Time Ordering and Ties in Censoring and Event Data

Why Time Ordering Matters

A survival record may show an event and censoring at the same recorded time. This can happen because time is recorded only to the nearest day, because events and follow-up are assessed at scheduled visits, or because a data system records both processes using a shared timestamp. The record alone then does not tell us which happened first. Yet the ordering determines whether the individual belongs in the event-time risk set, contributes an event, or is treated as censored before the event.

The issue matters for inverse-probability-of-censoring-weighted (IPCW) survival estimates. A weighted event hazard divides weighted event mass by weighted risk-set mass. Changing which tied records enter either quantity can change the hazard and therefore the survival curve. A convention must be stated before the estimate can be interpreted or reproduced.

Definition: At a recorded time $t$, an event-first convention treats the event as occurring before censoring: an event-censor tie is counted as an event, remains in the risk set just before $t$, and uses the censoring weight immediately before $t$. A censor-first convention treats censoring as occurring before the event: a tied record is censored at $t$, is removed before the event risk set is formed, and is not counted as an event at $t$. These are rules for interpreting tied records, not claims about the unobserved true ordering.

For a discrete-time analysis, let $t_1,\ldots,t_K$ be the ordered event times. Write $R(t)$ for the total IPC-weighted mass at risk just before an event calculation at $t$, and $D(t)$ for the weighted event mass included at $t$. With positive weights, the weighted hazard is $D(t)/R(t)$, and the corresponding product-limit estimate multiplies the conditional survival factors over event times.

\widehat h(t)=\frac{D(t)}{R(t)}, \qquad \widehat S(t_k)=\prod_{j=1}^{k}\left\{1-\widehat h(t_j)\right\}.

The weights are based on the censoring survival probability conditional on the observed history used in the censoring model. For a subject with history $H$, write $G(t-\mid H)$ for the probability of remaining uncensored through the instant just before $t$, and $G(t\mid H)$ for the probability of remaining uncensored strictly beyond $t$. At a discrete censoring jump these can differ. Event-first uses $1/G(t-\mid H)$ for an event at $t$; censor-first uses $1/G(t\mid H)$ for events eligible after the censoring decision at $t$. Positivity requires the relevant probabilities to be greater than zero, and the censoring model must correctly represent the conditional censoring probabilities for the IPCW interpretation to apply.

Calculating the Effect of a Tie Rule

A useful calculation separates four weighted quantities at a time $t$: $R$, the full risk-set mass before resolving ties; $D$, the event mass unambiguously recorded as an event; $Q$, the mass tied as both event and censoring; and $C$, the mass censored at $t$ without an event. For this calculation, take the weights as already specified for the convention being illustrated. Event-first counts the tied mass $Q$ as event mass. Censor-first removes both $Q$ and $C$ before forming the event risk set and counts only $D$ as events.

\widehat h_{\mathrm{event\ first}}=\frac{D+Q}{R}, \qquad \widehat h_{\mathrm{censor\ first}}=\frac{D}{R-Q-C}, \quad R>Q+C.

Worked Example: One Tied Time With Weighted Records

At one event time, suppose the pre-resolution weighted risk mass is $R=10$. Clear events have mass $D=2$, event-censor ties have mass $Q=3$, censor-only records have mass $C=1$, and the remaining mass is $10-2-3-1=4$. Calculate the hazard and survival factor under each convention.

Event-first. The tied mass is included as events, so the event mass is $D+Q=2+3=5$. The full risk mass is 10. Thus $\widehat h=5/10=0.5$, and the one-time survival factor is $1-0.5=0.5$.

Censor-first. The tied mass and censor-only mass are removed before the event calculation. The remaining risk mass is $10-3-1=6$, while the event mass is 2. Thus $\widehat h=2/6=1/3$, and the survival factor is $1-1/3=2/3$.

The event-first estimate has a smaller survival factor here because it counts the tied mass as events. This is a calculation under the stated rules, not evidence that either rule recovered the true chronological ordering.

Result: With the quantities above and $R>Q+C$, the event-first hazard minus the censor-first hazard is $$ \frac{D+Q}{R}-\frac{D}{R-Q-C} = \frac{QR-(D+Q)(Q+C)}{R(R-Q-C)}. $$ Its sign is not fixed. Therefore, event-first need not always produce a larger estimated hazard than censor-first.

To verify the result, put the two fractions over their common denominator. The numerator is $(D+Q)(R-Q-C)-DR$. Expanding gives $DR+QR-(D+Q)(Q+C)-DR=QR-(D+Q)(Q+C)$. The denominator is positive by $R>Q+C$, so the sign depends on that numerator. For instance, the values in the example give $3(10)-(2+3)(3+1)=30-20=10>0$, whereas other valid masses can make it negative. This establishes both the identity and why the direction of the change cannot be assumed. $\square$

From One Hazard to a Survival Curve

The same convention must be applied at every event time in the product-limit calculation. A change at an early time can affect both the survival factor at that time and the weighted risk mass carried forward to later times. The following derivation assumes that the stated weighted masses are updated consistently after each event and censoring decision.

Worked Example: Propagating a Tie-Rule Difference

Suppose at $t_1$ the weighted risk mass is 20, with clear event mass 4, tied event-censor mass 2, and censor-only mass 2. At $t_2$, the event mass is 3 and the risk mass under either rule is 12. There are no ties at $t_2$. Calculate the survival estimates through $t_2$.

Event-first at $t_1$. The hazard is $(4+2)/20=6/20=0.30$, so the first survival factor is $0.70$. After removing event mass $4+2=6$ and censor-only mass 2, the weighted mass carried forward is $20-6-2=12$. At $t_2$, the hazard is $3/12=0.25$ and the survival factor is $0.75$. Therefore $\widehat S(t_2)=0.70(0.75)=0.525$.

