Diagnosing Imputed Survival Covariates Across Follow-Up

Why Diagnose Covariates in Survival Time

A survival covariate is not just a column of values. Its meaning depends on when it was measured and whether the person was still under observation and at risk at that time. A plausible imputation for a covariate measured early in follow-up may be implausible later; a distribution pooled across all visits can conceal that problem. Diagnostics should therefore compare observed and imputed values in time-specific groups and, where relevant, in the risk sets used by the survival analysis.

These comparisons are checks on the behavior of the imputation model, not tests that certify it as correct. Observed values and missing values can differ for legitimate reasons, including chance, covariate-dependent missingness, and differences in the people still under follow-up. The goal is to find patterns that merit investigation while preserving the time ordering and eligibility rules of the analysis.

For a person with entry time \(L_i\), event or censoring time \(T_i\), and event indicator \(\delta_i\), define observed follow-up as \(\tilde T_i=\min(T_i,C_i)\) when \(C_i\) is the censoring time. Using the convention that a person is at risk immediately before an event at time \(t\), risk-set membership is

The strict inequality at entry and inclusive inequality at exit are a stated time convention; a discrete-time analysis may instead define eligibility by its interval-specific rules. Apply the convention used by the analysis consistently. A covariate value should be compared at the time to which it belongs, not assigned to an interval merely because it appears on the same row after data reshaping.

What to Compare, and Why

For each follow-up interval or meaningful time band, tabulate the number of observed and imputed values and compare their distributions. For a continuous covariate, useful summaries include means, medians, quantiles, ranges, and plots of the distribution. For a binary or categorical covariate, compare category counts and proportions. Examine these summaries separately by event status when that distinction is relevant and well represented. Then inspect the risk-set summaries at selected analysis times, using only records eligible to contribute at those times.

An observed–imputed comparison is most informative when the groups are comparable on predictors of the covariate and its missingness. Under a missing-at-random assumption conditional on observed information \(Z\), the indicator \(R\) of observation is conditionally independent of \(X\) given \(Z\). With positive probability of both observation and missingness at the relevant \(Z\) values, this implies:

This gives a justification for looking for compatibility between observed and imputed distributions after accounting for relevant information. In practice, plots and tables use finite samples and often broad strata, so exact agreement is not expected. Nor does marginal agreement establish the conditional relationship above: a model can match an overall mean while missing a time trend, subgroup difference, or tail behavior.

Where several imputations were generated, calculate the diagnostic summaries within each completed data set. Look for a pattern that is stable across imputations, and note whether conclusions change materially from one completed data set to another. Do not concatenate the \(m\) completed data sets as if they contained \(m\) independent copies of the people; that would give artificial weight to the same individuals. Nor should the imputed values from different draws be averaged into a single completed data set for these checks.

Event Status and Risk-Set Checks

Event status can help reveal whether imputed covariates behave differently among people who experience an event and those censored without an observed event. This is a diagnostic stratification, not a claim that event status defines a causal subgroup. Interpret any difference in light of the variables used in imputation and the different follow-up histories represented in each group. A difference may be expected; an unexplained reversal, implausible concentration, or systematic absence of variation deserves investigation.

Risk-set checks answer a different question: among records eligible to contribute at a particular time, do the observed and imputed covariate values have plausible distributions? This matters because people leave the risk set over time through events, censoring, or other exit rules, and some may enter after time zero. A changing risk-set distribution can arise from this changing membership, even when each person's covariate trajectory is reasonable. Do not interpret that compositional change by comparing risk-set summaries as if they were repeated measurements on an unchanged group.

Diagnostics, Alternatives, and Limits

Use tables and visual displays to make the comparisons easy to inspect. A compact table can report the number observed, number imputed, and separate summaries for each time-by-status stratum. For continuous covariates, overlay or juxtapose observed and imputed distributions using histograms, density displays, or boxplots, while retaining the time and status labels. For categorical covariates, show counts as well as proportions; the denominator is essential, especially in sparse cells. Repeat the check for several completed data sets rather than relying on a single draw.

Warning patterns include an imputed distribution outside the plausible range, a sharp change unsupported by observed trajectories, a trend that reverses across time, unexpected pile-up at a boundary, or large differences concentrated in an event-status or risk-set stratum. Check first that the units and measurement occasions match, that risk-set membership is correct, and that the counts are adequate. Then investigate whether the imputation model includes the relevant time structure, predictors, and survival information. An unusual pattern can also reflect sparse data or genuine selection into the risk set, so it is not automatically a modeling failure.

These descriptive checks complement, rather than replace, other approaches. Reviewing observed data alone cannot reveal the distribution of unobserved values. Model-based predictive checks ask whether simulated or imputed data reproduce features targeted by the model, but depend on the chosen model and checking statistics. Comparing survival-model estimates across completed data sets assesses inferential stability, not whether every imputed covariate trajectory is plausible. Formal tests of observed–imputed differences are generally a poor default: the groups are not randomly assigned, the imputed values depend on the observed data, and many time-by-status comparisons create opportunities for chance discrepancies.

Finally, diagnostics cannot verify the unobserved values directly or prove the missingness assumption. Similar summaries may coexist with incorrect conditional relationships, while different summaries may be justified by measured predictors. Use clinical and measurement knowledge to assess plausible ranges and trajectories, and use the diagnostics to identify specific features requiring explanation. A covariate value assigned after a person has left observation or after an event when it should no longer be defined is a data-structure problem, not something a reassuring pooled mean can fix.

A Short Derivation for Conditional Comparisons

The conditional comparison follows directly from the MAR condition. For a covariate value \(x\) and observed predictors \(z\), MAR states that observation status adds no information about \(X\) once \(Z\) is known. In conditional-distribution notation, \(X\) and \(R\) are independent given \(Z\). Therefore, for predictor values where both observed and missing records occur,

If an imputation procedure correctly represents this conditional distribution, its draws for records with \(R=0\) should reproduce the relevant conditional distribution over repeated draws, apart from finite-sample and Monte Carlo variation. This is a justification for checking conditional patterns, not a proof that a particular imputation model is correct. If important predictors are omitted, the chosen time strata are too broad, or the model is incompatible with the data structure, the practical comparison may fail to expose the problem.