Why an Intermediate Missing Visit Needs Special Treatment
A time-varying covariate is measured repeatedly on the same participant. If a measurement is missing at one visit but recorded at an earlier and a later visit, the missing value is intermittent: there is observed information on both sides of the gap. Those two neighboring measurements can help predict the missing value. They do not, however, turn the visits into independent participants. A model that treats each visit as a separate person can understate dependence and produce imputations that fail to reflect the observed within-person pattern.
The task here is to impute one repeatedly measured covariate in a data set organized by participant and visit. For survival analyses, the resulting values must remain attached to the correct participant and time interval. The survival-record structure and interval splitting established in the earlier material on Survival Records for Chained-Equation Imputation and Splitting Follow-Up Into Imputation Intervals provide the relevant framework.
A scheduled measurement that is missing between observed visits is different from a measurement that was never defined or could not be observed because follow-up had ended. For example, a covariate value after death is not an ordinary missing visit to be filled in. The target must be a meaningful measurement at a time when the participant was under observation and the covariate could have been recorded.
Represent the Visits and Their Timing
Use one record for each participant-visit or participant-interval, following the analysis data structure. Record the participant identifier, visit time, the covariate value, and any relevant predictors. A participant identifier indicates which records belong together; it is not a quantitative covariate whose numerical value should be used as a slope. In the survival setting, a visit measurement should be aligned with the start and end times of the interval for which it is intended to apply.
| Participant | Visit time (months) | Covariate value | Interpretation |
|---|---|---|---|
| A | 0 | 2 | Observed baseline measurement |
| A | 3 | 4 | Observed measurement |
| A | 6 | Missing | Intermediate visit to impute |
| A | 9 | 7 | Observed later measurement |
Worked Example: Locating an Intermittent Gap in Long-Format Data
Suppose participant A has visits at months 0, 3, 6, and 9, with covariate measurements 2, 4, missing, and 7. The missing measurement is at month 6. The immediately preceding observed value is 4 at month 3, and the subsequent observed value is 7 at month 9. Thus, the gap is intermittent and both measurements can enter the conditional imputation model.
There are four visit records for participant A, not four independent participants. If the month-6 measurement applies from month 6 until the next visit at month 9, its imputed value belongs to that interval under the study's measurement convention. It must not be assigned to the month-3 interval merely because the earlier measurement was observed.
Conditional Model and Assumptions
For a continuous covariate, one useful conditional model includes visit time, observed covariates, and the measurements immediately before and after the missing visit. A random participant intercept can represent persistent differences between participants. One illustrative model is
Here \(t_{ij}\) is visit time, \(\mathbf{Z}_{ij}\) contains other observed predictors, and \(b_i\) is a participant-specific random intercept. The equation is an example of a model form, not a universal specification. Neighboring measurements may be unavailable, the covariate may be categorical, or time trends may be nonlinear; the conditional model must suit the covariate and visit process.
The interpretation of the normal model requires that, conditional on its predictors and participant effect, the residual distribution is adequately represented by a normal distribution with suitable variance. The random-intercept specification assumes participant effects follow the stated distribution and are independent of the residuals, conditional on included predictors. If important serial dependence remains after conditioning on these terms, the model needs a richer dependence structure or additional time-history predictors.
For imputation to be justified under a missing-at-random assumption, the probability that a visit value is missing must be explainable by observed information included in the imputation model. That information may include prior and subsequent observed measurements, time, baseline covariates, treatment, and relevant observed outcome or survival information. There must also be adequate overlap: for the predictor patterns represented by missing visits, the model needs enough observed measurements to estimate plausible conditional distributions. These assumptions are not guaranteed by using MICE; they require scientific assessment and diagnostics.
During imputation, draws should reflect uncertainty in both the model parameters and the missing value given those parameters. In a multilevel implementation, participant-level dependence must be represented in the imputation procedure itself. Adding a participant identifier as an ordinary numeric predictor, or using a standard error adjustment only in the later analysis, does not by itself produce appropriate dependent imputations.
Worked Example: A Continuous Intermediate Measurement
Consider a hypothetical fitted conditional model for a biomarker measured in units per litre:
For a missing month-3 value, suppose the observed month-0 value is 4, the observed month-6 value is 8, the time value is \(t_{ij}=3\) in months, and a particular parameter draw gives \(b_i=0\). The conditional mean is
Suppose the residual standard deviation for that parameter draw is 1.2 units per litre. A predictive draw can be written \(5.1 + 1.2z\), where \(z\) is drawn from a standard normal distribution. If the illustrative draw is \(z=-0.5\), the imputed value is \(5.1 + (1.2)(-0.5)=4.5\) units per litre. This is one plausible draw, not a claim that the missing value is known to equal 4.5. In a proper multiple-imputation procedure, parameter uncertainty and the participant effect are also drawn rather than fixed at their illustrative values.
