Intermittent Missing Visits Before Permanent Dropout

Why the Timing of Missingness Matters

A missing measurement does not by itself establish that a participant has dropped out. A visit may be missed and followed by a later observed visit; that is an intermittent missing measurement. Permanent dropout has a different pattern: after some visit, no further outcomes are observed. In a reference-based analysis, this distinction determines which predictive assumption applies to each missing value.

Applying a post-dropout reference rule to every missing measurement can therefore change the assumed data distribution. It can assign reference behavior to an outcome at a visit that occurred before the participant permanently left the study, even when later treatment-arm outcomes were observed. The construction below keeps the two kinds of missingness separate.

Definition: Let $Y_{ij}$ denote participant $i$'s outcome at visit $j$, and let $R_{ij}=1$ when that outcome is observed and $R_{ij}=0$ when it is missing. For a participant with at least one missing visit, define $d_i$ as the first visit such that $R_{ij}=0$ for every $j\ge d_i$. Visits $j<d_i$ with $R_{ij}=0$ are intermittent missing visits if a later visit before $d_i$ is observed. Visits $j\ge d_i$ are post-dropout missing visits. The first missing visit is not necessarily the dropout visit.

For example, the sequence $(1,0,1,0,0)$ has an intermittent gap at visit 2, followed by permanent dropout beginning at visit 4. By contrast, $(1,0,0,0,0)$ has dropout beginning at visit 2. If the only missing outcome is at the final scheduled visit, there is no later visit with which to distinguish a one-time missed visit from permanent dropout. That case needs an operational pattern definition; it cannot be classified from a subsequent observed outcome.

Prediction in Two Stages

Suppose a participant is assigned to the treatment arm and has intermittent missing visits before dropout. Let $O_i$ be the observed outcomes before dropout, $M_i$ the missing outcomes before dropout, and $P_i$ the outcomes at and after dropout. A common reference-based construction has two parts: predict $M_i$ from the treatment-arm outcome model, then predict $P_i$ using the selected reference-based restriction, conditional on the participant’s complete history through the pre-dropout period.

Key construction: For a treatment-arm participant with dropout visit $d_i$, the predictive distribution is factored as $$ p(Y_{i,M_i},Y_{i,P_i}\mid Y_{i,O_i},A_i=T,d_i) = p_T(Y_{i,M_i}\mid Y_{i,O_i}) \, p_g(Y_{i,P_i}\mid Y_{i,O_i},Y_{i,M_i},d_i). $$ Here $T$ denotes the treatment arm, and $g$ denotes the chosen post-dropout rule, such as jump-to-reference (J2R) or copy-reference. The first factor predicts intermittent gaps under the treatment-arm model. The second applies the selected reference-based conditional distribution only to outcomes from dropout onward.

The factorization expresses a coherent sequence of predictions, not two unrelated analyses. The post-dropout prediction can depend on an intermittent value that was itself predicted in the first stage. In particular, if $Y_{i,M_i}$ is uncertain, the predictive distribution for $Y_{i,P_i}$ must account for that uncertainty. Substituting a single predicted mean for the intermittent value generally understates uncertainty.

This construction relies on assumptions. The treatment-arm model must describe the relevant pre-dropout outcome distribution, including the dependence among visits used to predict an intermittent gap. The usual missing-at-random assumption for intermittent missingness is that, conditional on the observed information in the model, whether a pre-dropout measurement is missing does not additionally depend on its unobserved value. This does not imply that missingness is unrelated to observed outcomes. The post-dropout distribution is not identified from the participant’s unobserved outcomes; it is specified by the sensitivity restriction $g$. These assumptions are distinct and should be reported separately.

Under a multivariate normal model, conditional predictions are obtained from the appropriate joint mean vector and covariance matrix. If a missing outcome $Y_M$ and observed outcomes $Y_O$ have joint mean vectors $\mu_M,\mu_O$ and covariance blocks $\Sigma_{MM},\Sigma_{MO},\Sigma_{OO}$, then

Y_M\mid Y_O=y_O \sim N\!\left( \mu_M+\Sigma_{MO}\Sigma_{OO}^{-1}(y_O-\mu_O), \; \Sigma_{MM}-\Sigma_{MO}\Sigma_{OO}^{-1}\Sigma_{OM} \right).

