Survival and Time-Varying Data · Tutorial 389 of 497

Pairing Event Status With Cumulative Hazard

Learn how event status and cumulative hazard jointly inform a missing covariate, and how to recognize the distortions that can arise when either is omitted.

Advanced 12 min read

What You'll Learn

Derive the survival-likelihood contribution to a missing covariate’s conditional distribution.
Distinguish the information carried by event status from that carried by cumulative hazard.
Calculate conditional odds using both pieces of survival information.
Identify how omitting event status or cumulative hazard can distort imputation.
Recognize assumptions and limitations of using these predictors in a MICE model.

Why the Two Survival Predictors Belong Together

When a covariate is missing, its observed associations with other variables may not be enough to impute it well. Survival outcomes provide additional information: an event and a censoring time constrain plausible covariate values in different ways, and the length of observed follow-up matters as well. For this reason, a covariate imputation model for survival data generally needs both an event indicator and cumulative-hazard information evaluated at the person’s observed follow-up time.

These are distinct pieces of information. Event status indicates whether the observed follow-up ended in an event or censoring. Cumulative hazard summarizes the amount of event risk accumulated by the end of that follow-up. One cannot generally replace the other with no loss of information. The calculation below shows how they enter together under a proportional-hazards model.

Definition: For individual $i$, let $t_i$ be the observed follow-up time and let $\delta_i=1$ if the event was observed and $\delta_i=0$ if follow-up was censored. Let $H_0(t)$ be a baseline cumulative hazard function. Under a proportional-hazards model with linear predictor $\eta_i$, the event-time likelihood contribution, apart from factors not involving the covariates, is $$ \left\{h_0(t_i)\exp(\eta_i)\right\}^{\delta_i} \exp\left\{-H_0(t_i)\exp(\eta_i)\right\}. $$ The event indicator selects the hazard factor when an event occurs; the cumulative-hazard term contributes for both events and censored observations.

In a MICE covariate model, a common practical approach is to include $\delta_i$ and an estimate of cumulative hazard at $t_i$, such as a Nelson–Aalen estimate, as predictors of the incomplete covariate. They are not a single combined variable: they carry complementary information. The earlier material on using cumulative hazard in covariate imputation models introduced the cumulative-hazard predictor; here the emphasis is on why it must be paired with event status.

Deriving the Joint Contribution

Consider a binary covariate $X_i$, with $X_i=1$ versus $X_i=0$. Suppose its hazard ratio is $\exp(\beta)$, and temporarily hold all other covariates fixed. Write $A_i$ for the cumulative baseline hazard through $t_i$, multiplied by the hazard-ratio contribution of those other covariates. Thus $A_i$ is nonnegative and does not depend on $X_i$. The survival-likelihood contribution for $X_i=x$, with terms common to both values suppressed, is $$ L_i(x)\ \propto\ \exp(\delta_i\beta x)\exp\{-A_i\exp(\beta x)\}. $$

Comparing $X_i=1$ with $X_i=0$, the ratio of these likelihood contributions is

\frac{L_i(1)}{L_i(0)} =\exp\left\{\delta_i\beta-A_i\big(\exp(\beta)-1\big)\right\}.

Bayes’ rule says that the conditional odds for $X_i=1$ equal its odds from the covariate model given the other observed predictors, multiplied by this likelihood ratio. The formula separates the two survival contributions: $\delta_i\beta$ depends on event status, while the cumulative-hazard component depends on $A_i$. A MICE regression that includes both predictors is a practical way to convey both kinds of information to the imputation model.

Important qualification: Including event status and cumulative hazard as predictors is generally motivated by the survival likelihood, but a regression with only linear terms in these predictors is not automatically the exact conditional distribution. The exact relationship can be nonlinear and can depend on covariates, interactions, and the model for $X_i$. The two predictors are important ingredients, not a guarantee of perfect model compatibility.

