Cause-Specific Hazards as Imputation Predictors

Why Include Competing-Risk Information?

When a covariate is incomplete, an imputation model should preserve its relationships with the other observed information relevant to the analysis. In survival data, that information can include whether an event occurred, which cause occurred, and how long the participant remained under observation. With competing risks, a single event indicator or follow-up time may not distinguish the different event processes. Cause-specific event indicators and cumulative-hazard summaries provide a compact way to represent those outcome features as predictors in an imputation model.

These summaries are predictors, not imputed outcomes and not causal effects. They do not by themselves guarantee that an imputation model is correctly specified or that missingness is ignorable. Their purpose is to make relevant observed survival information available to the conditional models used by MICE.

Definitions and Construction

Suppose each participant is followed from a common time origin until the first event or censoring. Let \(Y_i\) be the observed follow-up time for participant \(i\), and let \(J_i\) identify the observed outcome: \(J_i=k\) if cause \(k\) occurs, and \(J_i=0\) if the participant is censored before an event. For \(K\) mutually exclusive causes, define one indicator for each cause. A censored participant has zero for every cause indicator.

For observed event times \(t_1,\ldots,t_m\), let \(Y(t_j)\) be the number at risk immediately before \(t_j\), and \(d_k(t_j)\) the number of cause-\(k\) events at that time. The Nelson–Aalen estimate and an individual’s cumulative-hazard predictor evaluated at observed follow-up are:

In practice, risk sets must reflect the study’s entry and exit rules. With delayed entry, for example, a participant contributes to a risk set only after entering follow-up. Tied event times require a consistent convention for defining the risk set. The earlier material on left truncation and cumulative-hazard calculations addresses those risk-set details.

The predictor set for an individual can include the vector \((D_{i1},\ldots,D_{iK})\) and the corresponding vector \((\widehat H_{i1},\ldots,\widehat H_{iK})\). The indicators identify the observed event type; the cumulative hazards encode how the observed follow-up time lies within each cause-specific event process. They are complementary rather than interchangeable.

Assumptions and a Short Justification

The construction is most interpretable when event causes are recorded correctly and are mutually exclusive for the first event under study, follow-up times share a well-defined origin, and the risk sets correctly represent who remains under observation and event-free. Estimation of the underlying cause-specific cumulative hazards also relies on censoring being independent of event occurrence, at least conditional on relevant observed information. If censoring is informative and not adequately accounted for, the estimated hazards may not summarize the event process of interest reliably.

As imputation predictors, these summaries are useful when the observed event information helps explain the incomplete covariate. A standard MAR formulation says that, conditional on the observed data included in the imputation model, the probability that a covariate is missing does not depend on its unseen value. Including event type and follow-up information can make that conditioning set more informative. It does not establish MAR, nor does it remove the need for a suitable conditional model for each incomplete variable.

The Nelson–Aalen form follows from accumulating estimated hazard increments. At an event time \(t_j\), the observed cause-\(k\) event fraction among those at risk is \(d_k(t_j)/Y(t_j)\). Summing these increments over times up to \(t\) gives an estimator of the cumulative hazard:

This is a step-function summary: it increases only at observed event times for cause \(k\). A participant’s predictor is obtained by evaluating that estimated function at the participant’s observed follow-up time. It is therefore a summary of the observed event history, not a prediction of that participant’s counterfactual event time.

Using the Summaries in MICE

When a covariate \(X\) is incomplete, its conditional imputation model can include observed covariates, the cause indicators, and the cause-specific cumulative-hazard summaries. The model depends on the type of \(X\): for a binary covariate, for example, a logistic conditional model may be used. In each MICE iteration, the fitted conditional model supplies a predictive distribution for missing values; a value is drawn from that distribution rather than replaced by a fitted mean or probability.

The numerical example below illustrates the predictor calculation for a binary \(X\). Coefficients are hypothetical and are treated as already estimated. This illustration does not claim that the model is correctly specified; that requires substantive and diagnostic assessment.

Model Choices, Diagnostics, and Limitations

Include information for every relevant cause rather than collapsing distinct causes into a single event indicator when their relationships with the incomplete covariate may differ. With \(K\) causes, this generally means \(K\) cause indicators and \(K\) cumulative-hazard summaries. If a cause is very rare, however, its indicator or hazard summary may be too sparse to estimate reliably. In that situation, consider a scientifically defensible simplification and examine whether conclusions are sensitive to it.

The predictors can be entered as separate terms, but a linear term is not automatically adequate. Consider whether the relationship between a continuous incomplete covariate and a cumulative-hazard summary is plausibly linear on the model’s link scale. Flexible terms may be appropriate when supported by the data. Also examine possible redundancy: event indicators, follow-up time, and cumulative hazards can be strongly associated. Near-collinearity can destabilize coefficients even though these variables encode distinct aspects of the observed outcome.

Useful checks include confirming that event indicators match the recorded causes, checking that cumulative-hazard summaries are nonnegative and nondecreasing with follow-up time, and reviewing the distribution of each summary by event type and missingness status. Inspect model convergence, extreme fitted probabilities, sparse event-by-predictor patterns, and whether the imputed covariate distributions preserve plausible relationships with event information. A predictor that is constant, nearly constant, or numerically unstable contributes little and can make fitting harder.

A common alternative is to include only follow-up time and a single event indicator, or only a categorical outcome type. These may be simpler, but can discard differences among cause-specific processes or fail to represent how the estimated event experience changes over time. Conversely, adding every event-time detail is not always better: a large set of event-time indicators may be sparse and unstable. Cause-specific cumulative hazards offer a compact summary, but they can smooth over features of the event process. Compare plausible specifications using substantive knowledge and diagnostics rather than treating one encoding as universally sufficient.

• Not an event-time imputation method. These predictors summarize observed outcome information; they do not generate or recover a censored event time.
• Not a correction for informative censoring by themselves. If censoring depends on unobserved event risk, the summaries may be biased unless the censoring mechanism is addressed.
• Not a guarantee of compatibility. Adding these predictors does not ensure that all chained conditional models arise from a single joint distribution.
• Not causal variables. A coefficient for a cause indicator in a covariate imputation model describes a conditional association, not a causal effect of the event.