For intermittent gaps, the parameters in this calculation come from the treatment-arm model. For post-dropout outcomes, use the conditional distribution prescribed by the chosen reference-based rule. For J2R, that post-dropout conditional distribution follows the reference-arm predictive distribution. Copy-reference instead specifies a reference-based post-dropout evolution that carries forward the treatment-arm level at dropout. In either case, the rule begins at dropout; it does not retroactively redefine intermittent pre-dropout outcomes.

Worked Example: Classifying the Missing-Visit Pattern

Consider a hypothetical treatment-arm participant scheduled for five visits, with observation indicators $(1,0,1,0,0)$. Identify the intermittent and post-dropout missing visits.

The first missing visit is visit 2, but visit 3 is observed. Thus visit 2 is intermittent missingness, not permanent dropout. Visit 4 is the first visit after which all scheduled outcomes are missing, so $d_i=4$.

The sets are $M_i=\{2\}$ and $P_i=\{4,5\}$. The observed pre-dropout outcomes are at visits 1 and 3, so $O_i=\{1,3\}$. The prediction for visit 2 should use the treatment-arm model conditional on the observed pre-dropout outcomes. A reference-based rule is then applied to visits 4 and 5, conditional on the history that includes the imputed visit 2 and observed visit 3.

Conditional Prediction for an Intermittent Gap

A key practical detail is that all observed pre-dropout outcomes can contribute to the prediction of an intermittent gap. Conditioning only on the outcome immediately before the gap may discard information from later observed visits. The treatment-arm conditional model should use the observed history specified by the longitudinal model.

Worked Example: Predicting a Pre-Dropout Missing Visit

Suppose a hypothetical treatment-arm participant has observed outcomes $Y_{i1}=11$ and $Y_{i3}=14$, while visit 2 is missing and visit 3 is observed. Under a fitted multivariate normal treatment-arm model, assume the resulting conditional distribution is $$ Y_{i2}\mid Y_{i1}=y_1,Y_{i3}=y_3,A_i=T \sim N\!\left(12+0.2(y_1-10)+0.3(y_3-13),\,6.25\right). $$ The outcome is measured in points.

Substituting the observed values gives $$ E(Y_{i2}\mid Y_{i1}=11,Y_{i3}=14,A_i=T) = 12+0.2(11-10)+0.3(14-13) = 12+0.2+0.3 = 12.5\text{ points}. $$ The conditional variance is $6.25$ points squared, so the conditional standard deviation is $\sqrt{6.25}=2.5$ points. Thus $12.5$ is the model-based conditional mean, not a known value for the missing outcome. A predictive draw, when required, must retain the conditional spread as well as the mean.

The assumption check here is specific: the stated conditional distribution must be the treatment-arm model’s conditional law for visit 2 given visits 1 and 3, and the intermittent missingness assumption must be defensible given the observed information. The observed visit 3 makes visit 2 intermittent; it does not make visit 2 a post-dropout outcome.

Carry Uncertainty into the Post-Dropout Prediction

After imputing an intermittent gap, the post-dropout conditional distribution is evaluated using the resulting complete pre-dropout history. If the intermittent outcome is not fixed at one draw, the post-dropout predictive distribution averages over its uncertainty. The law of total expectation and the law of total variance give a transparent way to see the consequences.

Worked Example: A J2R Prediction After an Intermittent Gap

Continue with the hypothetical participant above. Suppose dropout begins at visit 4. Let the treatment-arm conditional distribution for the missing visit be $Y_{i2}\mid Y_{i1}=11,Y_{i3}=14\sim N(12.5,6.25)$, as in the preceding example. Suppose the reference-arm conditional model used for J2R is $$ Y_{i4}\mid Y_{i1},Y_{i2},Y_{i3} \sim N\!\left( 15+0.1(Y_{i1}-10)+0.25(Y_{i2}-12)+0.5(Y_{i3}-13), \,4 \right). $$ The variance $4$ is the conditional variance under that reference predictive model.

