Delayed Entry in Cause-Specific Cox Regression

Why Entry Time Changes the Risk Set

In a cohort with delayed entry, a person is observed only after reaching an entry time on the study’s time scale. For example, follow-up might begin at disease onset, while a person is recruited only after surviving long enough to enroll. At an event time, that person belongs in a Cox risk set only if they have already entered and remain free of every event and censoring up to that time. Treating everyone as observed from time zero can therefore put people into risk sets when they were not yet under observation.

This tutorial concerns the cause-specific Cox model. For a selected cause \(k\), failures from other causes remove people from later risk sets, but those people remain at risk for cause \(k\) until their competing event occurs. The central task is to represent delayed observation without changing the target time scale or the cause-specific hazard being modeled.

Definition: Let \(L_i\) be subject \(i\)'s entry time, measured from the chosen time origin. Let \(X_i\) be the observed time of the first event or censoring, and let \(\Delta_i\) identify the event cause, with \(\Delta_i=0\) for censoring. A subject is observed only if they are event-free at entry, so their observed interval begins at \(L_i\) and ends at \(X_i\). With no entry or event ties, subject \(i\) is in the risk set immediately before time \(t\) exactly when \(L_i<t\leq X_i\).

The strict inequality \(L_i<t\) means a subject whose entry time equals \(t\) is not treated as having been at risk immediately before \(t\). If entry and failure are recorded at the same time, the analysis must specify their ordering; that convention should be applied consistently. The examples below have no such ties.

Model and Left-Truncated Risk Sets

Let \(Z_i\) be a vector of baseline covariates. For cause \(k\), the cause-specific proportional hazards model is

\lambda_{ik}(t\mid Z_i) = \lambda_{0k}(t)\exp(\beta_k^\top Z_i).

Here \(\lambda_{ik}(t\mid Z_i)\) is the instantaneous rate of cause \(k\) at time \(t\), among subjects who have experienced no event by \(t\), conditional on their covariates. The baseline hazard \(\lambda_{0k}(t)\) is unspecified. The parameter \(\beta_k\) describes covariate associations with the cause-specific hazard; for a one-unit increase in a covariate, the corresponding hazard ratio is the exponential of its coefficient, holding the other modeled covariates fixed.

For an event time \(t\), the eligible risk set is

R(t)=\{j:L_j<t\leq X_j\}.

This set includes people who will later have a competing event, because immediately before \(t\) they have not yet had any event. It excludes people who entered after \(t\), as well as people who had already failed from any cause or had been censored before \(t\). For a cause \(k\) event, the risk set is not restricted to people whose eventual event cause is \(k\).

Assumptions: The time origin and event-cause definitions are meaningful for all subjects, including those who enter late. The cause-specific proportional hazards model is correctly specified over the analyzed time range. Conditional on the modeled covariates, entry is independent of the event process in the source population, or satisfies the corresponding quasi-independence condition over the observable region \(L_i<T_i\). In practical terms, among people with the same modeled covariates who are still event-free at \(t\), entry and inclusion in observation must not select people according to their unobserved future event experience. The usual condition for censoring to be handled as non-informative is also required.

Conditional independence may need to be stated given additional design variables or strata if entry depends on them. If entry is associated with prognosis even after conditioning on the variables in the model, the observed risk set can be a selected subset of the source population. The ordinary left-truncated partial likelihood does not, by itself, correct that selection. The independent-truncation condition is generally a design and scientific assumption; observed data alone cannot establish it.

Why the Partial Likelihood Uses This Set

Suppose, first, that failures from cause \(k\) occur at distinct times. At a time \(t\) when subject \(i\) has a cause \(k\) event, each member \(j\) of \(R(t)\) has an instantaneous cause-\(k\) rate proportional to \(\exp(\beta_k^\top Z_j)\). Conditional on one cause-\(k\) failure occurring at that time, the relative probability that \(i\) is the failing subject is therefore its relative rate among eligible subjects:

P(i\text{ has the cause-}k\text{ failure at }t\mid \text{one cause-}k\text{ failure in }R(t)) = \frac{\exp(\beta_k^\top Z_i)} {\sum_{j\in R(t)}\exp(\beta_k^\top Z_j)}.

