace them. Yet they can still contain information about the participant’s later outcomes: people with higher-than-average outcomes at one visit may tend to have higher outcomes at the next. A prediction that uses this association is conditional on the observed history, rather than being an unconditional draw from the reference arm’s post-dropout distribution.

This conditional prediction is one way treatment history can be carried into post-dropout predictions. The extent of carryover depends on the cross-visit covariance in the predictive model. It is not a rule to add an observed treatment effect directly to each future reference mean, nor does it make any observed outcome missing.

Model and Conditional Distribution

Let $Y_i=(Y_{i1},\ldots,Y_{iK})^\mathsf{T}$ be the outcome vector for participant $i$ over $K$ visits. Suppose the participant is assigned to the active arm and is observed through visit $d$, then drops out. Let $O$ index the observed visits and $M$ the missing post-dropout visits. For simple monotone dropout, $O=\{1,\ldots,d\}$ and $M=\{d+1,\ldots,K\}$.

For a jump-to-reference (J2R) mean specification, the observed portion of the hybrid mean vector uses the active-arm means, while the missing portion uses the reference-arm means. This specifies the means for the predictive distribution; it does not change the data already recorded for the participant.

Definition: Conditional carryover of observed treatment history means predicting missing post-dropout outcomes from their reference-based marginal means, adjusted for the observed outcomes through the cross-visit covariance. Under a multivariate normal working model, $$ Y_{i,M}\mid Y_{i,O}=y_{i,O}\sim N\left(\mu_M^{R}+\Sigma_{MO}\Sigma_{OO}^{-1}(y_{i,O}-\mu_O^{A}),\ \Sigma_{MM}-\Sigma_{MO}\Sigma_{OO}^{-1}\Sigma_{OM}\right). $$ Here $\mu_O^{A}$ is the vector of active-arm means at observed visits, $\mu_M^{R}$ is the vector of reference-arm means at missing visits, and the $\Sigma$ blocks come from the specified joint covariance model. The observed values $y_{i,O}$ are conditioned on, not imputed.

The formula requires a multivariate normal predictive model, a compatible covariance matrix, and an invertible $\Sigma_{OO}$ (or an appropriately specified generalized calculation when the observed block is singular). The covariance model must specify the associations between observed and missing visits as well as the within-block covariances. The hybrid mean and covariance together define the prediction; mean rules alone do not determine conditional carryover.

The vector $y_{i,O}-\mu_O^{A}$ is the participant’s observed deviation from the active-arm mean history. The multiplier $\Sigma_{MO}\Sigma_{OO}^{-1}$ translates that deviation into a prediction adjustment. If the cross-covariance block $\Sigma_{MO}$ is zero, there is no adjustment and the conditional mean is just $\mu_M^{R}$. When the cross-visit associations are positive, higher observed outcomes relative to the active mean generally raise predictions, though the direction and size depend on the full covariance structure.

Why the Conditional Formula Holds

The conditional normal result follows from the joint normal distribution. Write the centered observed and missing vectors as $X_O=Y_O-\mu_O$ and $X_M=Y_M-\mu_M$, and define $B=\Sigma_{MO}\Sigma_{OO}^{-1}$. Then $X_M-BX_O$ has zero covariance with $X_O$:

\operatorname{Cov}(X_M-BX_O,X_O) =\Sigma_{MO}-B\Sigma_{OO} =\Sigma_{MO}-\Sigma_{MO}\Sigma_{OO}^{-1}\Sigma_{OO} =0.

For jointly normal vectors, zero covariance implies independence. Therefore, given $X_O$, the conditional distribution of $X_M$ is the distribution of $BX_O+(X_M-BX_O)$. Its mean is $BX_O$, and its covariance is

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

In the reference-based application, use the specified hybrid means $\mu_O^{A}$ and $\mu_M^{R}$ in this conditional result. This derivation explains both the adjustment to the mean and the reduction in predictive covariance after conditioning on observed history.

