Why the Location of Missingness Matters
In survival data, a covariate can describe a subject once at study entry or can change during follow-up. A missing baseline measurement and a missing value at one follow-up interval are therefore different problems: the first leaves an entire subject’s baseline characteristic unknown, while the second leaves one piece of a subject’s evolving history unknown. Treating them as the same kind of missing value can distort both the imputation and the survival analysis that follows.
The distinction is especially important when follow-up has been represented in start-stop intervals. A single subject may then occupy several rows, even though a baseline covariate has only one value. A time-varying covariate, by contrast, may have a different value in each interval. The number of rows is not the number of independent covariate values.
Write \(\tilde T_i=\min(T_i,C_i)\) for observed follow-up, where \(T_i\) is event time and \(C_i\) is censoring time, and let \(\Delta_i=I(T_i\le C_i)\) indicate whether the event was observed. The observed survival record can contribute information to covariate imputation: \(\tilde T_i\) and \(\Delta_i\), or a suitable cumulative-hazard summary such as \(\hat H(\tilde T_i)\), may be associated with the covariate being imputed. That information is useful only when it is included in a model appropriate to the covariate’s level and timing.
Two Missingness Structures
For a baseline covariate, one missing value belongs to a subject, even if that value appears on many interval rows in a long-format data set. The imputation target is \(X_i\), not a separate \(X_{ij}\) for every interval. Once imputed, the same completed baseline value is carried across that subject’s rows.
For a time-varying covariate, each \(Z_{ij}\) belongs to a particular interval. One observed value in a later interval does not make an earlier missing interval value observed, nor does an unknown value in one interval imply that the entire subject’s trajectory is unknown. The interval, the measurements around it, and the survival process relevant to that interval all matter.
| Feature | Baseline covariate | Time-varying covariate |
|---|---|---|
| Index of a value | Subject \(i\) | Subject \(i\), interval \(j\) |
| Meaning of one missing entry | The subject’s baseline value is unknown | The value for one interval is unknown |
| Representation in interval data | Repeated across rows, but still one value | May have a distinct value in each row |
| Relevant survival information | Subject-level outcome information over observed follow-up | Information aligned to the interval and the covariate’s time pattern |
Worked Example: One Baseline Value Across Several Rows
Suppose a hypothetical subject has three start-stop intervals, from 0 to 2, 2 to 5, and 5 to 8 months. The baseline covariate is smoking status at entry; it is missing. The subject’s event is observed at month 8.
In long format, the three rows might each display a blank for baseline smoking status. These blanks do not represent three independent missing smoking measurements. They are three appearances of the same unknown \(X_i\). If an imputation produces “current smoker,” the completed value is current smoker on all three rows. If it produces “not current smoker,” that value is likewise repeated on all three rows.
The subject’s observed event indicator and follow-up summary can help predict baseline smoking status because they summarize how the subject was observed over follow-up. But the event at month 8 does not turn the three row-level blanks into three distinct baseline targets. The imputation unit remains the subject.
What Survival Information Contributes
Imputation models should preserve associations relevant to the analysis. If a covariate is associated with survival, omitting survival information from its imputation model can weaken that association in completed data. Earlier material on using cumulative hazard in covariate imputation motivates including a survival summary alongside event status. For a baseline covariate, the outcome summary is naturally subject-level: the observed event or censoring record belongs to the subject, as does \(X_i\).
A time-varying value requires more care. Its meaning depends on when it applies. In interval data, the covariate value is linked to a specific period at risk, so a survival summary used with that value must be aligned with the period and the intended imputation model. A single terminal summary copied mechanically onto every interval is not automatically an adequate representation of interval-specific information. Nor should one assume that the same predictors or functional form are suitable for baseline and time-varying covariates.
The following formulas are illustrative conditional models, not universal prescriptions. Suppose baseline \(X_i\) is continuous and a linear conditional model is adequate given observed predictors \(W_i\), event status, and a cumulative-hazard summary. One possible specification is:
For a time-varying value, an illustrative model could instead condition on interval-relevant predictors and survival information represented for that interval. For example, \(D_{ij}\) can indicate an event in interval \(j\), and \(\hat H(t_{ij}^{\mathrm{end}})\) can denote a cumulative-hazard summary evaluated at that interval’s end:
The second expression is appropriate only if its conditional normal form and predictors are defensible for the application. It illustrates the different indexing: one \(X_i\) and one subject-level survival record in the first model, versus \(Z_{ij}\) and interval-indexed information in the second. Categorical covariates require suitable categorical conditional models rather than a normal model.
Worked Example: A Baseline Imputation Prediction
Consider a hypothetical continuous baseline marker \(X_i\), measured in units, with a fitted conditional mean model \(E(X_i\mid\Delta_i,\hat H)=10+2\Delta_i+4\hat H\). For a subject with an observed event (\(\Delta_i=1\)) and cumulative-hazard summary \(\hat H=0.30\), the predicted conditional mean is:
This is a conditional mean, not a completed value by itself. In multiple imputation, a value is drawn from a predictive distribution that also reflects residual and parameter uncertainty. For example, if the assumed residual standard deviation were 2 units, a predictive draw would vary around 13.20 rather than assigning every such subject exactly 13.20. The model’s linear form and its predictors must be checked against the data; the arithmetic alone does not establish that the model is suitable.
