Delayed Entry in Weighted Survival Risk Sets

Why Delayed Entry Changes the Risk Set

In survival data, a person may become observable after the time origin used to measure the outcome. For example, age may be the time scale, but a participant may enroll in a study at age 50. That participant cannot contribute to a risk set at age 40, even if the analysis file records their eventual event or censoring time. Treating them as under observation from the origin creates immortal or unobserved person-time in the data representation and can distort the fitted survival model.

This situation is called delayed entry, or left truncation. It is distinct from ordinary right censoring: entry determines when a person first becomes eligible to contribute, while censoring determines when observed follow-up ends. Treatment weights address treatment-confounding mechanisms; they do not repair an incorrectly constructed risk set or automatically correct selection induced by delayed entry.

Definition: For individual $i$, let $E_i$ be the entry time, $T_i$ the event time, and $C_i$ the censoring time. Let $X_i=\min(T_i,C_i)$, and let $\Delta_i=I(T_i\leq C_i)$ indicate an observed event. With the usual convention that an individual entering exactly at time $t$ is not at risk immediately before $t$, the delayed-entry risk indicator is $$ Y_i(t)=I(E_i<t\leq X_i). $$ The observed event counting process is $N_i^{\mathrm{obs}}(t)=I(E_i<X_i\leq t,\Delta_i=1)$. Thus an event at $t$ is included only if the individual had entered before $t$.

The strict inequality at entry reflects the risk set immediately before an event time. If a study's time recording or event convention differs, entry and event ties need a consistent rule; the formula above states the convention used here. The time origin must also be meaningful and common across individuals. Resetting each person's clock to zero at enrollment changes the time scale and is not merely a data-cleaning choice.

Weighted Cox Risk Sets With Left Truncation

Suppose a marginal structural Cox model uses covariate vector $Z_i(t)$, parameter $\beta$, and positive treatment weight $W_i(t)$. For a given event time $t$, only people with $Y_j(t)=1$ belong in its denominator. For simplicity, first suppose observed event times are distinct. The weighted partial likelihood contribution for event subject $i$ at time $T_i$ is

\left[ \frac{\exp\{\beta^\mathsf{T}Z_i(T_i)\}} {\displaystyle\sum_{j=1}^n Y_j(T_i)W_j(T_i) \exp\{\beta^\mathsf{T}Z_j(T_i)\}} \right]^{W_i(T_i)}.

The event-subject weight is an exponent, not just a multiplicative factor in the numerator. The full weighted partial likelihood is the product of these contributions over observed events. This is a weighted estimating procedure; when weights are not integer counts, it should not be interpreted as an ordinary likelihood for independently replicated observations.

L_w(\beta)= \prod_{i:E_i<T_i,\Delta_i=1} \left[ \frac{\exp\{\beta^\mathsf{T}Z_i(T_i)\}} {\displaystyle\sum_{j=1}^n Y_j(T_i)W_j(T_i) \exp\{\beta^\mathsf{T}Z_j(T_i)\}} \right]^{W_i(T_i)}.

A useful equivalent expression is the weighted Cox score. Define

\bar Z_w(t;\beta)= \frac{\displaystyle\sum_{j=1}^nY_j(t)W_j(t) \exp\{\beta^\mathsf{T}Z_j(t)\}Z_j(t)} {\displaystyle\sum_{j=1}^nY_j(t)W_j(t) \exp\{\beta^\mathsf{T}Z_j(t)\}}.

Then the score is

U_w(\beta)= \sum_{i=1}^n\int W_i(t)\{Z_i(t)-\bar Z_w(t;\beta)\}\,dN_i^{\mathrm{obs}}(t).

Result: Under the distinct-event-time convention, differentiating the weighted partial log-likelihood gives the score above. Delayed entry enters through $Y_j(t)$: a person contributes to the weighted risk-set mean only after entry and through the end of observed follow-up.

To verify the result, take logarithms of $L_w(\beta)$. For each observed event $i$, the contribution is

W_i(T_i)\left[ \beta^\mathsf{T}Z_i(T_i)- \log\left\{\sum_{j=1}^nY_j(T_i)W_j(T_i) \exp\{\beta^\mathsf{T}Z_j(T_i)\}\right\} \right].

Weights and risk indicators do not depend on $\beta$. Differentiating the first term gives $W_i(T_i)Z_i(T_i)$. Differentiating the log-sum gives

W_i(T_i) \frac{\displaystyle\sum_jY_j(T_i)W_j(T_i) \exp\{\beta^\mathsf{T}Z_j(T_i)\}Z_j(T_i)} {\displaystyle\sum_jY_j(T_i)W_j(T_i) \exp\{\beta^\mathsf{T}Z_j(T_i)\}} = W_i(T_i)\bar Z_w(T_i;\beta).