Worked Example: One Observed Visit and One Missing Visit

Suppose a hypothetical trial has two visits. The active-arm mean at visit 1 is 10 points, the reference-arm mean at visit 2 is 8 points, and the covariance model gives variance 9 at visit 1, variance 16 at visit 2, and covariance 7.2 between visits. A participant’s observed visit-1 score is 13. Assume the joint normal model and these covariance values apply to the predictive distribution.

For this two-visit calculation, $\Sigma_{OO}=9$, $\Sigma_{MO}=7.2$, and $\Sigma_{MM}=16$. The conditional mean is

8+\frac{7.2}{9}(13-10)=8+0.8(3)=10.4\text{ points}.

The conditional variance is

16-\frac{7.2^2}{9}=16-5.76=10.24\text{ points}^2,

so the conditional standard deviation is $\sqrt{10.24}=3.20$ points. The prediction is centered 2.4 points above the reference marginal mean because the participant’s observed score was 3 points above the active-arm mean and the cross-visit regression coefficient is 0.8. The recorded visit-1 score remains 13 points; it is not replaced by an imputed value.

Worked Example: Conditioning on Two Observed Visits

Consider a hypothetical three-visit setting in which a participant drops out after visit 2. The active-arm means at visits 1 and 2 are 10 and 12 points, and the reference-arm mean at visit 3 is 9 points. Assume the covariance blocks are $\Sigma_{OO}=\begin{pmatrix}4&2\\2&4\end{pmatrix}$, $\Sigma_{MO}=\begin{pmatrix}1&2\end{pmatrix}$, and $\Sigma_{MM}=4$, all in point-squared units. The observed outcomes are $y_O=(12,14)^\mathsf{T}$ points. These blocks form a valid positive-definite joint covariance matrix.

The observed residual vector relative to the active means is $(2,2)^\mathsf{T}$ points. The inverse observed covariance block is

\Sigma_{OO}^{-1} =\frac{1}{12}\begin{pmatrix}4&-2\\-2&4\end{pmatrix}.

Multiplying this inverse by the residual vector gives $(1/3,1/3)^\mathsf{T}$. Thus the conditional mean at the missing visit is

9+\begin{pmatrix}1&2\end{pmatrix} \begin{pmatrix}1/3\\1/3\end{pmatrix} =9+1=10\text{ points}.

For the conditional variance, first compute $\Sigma_{OO}^{-1}\Sigma_{OM}=(0,1/2)^\mathsf{T}$. Then

4-\begin{pmatrix}1&2\end{pmatrix} \begin{pmatrix}0\\1/2\end{pmatrix} =4-1=3\text{ points}^2.

The prediction uses both observed visits and their covariance. Simply applying separate one-visit adjustments would not, in general, give this result because the observed outcomes are correlated with each other.

Worked Example: Several Missing Visits After Dropout

In a hypothetical three-visit trial, suppose dropout occurs after visit 1. The observed active-arm mean is 10 points and the reference means for visits 2 and 3 are 8 and 7 points. Let the observed visit-1 variance be 4, the missing-visit covariance block be $\begin{pmatrix}4&2\\2&9\end{pmatrix}$, and the covariance between the missing outcomes and visit 1 be $\begin{pmatrix}1\\3\end{pmatrix}$. The participant’s observed visit-1 outcome is 13 points. Assume these blocks form a valid joint covariance model.

The observed residual is $13-10=3$ points. The conditional mean vector for visits 2 and 3 is

\begin{pmatrix}8\\7\end{pmatrix} +\begin{pmatrix}1\\3\end{pmatrix}\frac{1}{4}(3) = \begin{pmatrix}8.75\\9.25\end{pmatrix} \text{ points}.

The conditional covariance matrix is

\begin{pmatrix}4&2\\2&9\end{pmatrix} -\begin{pmatrix}1\\3\end{pmatrix} \frac{1}{4} \begin{pmatrix}1&3\end{pmatrix} = \begin{pmatrix}3.75&1.25\\1.25&6.75\end{pmatrix} \text{ points}^2.