The common baseline hazard cancels from this ratio. Multiplying these contributions over cause-\(k\) failures gives the left-truncated partial likelihood:

L_k(\beta_k) = \prod_{i:\Delta_i=k} \frac{\exp(\beta_k^\top Z_i)} {\displaystyle\sum_{j:L_j<X_i\leq X_j}\exp(\beta_k^\top Z_j)}.

The risk set in each denominator depends on the event time \(X_i\). If several cause-\(k\) failures are tied, a method for handling tied failures is needed; it must use the delayed-entry risk set at that time. The distinct-time formula above makes the role of entry explicit.

Worked Example: Constructing Two Risk Sets

Consider hypothetical follow-up measured in years from disease onset. The table gives entry time \(L\), observed exit time \(X\), exit status, and a binary covariate \(Z\). The cause of interest is cause 1.

Subject	\(L\)	\(X\)	Exit status	\(Z\)
A	0	4	Cause 1	0
B	2	6	Cause 2	1
C	5	9	Censored	1
D	0	7	Cause 1	1
E	3	8	Cause 1	0

At \(t=4\), A, B, D, and E have entered and have not yet exited: \(R(4)=\{A,B,D,E\}\). C is excluded because \(L_C=5>4\). The event is A’s cause 1 failure. Subject B belongs in the set even though B later has a cause 2 event.

At \(t=7\), A and B have already exited. C, D, and E have entered and remain under observation through 7, so \(R(7)=\{C,D,E\}\). D has a cause 1 event at 7. Under the model, the two partial-likelihood contributions are

L(\beta)= \frac{\exp(0\beta)}{1+2\exp(\beta)} \cdot \frac{\exp(\beta)}{1+2\exp(\beta)} = \frac{\exp(\beta)}{\{1+2\exp(\beta)\}^2}.

The first denominator has two subjects with \(Z=0\) and two with \(Z=1\); the second has one with \(Z=0\) and two with \(Z=1\). Both denominators reflect who was actually at risk immediately before the respective event times.

Estimating a Covariate Effect

The partial likelihood is maximized over \(\beta_k\), just as in the ordinary cause-specific Cox model, but with the delayed-entry risk set in every denominator. The following small hypothetical calculation illustrates how the entry times affect the estimate.

Worked Example: Maximizing a Delayed-Entry Partial Likelihood

Use the data in the previous example, with one binary covariate \(Z\) and coefficient \(\beta\). Write \(q=\exp(\beta)\). The partial likelihood is

L(q)=\frac{q}{(1+2q)^2},\qquad q>0.

It is convenient to maximize the log likelihood with respect to \(\beta\). Differentiating gives

\ell(\beta)=\beta-2\log(1+2\exp(\beta)), \qquad \frac{d\ell}{d\beta} = 1-\frac{4\exp(\beta)}{1+2\exp(\beta)}.

Set the derivative to zero and solve:

1-\frac{4q}{1+2q}=0 \quad\Longrightarrow\quad 1+2q=4q \quad\Longrightarrow\quad \hat q=0.5, \qquad \hat\beta=\log(0.5)\approx -0.6931.

The estimated cause-specific hazard ratio for \(Z=1\) versus \(Z=0\) is \(0.5\) in this hypothetical calculation. It describes the fitted relative cause 1 hazard at a given time among subjects still event-free, under the model. With only two cause 1 events, this estimate would be highly uncertain; the example illustrates the calculation, not a basis for substantive conclusions.

Checking the Entry Specification

Before interpreting an estimate, inspect the time origin, entry records, and risk sets. A practical check is to list the subjects in each event-time risk set and verify for every included subject that \(L_i<t\leq X_i\). Also verify that a subject with a competing event is removed only from risk sets after that event, not before it. Risk sets that unexpectedly become empty, or event times at which the event subject is absent, usually indicate a data or time-ordering error.

