Checking Survival Predictors in Covariate Imputation

Why Check Survival Predictors After Imputation?

When a covariate is incomplete in a survival analysis, its imputation model should retain information about the observed survival outcome. Event status and cumulative-hazard summaries can carry different information: event status distinguishes observed events from censoring, while cumulative hazard summarizes the event experience associated with follow-up. Including them as predictors is not enough by itself. The imputed values should also show sensible relationships with those predictors.

This tutorial focuses on checking those relationships after chained-equation imputation. The checks are diagnostic, not a test that proves the imputation model is correct. In particular, close agreement between observed and completed-data associations cannot establish that the missingness mechanism is ignorable.

The checks assume that event status, follow-up time, and the hazard summaries were constructed consistently from the observed survival records, and that the selected cumulative-hazard evaluation times are appropriate for the analysis. Estimating a population cumulative hazard also relies on suitable censoring assumptions for the estimator used. The imputation model must be specified for the covariate type, include the survival predictors needed by the analysis, and represent important nonlinearities or interactions when they are plausible. Conventional MICE inference also relies on an adequate missingness assumption, commonly that missingness is conditionally independent of the missing values given the variables in the imputation model. Diagnostics cannot verify that assumption.

What Relationships Should Be Examined?

Start with the scientific and statistical structure of the analysis. If a covariate is expected to be associated with event occurrence, follow-up experience, or both, examine whether that relationship appears in the completed data. Useful summaries include event proportions by covariate group, differences in mean cumulative hazard by group, and regression relationships between a continuous covariate and survival predictors. Use the same definitions and scales across observed and completed records.

Comparisons need care. The observed covariate values form a selected subset whenever missingness is present, so their associations need not equal the associations in the full population. Compare observed records with imputed records as a diagnostic of possible incompatibility, not as two random samples that must match exactly. Also examine each completed dataset, rather than relying only on a single average: averaging can hide an imputation that produces an implausibly weak or strong relationship.

For a continuous covariate, compare its distribution across event-status groups and examine its relationship with cumulative hazard. A difference in group means can be a transparent first check, but it is not a substitute for checking distributions or conditional associations. If the relationship is nonlinear, a single slope or mean difference may conceal important departures. When the substantive analysis includes interactions or nonlinear terms, the diagnostic should look for their preservation as well.

A Conditional-Mean Justification

The reason to check both event status and cumulative hazard can be made explicit using a simple model. Suppose, for illustration, that a continuous covariate has conditional mean

where \(Z_i\) denotes other predictors. This linear form is only an illustration; the appropriate imputation model depends on the covariate type and data structure. At fixed \(H_i(t_k)\) and \(Z_i\), the model implies a difference of \(\beta_\delta\) in the conditional mean of \(X_i\) between event and censored records. At fixed event status and \(Z_i\), changing cumulative hazard from \(h_1\) to \(h_2\) changes the conditional mean by \(\beta_H(h_2-h_1)\).

If the imputation model omits a survival predictor whose conditional association with \(X_i\) is nonzero, it cannot represent that part of the conditional mean. In the simple case with no \(Z_i\), omitting the outcome predictors makes the imputed draws independent of them under the imputation model. Consequently, among records whose \(X_i\) is imputed, the model may fail to reproduce the association present in the data. This is why post-imputation checks should consider both the association in imputed records and the overall completed-data association.

Diagnostics, Alternatives, and Failure Modes

Use several views of the imputed data rather than a single number. For each completed dataset, inspect covariate distributions separately for observed and imputed records; compare event proportions or cumulative-hazard summaries across covariate groups; and check whether these patterns vary substantially across imputations. For iterative algorithms, inspect trace summaries such as the mean or proportion of imputed values over iterations. A stable trace does not establish model adequacy, but a persistent trend or chains settling at markedly different levels is a warning that the draws may not have stabilized.

Also look for signs that the imputation model is being asked to extrapolate. If one covariate group has very few events, or if particular combinations of event status and cumulative hazard are absent among records with observed covariates, estimates for those combinations may be weakly supported. Extreme imputed values, implausible category frequencies, or sharply different associations among imputations are further reasons to investigate. Recheck predictor coding, the treatment of nonlinear relationships, the choice and construction of hazard summaries, and whether the model has enough information for the patterns of missingness.

The main alternatives are not interchangeable. Omitting survival information is simpler but can lose outcome-related structure. Including event status alone preserves a status association but may not represent variation in follow-up experience. Including a cumulative-hazard summary alone may fail to distinguish events from censoring adequately. Including both, when justified by the analysis and correctly specified, is a common strategy; it still requires diagnostics. A substantive joint-modeling approach is another broad alternative, but it involves a different modeling framework and is not validated merely by passing the checks described here.

These diagnostics have important limits. The observed subset may differ systematically from the records with missing covariates, so exact agreement is not generally expected. Marginal agreement can also hide a distorted conditional association, and agreement in one completed dataset can hide instability in others. Most importantly, no plot, summary, or association check can establish that the missingness mechanism is ignorable or that unmeasured predictors are irrelevant.