Censor-first at $t_1$. The tied and censor-only masses are excluded before the event calculation. The hazard is $4/(20-2-2)=4/16=0.25$, so the first survival factor is $0.75$. After removing the clear event mass 4 and the two censored masses 2 and 2, the carried-forward mass is $20-4-2-2=12$. The hazard at $t_2$ is again $3/12=0.25$. Hence $\widehat S(t_2)=0.75(0.75)=0.5625$.

The later hazard is the same in this constructed example, but the earlier survival factor is not. The difference therefore remains in the product at $t_2$.

Derivation: If the weighted hazard at each event time $t_j$ is the fraction $D(t_j)/R(t_j)$, then the fraction of weighted risk mass not experiencing the event at that time is $1-D(t_j)/R(t_j)$. Applying that conditional survival factor in sequence gives $$ \widehat S(t_k) = \prod_{j=1}^{k}\left(1-\frac{D(t_j)}{R(t_j)}\right). $$ This is a finite product identity; it requires no approximation or interchange of limits. Changing a tie rule changes the corresponding $D(t_j)$, $R(t_j)$, or both, so it can change the product. $\square$

Weights Before and After a Censoring Jump

Tie handling can affect not only risk-set membership but also the appropriate censoring weight. Let the conditional probability of remaining uncensored immediately before $t$ be $G(t-\mid H)=0.8$. Suppose the conditional probability of censoring at $t$, among those uncensored just before $t$, is $0.25$. Then the probability of remaining uncensored strictly beyond $t$ is $G(t\mid H)=0.8(1-0.25)=0.6$.

Worked Example: Pre-Jump and Post-Jump Weights

For an event at $t$ with history $H$, calculate the event-first and censor-first inverse-censoring weights using the probabilities above.

Event-first. The event is counted before the censoring jump at $t$, so the relevant probability is $G(t-\mid H)=0.8$. The weight is $1/0.8=1.25$.

Censor-first. The event is eligible only after surviving the censoring decision at $t$, so the relevant probability is $G(t\mid H)=0.6$. The weight is $1/0.6=5/3$, approximately $1.667$.

The larger post-jump weight reflects that fewer individuals remain uncensored beyond $t$. It is not appropriate to choose between 1.25 and 1.667 by looking for the more convenient estimate; the weight must match the time-ordering definition of the event and risk set.

Assumptions, Diagnostics, and Alternatives

A tie convention resolves an ambiguity in recorded data; it does not remove the usual requirements for an IPCW interpretation. In particular, the censoring model must condition on an adequate observed history, the relevant conditional probabilities must be positive, and the censoring weights must correspond to the observation rule used to define eligibility at each event time. The earlier material on IPCW survival estimation supplies the broader identification framework; the additional requirement here is that the time ordering and recorded tie coding agree with the analysis.

Useful checks include tabulating the number of event-only, censor-only, and tied records at each time; confirming how the source system defines simultaneous records; and recomputing the estimate under both conventions when ties are present. Compare not only the final survival estimate but also event counts, weighted risk totals, censoring probabilities, and the distribution of weights around tied times. A mismatch between a stated convention and the software's risk-set construction is a substantive error, even if the resulting curve looks plausible.

If event times are genuinely continuous and measured without rounding, exact event-censor ties may have probability zero under common continuous-time models. That observation does not justify ignoring ties in data recorded on days or visits: the recorded times are discrete, and the actual order may remain unknown. When timestamps contain finer ordering information, using that information is preferable to imposing an arbitrary order. When it does not, report the convention and consider a sensitivity analysis using the alternative rule.

These conventions are not interchangeable with modeling the censoring process more accurately. A better censoring model can improve the estimated weights, but cannot determine whether an ambiguous tied event preceded censoring. Conversely, changing tie order does not correct a misspecified censoring model, lack of positivity, or unmeasured predictors of censoring. The convention is also a limitation: both estimates can be sensitive to the recording resolution and to how ties are classified.

Common pitfall: “Ties are rare, so I can count every tied record as an event and use the same risk set under either convention.” This is wrong because a tied record's classification changes both event mass and, under censor-first, risk-set membership; a censoring jump can also change which inverse probability belongs in the weight. State the convention, apply it consistently, and check the alternative when ties occur.

Classify each record at a tied time.
Separate clear events, event-censor ties, censor-only records, and individuals still at risk.

Apply one ordering rule.
Determine which records contribute to the event count and which remain in the weighted risk set.

Match the weight to the rule.
Use the pre-jump or post-jump censoring probability as required by the event eligibility definition.

Recalculate and compare.
Recompute hazards and the product-limit curve under the alternative convention if ties are consequential.

Check Your Understanding

Answer each question, then compare with the solutions below.

With $R=12$, $D=3$, $Q=2$, and $C=1$, calculate the event-first and censor-first hazards.
Why can the event-first hazard be smaller than the censor-first hazard for some weighted masses?
If $G(t-\mid H)=0.9$ and the conditional censoring probability at $t$ is $0.2$, find $G(t\mid H)$ and both inverse weights.
What must be checked before interpreting differences between two tie-rule survival curves?

Solutions. (1) Event-first gives $(3+2)/12=5/12$. Censor-first gives $3/(12-2-1)=3/9=1/3$. (2) The sign is determined by $QR-(D+Q)(Q+C)$; if this quantity is negative, the event-first hazard is smaller. (3) $G(t\mid H)=0.9(1-0.2)=0.72$; the pre-jump weight is $1/0.9=10/9$, and the post-jump weight is $1/0.72=25/18$. (4) Check that tie classification, risk-set construction, and pre- or post-jump weights match the stated rule, and that the censoring model and positivity assumptions are adequate.