The independent-truncation assumption deserves a separate scientific review. Ask why and when people entered, and whether entry could depend on prognosis after accounting for the modeled covariates. Compare entry patterns across relevant covariate groups and check whether the analysis has adequate overlap in entry times and covariates. Such checks can reveal implausible specifications or sparse support, but cannot prove that truncation is independent.

If entry is informative, possible responses depend on the design and target question: measure and adjust for relevant predictors of entry, use an appropriate model for the selection or truncation process, or redefine the target population and time origin. A landmark analysis, for example, conditions on survival and eligibility at a chosen time and answers a different question. Simply treating entry as time zero instead changes the timescale; it is not a general correction for delayed entry.

Worked Example: The Cost of Including Someone Before Entry

In another hypothetical cohort, a cause 1 event occurs at year 6. Subject A has \(L=0\), \(X=6\), and \(Z=1\); B has \(L=0\), \(X=8\), and \(Z=0\); C has \(L=7\), \(X=9\), and \(Z=0\); and D has \(L=5\), \(X=10\), and \(Z=1\). There are no other exits before year 6.

The correct set is \(R(6)=\{A,B,D\}\): C has not yet entered. At \(\beta=\log 2\), the covariate weights are 2 for \(Z=1\) and 1 for \(Z=0\). The partial-likelihood contribution for A’s event is therefore

\frac{\exp(\beta Z_A)} {\sum_{j\in R(6)}\exp(\beta Z_j)} = \frac{2}{2+1+2} = 0.400.

If C is incorrectly included, the contribution becomes \(2/(2+1+1+2)=0.333\). That altered denominator gives a different likelihood contribution because it claims C was at risk at year 6, although observation did not begin until year 7. This is not a harmless bookkeeping choice.

Common Pitfalls and Alternatives

Starting everyone’s follow-up at zero. This places late entrants into earlier risk sets and can bias the fitted model. Use actual entry times in the risk-set condition.
Resetting the clock to time since entry without changing the question. If the scientific timescale is time since disease onset, use that scale and delayed entry. Time since enrollment describes a different hazard process.
Removing competing-event subjects before their competing event. Cause-specific risk sets include anyone still free of all events immediately before \(t\), regardless of the cause they later experience.
Assuming independent entry because the risk-set formula was used. The formula handles observed entry times; it does not make informative truncation ignorable.

For a cause-specific Cox model, competing events end a subject’s contribution to future risk sets. Fine-Gray regression uses a different risk-set construction, so the formula here should not be transferred to that model. Likewise, left truncation is not left censoring: with left truncation, a person is unobserved before entry; with left censoring, the event has occurred by an observation time but its exact time is unknown.

Key takeaway: For a cause-specific Cox model with delayed entry, a subject contributes to the risk set at \(t\) only if they entered before \(t\) and remain under observation and event-free through \(t\). Valid estimation additionally requires the model and an appropriate independent-truncation condition, conditional on the covariates or design variables used.

Check Your Understanding

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

At \(t=5\), subject A entered at 1 and is censored at 8; B enters at 5 and has an event at 9; C entered at 0 and had a competing event at 4. Who is in the risk set immediately before \(t=5\), assuming no entry-event ties?
At a cause \(k\) event time, should a subject who later has a competing event be included in the risk set? Explain.
For a single cause \(k\) event at \(t\), the event subject has \(Z=1\), and the risk set has one other subject with \(Z=1\) and two with \(Z=0\). Write the partial-likelihood contribution at \(\beta=\log 2\).
Why does using time since enrollment as the time scale not automatically account for delayed entry when the intended time origin is disease onset?

Solutions. (1) Only A is included: \(1<5\leq8\). B is not included because \(L_B=5\), and C has already had a competing event. (2) Yes. Immediately before the event time, that subject remains event-free and contributes to the cause-specific risk set; the competing event removes them only from later risk sets. (3) The event subject’s weight is \(2\), and the denominator is \(2+2+1+1=6\), so the contribution is \(2/6=0.333\). (4) The time scale changes the meaning of the hazard and the comparisons between subjects. If the target is a hazard by time since disease onset, keep that origin and account for the later observation start through \(L_i\) in the risk sets.