Worked Example: An Interval-Specific Imputation Prediction
Now consider a hypothetical time-varying continuous covariate \(Z_{ij}\), in units, with an illustrative conditional mean \(E(Z_{ij}\mid Z_{i,j-1},D_{ij},\hat H(t_{ij}^{\mathrm{end}}))=1+0.5Z_{i,j-1}+3D_{ij}+2\hat H(t_{ij}^{\mathrm{end}})\). For one interval, suppose the previous interval’s value was 4 units, no event occurred in this interval (\(D_{ij}=0\)), and the interval-end cumulative-hazard summary is 0.10. Then:
The value 3.20 is the model-based conditional mean for that interval, not a claim that the subject’s covariate is known to equal 3.20. An imputation draw would include predictive uncertainty. In this example, the interval and the preceding measurement enter explicitly. This does not mean the displayed model is always the right choice: its time structure, conditional distribution, and survival predictors need substantive and empirical justification.
Assumptions, Diagnostics, and Common Failures
The missingness mechanism and the imputation model are separate considerations. Under a missing-at-random assumption, the probability that a value is missing may depend on observed information included in the analysis, but not on the missing value after conditioning on that information. For baseline missingness, this concerns the subject-level indicator \(R_i^X\); for time-varying missingness, it concerns the interval-level indicator \(R_{ij}^Z\). Neither assumption can be confirmed from observed data alone. If missingness still depends on the unseen value after conditioning on observed data, a missing-at-random analysis may be inadequate.
Other requirements are also substantive. The conditional model should represent the covariate’s distribution reasonably well, include observed predictors that explain both the covariate and its missingness, and respect the range and type of the variable. There must be enough observed data to estimate the model in the relevant groups or intervals. For time-varying covariates, the recorded interval must correspond to the period for which the value is intended to apply. A technically converged model cannot repair a mistaken time alignment or a lack of observed support.
Diagnostics should reflect the level at which each variable is defined. For a baseline covariate, count missing subjects, not blank repeated cells; check that completed values are constant across each subject’s interval rows; and compare observed and imputed distributions by event status and relevant survival summaries. For a time-varying covariate, inspect missingness and completed values by interval, check plausible ranges and within-subject patterns, and verify that each imputed value is assigned to the intended interval. Also examine whether the number of observed values becomes too small in particular intervals to support the chosen model.
A useful justification for treating baseline missingness once per subject follows from its data structure. The same baseline value is a single quantity \(X_i\); repeating it in long format does not create additional independent measurements. If an analyst instead treats each of \(m\) rows as a separate baseline target, the model can effectively count the same subject repeatedly. In a simplified setting where each row were incorrectly treated as an independent contribution with conditional density \(f(X_i\mid O_i)\), the contribution would be:
rather than the single subject-level contribution \(f(X_i\mid O_i)\). The repeated-row assumption is false because the displayed baseline values are the same quantity, not independent draws. The practical correction is to impute at the subject level and then carry the completed value onto that subject’s rows.
A common mistake is to see three missing entries for a baseline covariate in three interval rows and conclude that the subject has three missing values that should be imputed separately. That reasoning confuses the data layout with the measurement level. The correction is to identify the scientific variable first: if it was measured once at baseline, impute one subject-level value, then replicate it in the interval representation. For a time-varying covariate, retain interval indexing rather than collapsing all blanks into one subject-level missingness indicator.
How This Differs From Simpler Alternatives
Complete-case analysis removes subjects with a missing covariate and can waste information; its validity depends on the missingness mechanism and analysis target. Single-value replacement, such as filling every blank with a mean, fails to represent imputation uncertainty and can distort associations. For time-varying data, carrying a prior value forward may also misrepresent genuine change. These alternatives do not resolve the central distinction: baseline values belong to subjects, whereas changing covariate values belong to subject-intervals.
A chained-equation approach can accommodate different conditional models for different variable types, but it does not automatically respect the survival record’s structure. The imputation target, predictors, and row-level dependence must still be specified correctly. Nor does adding event status and a hazard summary guarantee a valid model: the summaries must be relevant, the model must be sufficiently supported by observed data, and the missingness assumptions must be plausible. When diagnostics reveal poor overlap, sparse intervals, implausible completed values, or inconsistent baseline values across rows, the imputation specification needs attention rather than more iterations alone.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A baseline covariate is missing in four interval rows for one subject. How many distinct baseline values are missing, and what should happen after imputation?
- One time-varying measurement is missing in interval 2, while the same subject has observed values in intervals 1 and 3. Is the whole subject’s covariate missing?
- Why can event status and a cumulative-hazard summary be useful predictors when imputing a baseline covariate?
- A fitted interval-specific model gives a conditional mean of 5.5 units and residual standard deviation 1.2 units. Is 5.5 necessarily the imputed value?
- Name one diagnostic that is specific to the repeated-row representation of baseline covariates and one relevant to time-varying covariates.
Solutions. (1) One baseline value is missing. Impute it once for the subject and repeat that completed value across all four rows. (2) No. The missing target is the value for interval 2; the subject has observed values in the other intervals. (3) They summarize the subject’s observed survival experience, which may be associated with the covariate; including them can help preserve that relationship, provided the model and assumptions are appropriate. (4) No. It is a conditional mean. A multiple-imputation draw should reflect predictive uncertainty, including residual uncertainty. (5) For a baseline covariate, verify the completed value is identical across a subject’s interval rows. For a time-varying covariate, check values and missingness by interval and confirm each value is assigned to the intended interval.