Given a value of the missing visit 2, the conditional mean at visit 4 is $$ 15+0.1(11-10)+0.25(Y_{i2}-12)+0.5(14-13) = 15.6+0.25(Y_{i2}-12). $$ Averaging this conditional mean over the treatment-model prediction for visit 2 gives $$ E(Y_{i4}\mid Y_{i1}=11,Y_{i3}=14) = 15.6+0.25(12.5-12) = 15.6+0.125 = 15.725\text{ points}. $$

The total predictive variance includes both the reference-model residual variance and variation transmitted from visit 2: $$ \operatorname{Var}(Y_{i4}\mid Y_{i1},Y_{i3}) = 4+(0.25)^2(6.25) = 4+0.390625 = 4.390625\text{ points}^2. $$ The predictive standard deviation is approximately $\sqrt{4.390625}=2.095$ points. This calculation assumes the stated conditional reference model and treatment-arm prediction are used in sequence. Using $Y_{i2}=12.5$ as if it were known would give the same conditional mean but variance $4$, omitting the uncertainty due to the missing visit.

Justification and Practical Checks

The two-stage formula follows from the conditional probability chain rule. For any joint distribution of the missing blocks given the observed data,

p(Y_M,Y_P\mid Y_O) = p(Y_M\mid Y_O)\,p(Y_P\mid Y_M,Y_O).

The reference-based construction assigns the treatment-arm model to the first conditional factor and the selected post-dropout restriction to the second. Consequently, a value at an intermittent visit is generated before it enters the conditioning history for post-dropout outcomes. This is why applying the post-dropout rule directly to every missing cell is not equivalent to the staged construction.

Several checks help detect a misapplied or poorly fitting model:

Check the visit sequence. Tabulate observation indicators by visit and verify that each intermittent gap is followed by an observed visit before the first permanent-dropout visit.
Check the model’s conditioning information. For each gap, confirm that the treatment-arm prediction conditions on the observed pre-dropout outcomes intended by the model, including later observed visits where appropriate.
Check predictive behavior. Compare model-based conditional predictions with observed outcomes in similar histories, and examine whether predicted values or spreads are implausible in the outcome’s units.
Check reference support. The post-dropout conditional predictions rely on reference-arm information for comparable histories. Sparse or poorly overlapping histories can make those predictions unstable or extrapolative.
Check uncertainty propagation. Ensure that variation in imputed intermittent values is carried into post-dropout predictions rather than replaced everywhere by a conditional mean.

The main alternative is a monotone-dropout construction, in which missingness begins once and continues for all later visits. It is adequate only when the observed pattern is monotone or when the analysis explicitly reduces the data to such a pattern. Another alternative is to treat every missing value under one common missing-data assumption. That may be a deliberate analysis choice, but it is not the same as applying a reference-based rule only after permanent dropout. The staged approach is useful precisely when an observed later visit distinguishes an intermittent gap from dropout.

The construction has limits. It does not determine the post-dropout distribution from observed data; the reference-based restriction is a sensitivity assumption. Nor does a correct pattern classification guarantee a correct outcome model. Misspecified means or covariance, weak reference-arm support, or a questionable missing-at-random assumption for intermittent gaps can all produce misleading predictions. These limitations should be reported rather than hidden by the numerical detail of an imputation.

Common Mistake

Pitfall: “Visit 2 is missing, so it is after dropout and should use the reference-arm rule.” This reasoning confuses the first missing visit with the first visit after which all measurements are missing. If visit 3 is observed, visit 2 is intermittent. Predict it under the treatment-arm model, then apply the chosen reference-based restriction only from the actual dropout visit onward.

Check Your Understanding

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

A participant’s five-visit pattern is $(1,1,0,1,0)$. Which visit is intermittent missing, and when does permanent dropout begin?
Why should an intermittent visit generally be predicted using the treatment-arm model rather than a post-dropout reference rule?
Suppose $Y_M\mid Y_O$ has variance $9$, and $Y_P\mid Y_M,Y_O$ has conditional mean $20+0.4Y_M$ and residual variance $5$. What is the predictive variance of $Y_P\mid Y_O$?
What can be concluded from a missing outcome at the final scheduled visit when there is no later measurement?

Solutions. (1) Visit 3 is intermittent because visit 4 is observed; permanent dropout begins at visit 5. (2) The later observed visit establishes that the participant had not permanently dropped out at the intermittent gap. The staged construction therefore uses the treatment-arm model before dropout and reserves the reference-based restriction for dropout onward. (3) By the law of total variance, $\operatorname{Var}(Y_P\mid Y_O)=5+(0.4)^2(9)=5+1.44=6.44$. This includes uncertainty transmitted from $Y_M$. (4) The pattern alone cannot distinguish a one-time missed final visit from permanent dropout, because there is no subsequent visit to establish whether observation resumed. An analysis must state its operational classification.