Conditional Models for Different Covariate Types
The outcome model for the missing covariate must match its scale. For a binary covariate, a logistic conditional model can use prior and subsequent values, time, and other observed predictors. If the missing measurement is ordinal or nominal, an ordinal or multinomial model may be more suitable. The same principle remains: condition on information from the same participant and preserve the repeated-measure structure.
Worked Example: A Binary Intermediate Measurement
Suppose a hypothetical binary exposure \(Y_{ij}\) indicates whether a participant used a particular medication at a visit, with 1 meaning yes and 0 meaning no. For illustration, let the fitted conditional model be
If the earlier visit is 1, the later visit is 1, and the missing visit occurs at \(t_{ij}=3\), the linear predictor is
The conditional probability of medication use is therefore
If the illustrative uniform draw is \(u=0.20\), then \(u<p\), so this imputation draw is 1. If instead \(u=0.60\), the draw is 0. The probability is not itself the imputed value; the stochastic draw retains uncertainty about the missing visit.
Why the Two-Sided Predictors Help
Under the conditional model, the missing measurement is predicted from its own time, participant structure, and observed information around the gap. A simple justification is that when both adjacent values are informative about \(Y_{ij}\), conditioning on them can reduce unexplained variation relative to a model using time and baseline variables alone. The predictive distribution still has residual uncertainty; interpolation of a mean is not equivalent to observing the value.
For example, if an individual tends to have similar covariate values at nearby visits, a prior value can indicate the participant's current level, while a subsequent value can indicate whether that level was changing. The model estimates these associations from observed data rather than imposing a fixed interpolation rule. The gain depends on the model being reasonable and on sufficient observed data across the relevant time and covariate patterns.
Using a later measurement as an imputation predictor is appropriate only for constructing a completed data set under the imputation assumptions; it does not relocate that later measurement to an earlier time. The imputed value remains attached to its scheduled visit, and the survival analysis uses covariates according to the time-dependent analysis definition.
Diagnostics, Alternatives, and Limitations
Check whether the imputation model is producing values consistent with the observed data and the repeated-measure structure. Useful checks include comparing observed and imputed distributions by visit time, plotting participant trajectories before and after imputation, and checking whether imputed values fall within plausible clinical ranges. For a binary covariate, inspect imputed proportions by time and by prior measurement. Investigate convergence and whether multiple imputed data sets show reasonable between-imputation variation.
Warning signs include implausibly flat trajectories, abrupt changes not supported by observed data, extreme continuous values, imputed probabilities near zero or one for many records, and strong differences between missingness patterns that the included predictors do not explain. Sparse observations around a gap, long consecutive gaps, or substantial missingness at the beginning or end of follow-up weaken the information available from both sides. A model cannot reliably use a subsequent measurement if none is observed.
The principal alternatives have different trade-offs. Deterministic filling, such as substituting a single predicted mean, omits uncertainty and can distort variability and associations. A row-wise regression that ignores participant clustering treats repeated visits as independent and can yield an imputation distribution that is too narrow or otherwise misspecified. A joint longitudinal model can represent the repeated measurements directly, but may require stronger distributional and computational commitments. Chained-equation imputation offers flexible conditional models, but its validity still depends on suitable model specification and a defensible missing-at-random assumption.
No observed-data diagnostic can prove that the missing-at-random assumption holds, because the unobserved values are not available for direct comparison. If missingness may depend on the unobserved covariate even after conditioning on recorded information, sensitivity analyses are needed. Imputation is also limited when a visit has no comparable observed cases, when the covariate has severe measurement error, or when the conditional distribution is poorly represented by the chosen model.
Common Mistake
- Wrong reasoning: “There are many visit rows, so I can fit an ordinary regression to all rows and treat each as independent.”
- Correction: Rows from one participant share characteristics and may be correlated. Use an imputation model that represents participant-level dependence, and include relevant prior and subsequent measurements without changing which visit each value describes.
- Another wrong reasoning: “The two observed neighbors determine the missing value, so their average is the imputation.”
- Correction: Their values can inform a conditional predictive distribution, but a deterministic average ignores model uncertainty and residual variation. Multiple imputation uses stochastic draws and retains uncertainty across completed data sets.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A covariate is recorded at months 0, 2, 4, and 6 as 3, missing, 5, and 6. Is the month-2 value intermittent, and which observations can directly serve as neighboring predictors?
- Why is participant ID not ordinarily entered as a numeric predictor with a single linear coefficient?
- In a binary imputation model, the predicted probability for a missing visit is 0.35 and the random uniform draw is 0.20. What value is imputed under the usual draw rule?
- Give one reason that a missing value after follow-up ends should not automatically be treated as an intermittent missing visit.
Solutions. (1) Yes. The month-2 value is between observed values at months 0 and 4, so those two measurements are direct neighboring predictors. (2) ID labels identify participants; numerical distances between labels have no substantive meaning, and a single slope does not represent within-participant dependence. (3) The imputed value is 1 because \(0.20<0.35\). (4) The covariate may not be defined or observable after follow-up ends, so there may be no eligible measurement to impute; the target and observation period must be specified first.