The two future outcomes are predicted jointly: their conditional covariance remains 1.25 point-squared. The conditional mean at visit 3 is above the visit-2 mean in this example even though the reference marginal means decline. That is a consequence of the specified cross-visit associations and the unusually high observed outcome, not a change to the reference means themselves.

Assumptions, Diagnostics, and Alternatives

Before using the formula, verify that the observed history is indexed correctly and that all means and covariance blocks refer to the same visits and outcome scale. The following checks are especially useful:

Covariance compatibility. The full covariance matrix must be symmetric and positive semidefinite; the observed block must be invertible for the displayed inverse formula. A negative conditional variance signals an invalid covariance specification or a calculation error.
History alignment. The conditioning vector must contain the participant’s actually observed outcomes, and the active means must correspond to those visits. Do not include a missing visit among the conditioning values.
Strength and direction of carryover. Inspect the relevant cross-visit covariances or conditional regression coefficients. Strong adjustments can occur when associations are high or the observed covariance block is near singular. Unexpected signs may be mathematically possible but should prompt a review of the covariance model and visit coding.
Distributional fit. The normal model makes the conditional mean and covariance sufficient to define the predictive distribution. Severe skewness, heavy tails, or outcome bounds can make normal predictive draws implausible, even if the covariance calculations are valid.

Conditional carryover differs from two useful alternatives. An unconditional reference prediction draws post-dropout outcomes from the reference marginal distribution and ignores the participant’s observed history. An active-arm continuation instead uses the active-arm post-dropout mean specification. Conditional J2R uses reference-arm means after dropout, while the observed active-arm history can adjust the participant-specific prediction through cross-visit association. These alternatives encode different assumptions; none is selected by the algebra alone.

The method also has limits. Conditional association does not establish that the active treatment’s causal effect persists after dropout. An observed high outcome may reflect stable participant characteristics, measurement variation, treatment response, or other factors captured by covariance. The prediction depends on the assumed reference means, covariance structure, and normality. It is also a predictive distribution, not a deterministic forecast: multiple imputation should represent its uncertainty rather than treating the conditional mean as the participant’s known future outcome.

Common Pitfall

Common mistake: “The participant scored above the active-arm mean, so add that entire difference to every reference mean after dropout.” This ignores how strongly each missing visit is associated with the observed history and can overstate or understate carryover. The adjustment is $\Sigma_{MO}\Sigma_{OO}^{-1}(y_O-\mu_O^A)$, which uses the full covariance structure. The observed outcomes stay fixed as observed values; only missing outcomes are predicted.

A related error is to set the conditional mean equal to the reference mean and then claim that the observed history has been used. Unless the cross-covariance block is zero, conditioning generally changes the mean and the predictive covariance. Conversely, a nonzero adjustment should not be described as replacing the observed measurements or as proof of causal treatment persistence.

Check Your Understanding

Answer each question, then compare with the solutions below.

A participant’s observed outcome equals the active-arm mean at all observed visits. What is the conditional mean for missing visits under the displayed model?
In a one-observed, one-missing visit model, the reference mean is 6, the active mean for the observed visit is 5, the observed value is 7, the observed variance is 4, the missing variance is 9, and the cross-visit covariance is 2. Find the conditional mean and variance.
What happens to the conditional mean when the cross-covariance block is zero?
Why does conditioning on observed history not mean that observed outcomes are imputed?

Solutions. (1) The residual vector $y_O-\mu_O^A$ is zero, so the adjustment is zero and the conditional mean is $\mu_M^R$. (2) The conditional mean is $6+(2/4)(7-5)=7$ outcome units. The conditional variance is $9-2^2/4=8$ outcome-units squared. (3) The adjustment is zero, so the conditional mean equals the reference mean vector. (4) The observed values enter as fixed conditioning values in the predictive distribution; only outcomes in $M$, the missing post-dropout visits, are predicted.