Subtracting and summing over event times yields the displayed score. In particular, the event subject's entire score contribution is multiplied by its own weight. Omitting the exponent on the event contribution would not yield this weighted score. Tied events require a specified tie-handling procedure; the simple product above is stated for distinct event times rather than silently imposing a tie approximation. $\square$

Worked Example: A Delayed Entrant Is Excluded From an Earlier Event

At time $t=5$, subject A has an event, entered at time 1, and has $Z_A=1$, $W_A(5)=2$. Subject B entered at time 2, remains event-free past 5, and has $Z_B=0$, $W_B(5)=1$. Subject C enters at time 5.5, and subject D was censored at time 4. Which subjects contribute to the risk-set denominator at the event time?

At $t=5$, $Y_A(5)=1$ because $1<5\leq X_A$, and $Y_B(5)=1$. Subject C has not entered yet, so $Y_C(5)=0$; subject D has already been censored, so $Y_D(5)=0$. At $\beta=\log 2$, the denominator is

2\exp(\log 2)+1\exp(0)=4+1=5.

The event-subject partial-likelihood contribution, with A's event weight used as a power, is $(2/5)^2=4/25$. The risk-set mean of $Z$ is $4/5$, so A's score contribution at this value of $\beta$ is

2\left(1-\frac45\right)=\frac25.

Neither C nor D belongs in the denominator, regardless of their covariate values or weights. C contributes only after entry; D contributes no later than censoring.

Representing Entry in Pooled Logistic Data

For pooled logistic survival models, each row represents an individual-interval combination in which the individual is observed and at risk at the start of the interval. Let $a_k$ and $b_k$ denote the left and right endpoints of interval $k$. A row is eligible only if the individual has entered by $a_k$, remains under observation through the interval start, and has not already had the event. Follow-up stops at the event interval or the censoring interval. Consequently, no rows are created for intervals before entry or after event or censoring.

Entry times should align with interval boundaries when possible. If a participant enters between prespecified boundaries, include the entry time as a cut point so that the first contributed interval begins at entry. Otherwise, a row covering time before entry falsely assigns that person risk time they did not contribute. If an interval is split, the hazard model must be specified for the resulting intervals; a probability for a full interval cannot automatically be reused as the probability for a shorter partial interval.

Write $D_{ik}=1$ when person $i$ has an event in an included interval and $D_{ik}=0$ otherwise. Let $R_{ik}$ indicate that the person contributes that row, $p_{ik}$ be the fitted event probability for the interval, and $W_{ik}$ the treatment weight used for that row. The weighted Bernoulli log-likelihood contribution is

\ell_w= \sum_i\sum_k R_{ik}W_{ik} \left[D_{ik}\log(p_{ik})+ (1-D_{ik})\log(1-p_{ik})\right].

Here $R_{ik}$ encodes delayed entry as well as continued observation and event-free status at interval start. The weight changes the included row's contribution; it does not make a pre-entry row eligible. In a standard pooled data set, each person contributes at most one event row, and no rows after that event.

Worked Example: Building Pooled Logistic Rows After Entry

Suppose a participant enters at time 2, has an event at time 3.4, and has treatment weight 2 throughout follow-up. Use intervals $[0,1)$, $[1,2)$, $[2,3)$, and $[3,4)$. The person has no event in $[2,3)$, and the event occurs in $[3,4)$. The correct rows are:

Interval	Included?	Event indicator	Reason
[0,1)	No	—	Before entry
[1,2)	No	—	Before entry
[2,3)	Yes	0	Entered and event-free
[3,4)	Yes	1	Event occurs in this interval

If the fitted probabilities in the two included intervals are $p_{i2}=0.10$ and $p_{i3}=0.20$, the person's weighted likelihood contribution is

\{(1-0.10)^2\}\{(0.20)^2\} =(0.90)^2(0.20)^2 =0.81(0.04)=0.0324.

There are no factors for $[0,1)$ or $[1,2)$, because the participant was not observed at risk then. Including either factor would model survival during time before entry as if it had been observed.