Worked Example: Updating Imputation Odds for an Observed Event

Suppose, hypothetically, that the covariate model gives $P(X=1)=0.30$ for an individual before incorporating survival information. Assume a proportional-hazards model with $\beta=\log(2)$, so $X=1$ doubles the hazard, and take $A=0.20$. The individual has an observed event, so $\delta=1$. Find the updated conditional probability of $X=1$.

The prior odds are $0.30/0.70=0.4285714$. The survival likelihood ratio is $$ \exp\{\delta\beta-A(\exp(\beta)-1)\} =\exp\{\log(2)-0.20(2-1)\} =\exp(0.4931472\ldots) =1.6374615\ldots. $$

Therefore the updated odds are $$ 0.4285714\ldots\times1.6374615\ldots =0.701769\ldots\approx0.7018. $$

Converting odds to probability gives $$ P(X=1\mid\text{survival information}) =\frac{0.701769\ldots}{1+0.701769\ldots} =0.41238\ldots\approx0.4124. $$

Under these assumptions, the event information raises the conditional probability from 0.30 to about 0.4124. This is an association implied by the specified model, not evidence that the covariate caused the event.

What Is Lost When One Predictor Is Omitted?

If event status is omitted, the imputation model cannot distinguish an event at a given follow-up time from censoring at that same time. If cumulative hazard is omitted, it cannot distinguish short follow-up from long follow-up or otherwise account for how much risk accumulated before observation ended. Either omission can make the conditional distribution of the missing covariate less consistent with the survival process.

Worked Example: Omitting Cumulative-Hazard Information

Suppose, hypothetically, that $P(X=1)=0.25$, $\beta=\log(2)$, and the individual is censored, so $\delta=0$. Compare two possible follow-up histories: one with $A=0.70$, the other with $A=0.10$. The event indicator is identical in the two histories, but the accumulated hazards differ.

For $\delta=0$, the likelihood ratio reduces to $\exp\{-A(2-1)\}=\exp(-A)$. With $A=0.70$, the updated odds and probability are $$ \frac{0.25}{0.75}\exp(-0.70) =0.3333333\ldots\times0.4965853\ldots =0.1655284\ldots, \qquad P(X=1\mid\cdot)=\frac{0.1655284\ldots}{1+0.1655284\ldots}=0.1420\ \text{(approximately)}. $$

With $A=0.10$, the corresponding calculation is $$ \frac{0.25}{0.75}\exp(-0.10) =0.3333333\ldots\times0.9048374\ldots =0.3016125\ldots, \qquad P(X=1\mid\cdot)=\frac{0.3016125\ldots}{1+0.3016125\ldots}=0.2317\ \text{(approximately)}. $$

Thus two censored individuals with the same event-indicator value can have different survival-based conditional probabilities. An imputation model that uses event status alone cannot represent this difference. In this hypothetical calculation, the longer accumulated hazard shifts the odds more strongly against $X=1$, as expected when $X=1$ raises the hazard.

Worked Example: Omitting Event Status

Now suppose $P(X=1)=0.40$, $\beta=\log(2)$, and $A=0.20$. Compare an observed event with censoring at the same cumulative hazard. The prior odds are $0.40/0.60=0.6666667\ldots$.

For the event, $\delta=1$, so the likelihood ratio is $\exp\{\log(2)-0.20\}=1.6374615\ldots$. The updated odds are $0.6666667\ldots\times1.6374615\ldots=1.091641\ldots$, giving probability $$ \frac{1.091641\ldots}{1+1.091641\ldots}=0.5219\ \text{(approximately)}. $$

For censoring, $\delta=0$, so the likelihood ratio is $\exp(-0.20)=0.8187308\ldots$. The updated odds are $0.6666667\ldots\times0.8187308\ldots=0.5458205\ldots$, giving probability $$ \frac{0.5458205\ldots}{1+0.5458205\ldots}=0.3531\ \text{(approximately)}. $$

The event and censored records therefore imply different conditional probabilities despite having the same $A$. A model using cumulative hazard alone would miss the event-specific hazard factor and could not reproduce this distinction.