Assumptions, Diagnostics, and Alternatives

Correct bookkeeping is necessary but does not by itself establish that a survival estimand is identified. Entry times and the time origin must be measured consistently; treatment weights must be appropriate for the intended treatment-adjusted analysis and well-defined over observed follow-up; and the relevant treatment positivity and exchangeability conditions must hold. For inference about survival from the population time origin, delayed-entry sampling also requires an appropriate assumption about truncation. A common sufficient formulation is that entry is independent of the event process conditional on measured variables used to condition or adjust the analysis. If entry depends on unobserved prognosis even after adjustment, the observed entrants may not represent the target population's survival experience. Treatment weighting alone does not remove that selection problem.

Useful diagnostics begin with the person-time representation. Tabulate entry and exit times, verify that every contributed interval starts at or after entry, and inspect risk-set sizes at observed events. For pooled data, confirm that each person has no pre-entry rows and no rows after event or censoring. For Cox data, compare the risk-set membership at selected event times against the entry and exit records. Extreme or missing weights, an unexpectedly empty risk set, or risk-set sizes that include people not yet enrolled are warning signs. Weight-distribution checks are relevant, but apparently reasonable weights cannot validate incorrect entry dates.

Two common alternatives require care. Resetting follow-up time to zero at enrollment uses time since entry rather than the original time scale, such as attained age; it changes the baseline hazard and the comparison being made. Restricting analysis to a landmark time can be appropriate for a landmark estimand, but it discards earlier follow-up and targets people who survived and remained observed to that landmark. Neither approach is a generic substitute for left-truncated risk sets on the original time scale.

Worked Example: Quantifying the Cost of Including a Future Entrant

At event time $t=6$, subject A has an event, entered at 1, and has $Z_A=1$, $W_A=2$. Subject B entered at 2 and remains at risk, with $Z_B=0$, $W_B=1$. Subject C has $Z_C=0$, $W_C=10$, but does not enter until 6.2. Evaluate the event's score contribution at $\beta=0$.

The correct risk set contains A and B, so $\bar Z_w=(2\cdot1+1\cdot0)/(2+1)=2/3$. The event score is $2(1-2/3)=2/3$.

If C is incorrectly included, the risk-set mean becomes $(2\cdot1+1\cdot0+10\cdot0)/(2+1+10)=2/13$, and the event score would be $2(1-2/13)=22/13$. This is not a small adjustment: the future entrant's large weight changes the score at a time when that person was not eligible to be at risk.

Common Mistakes

Starting everyone at the time origin. A later entrant must not contribute to earlier Cox risk sets or pooled logistic rows.
Using the event weight only as a numerator multiplier. A factor that is constant in $\beta$ does not weight the event-subject score. In the weighted partial likelihood shown here, the event contribution is raised to $W_i(T_i)$.
Using the wrong weight in the denominator. The risk-set mean uses each at-risk person's $W_j(t)$, and its membership is determined by delayed entry and observed follow-up.
Treating entry as ordinary censoring. Entry establishes when risk begins; censoring ends observed follow-up. They are different endpoints of the observation window.
Resetting the clock without changing the estimand. Time since entry and time since a population origin are different time scales, not interchangeable coding conventions.

Key takeaway: Delayed entry belongs in the at-risk indicator. The weighted Cox denominator and the pooled logistic rows must contain only people who have entered and remain event-free and observed at the relevant time. Correct weights cannot compensate for an incorrect risk set.

Check Your Understanding

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

A participant enters at time 4 and has an event at time 7. Which pooled logistic intervals may contribute if the interval boundaries include 4?
At an event time $t$, what condition determines whether a person who enters at time $t$ is included under the convention in this tutorial?
Why does multiplying the event numerator by $W_i(T_i)$ without raising the contribution to that weight fail to produce the stated weighted Cox score?
At $\beta=0$, a risk set has an event subject with $Z=1,W=2$ and one other subject with $Z=0,W=1$. Calculate the event score contribution.
Does treatment weighting, by itself, correct dependence between entry and prognosis? Explain.

Solutions. (1) Rows may begin in the interval starting at 4 and continue through the event interval containing time 7; there are no rows for intervals ending before 4 or after the event. (2) The person must have entered strictly before $t$, and must still be under observation through $t$, so $Y_i(t)=I(E_i<t\leq X_i)=1$. (3) A multiplicative event-weight factor outside the $\beta$-dependent ratio contributes only a constant to the log-likelihood. Its derivative does not multiply the event-subject term, whereas the power $W_i(T_i)$ multiplies the entire event score. (4) The weighted risk-set mean is $2/(2+1)=2/3$, so the event score is $2(1-2/3)=2/3$. (5) No. Treatment weights address treatment assignment under their assumptions; informative truncation can still select people according to prognosis and needs its own identification assumptions or adjustment.