Assumptions, Diagnostics, and Alternatives

The derivation relies on a correctly specified proportional-hazards model for the event process and on the event and follow-up records being correctly classified. It also treats the censoring mechanism as noninformative conditional on the variables included in the model: after conditioning on those variables, censoring should not carry additional information about the event time or missing covariate. If censoring is informative, simply adding event status and cumulative hazard does not solve that problem; the censoring process may require explicit modeling or additional predictors.

In applied MICE, cumulative hazard is estimated rather than known. The estimate is therefore a constructed predictor, and using it does not remove the need to inspect the imputation model. Useful warning signs include imputed values that differ implausibly between event and censored records, weak or absent variation in imputations across follow-up ranges, poor reproduction of observed covariate-survival associations, and strong sensitivity to reasonable changes in model specification. Such patterns can indicate omitted predictors, an inadequate functional form, or an incompatible imputation model.

The main alternatives each have a limitation. Using event status alone loses follow-up-duration information. Using cumulative hazard alone loses the event-specific contribution. Omitting both ignores survival information when imputing covariates. Using observed follow-up time alone may fail to represent its different meanings for an event and a censoring observation. A survival-model-based imputation approach can model the joint structure more directly, but it requires its own assumptions and implementation choices; it is not automatically preferable in every chained-equation analysis.

Record event status and follow-up.
For each record, retain the observed time $t_i$ and event indicator $\delta_i$, with the event coded separately from censoring.

Construct cumulative-hazard information.
Evaluate an appropriate cumulative-hazard estimate at the observed follow-up time, using the survival-data structure and covariates required by the analysis.

Include both in the covariate imputation model.
Use event status and cumulative hazard as predictors, then check whether the chosen functional form represents the relevant relationships adequately.

Assess the resulting imputations.
Check plausible ranges, conditional patterns, and sensitivity to model choices; the paired predictors improve the information supplied but cannot compensate for severe model misspecification.

Common Mistake and Key Takeaway

A common mistake is to treat the event indicator and cumulative hazard as interchangeable summaries of survival. The incorrect reasoning is that once an imputation model knows whether an event occurred, follow-up duration adds nothing; or, conversely, that cumulative hazard alone already identifies whether an event occurred. The correction is visible in the likelihood ratio: $\delta_i$ appears in the event factor, while $A_i$ appears in the cumulative-hazard factor. They answer different questions and generally both belong in the covariate imputation model.

Key takeaway: Survival-informed covariate imputation generally needs both the event indicator and cumulative hazard evaluated at observed follow-up. The event indicator distinguishes an observed event from censoring; cumulative hazard represents accumulated risk. Omitting either can erase information relevant to the missing covariate.

Check Your Understanding

Answer each question, then compare your reasoning with the brief solutions.

Under the stated proportional-hazards model, which part of the likelihood ratio changes when $\delta$ changes from 0 to 1 while $A$ stays fixed?
For $\beta=\log(2)$, $A=0.30$, and $\delta=0$, calculate the survival likelihood ratio for $X=1$ versus $X=0$.
Why can two censored people with different follow-up times have different conditional distributions for a missing covariate even though both have $\delta=0$?
Name one assumption in the derivation that could fail and explain why adding the two predictors alone would not repair that failure.

Solutions. (1) The event factor changes: the log likelihood ratio includes $\delta\beta$, which is zero for censoring and $\beta$ for an event. (2) The ratio is $\exp\{-0.30(2-1)\}=\exp(-0.30)=0.7408$ (approximately). (3) Their accumulated hazards $A$ can differ, changing the factor $\exp\{-A(\exp(\beta)-1)\}$. (4) For example, if censoring is informative conditional on the modeled variables, censoring itself may carry additional information; event status and cumulative hazard alone do not model that censoring mechanism.

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