Delayed Entry and Left Truncation

Why Delayed Entry Changes the Analysis

In some studies, a subject’s event-time clock starts at a meaningful origin, such as diagnosis or birth, but observation begins later. A subject might enter a study only after surviving event-free to a recruitment visit. The time between the origin and that visit is not ordinary follow-up: the subject had to remain eligible until entry to be observed at all.

This sampling feature is called delayed entry, or left truncation. It differs from right censoring. Right censoring means a subject is observed up to a time without an event being recorded; left truncation means subjects who experience the event before they can enter are absent from the observed sample. Ignoring delayed entry can therefore distort both likelihood contributions and the set of subjects considered at risk at a given time.

Definition: Let \(T_i\) be subject \(i\)'s event time measured from the common time origin, \(L_i\) the entry time, and \(C_i\) the right-censoring time after that origin. The observed follow-up endpoint is \(Y_i=\min(T_i,C_i)\), and \(\delta_i=I(T_i\le C_i)\) indicates an observed event. A subject is included only if \(T_i\ge L_i\) and observation begins at \(L_i\). The observed record must satisfy \(Y_i\ge L_i\).

The condition \(T_i\ge L_i\) is central: an included subject is known to have survived event-free until entry. The earlier material on event times, censoring, risk sets, and survival functions supplies the basic survival notation. Here the key change is that the event-time distribution must be interpreted among subjects who reached their entry times.

Likelihood Contributions Conditional on Entry

Suppose that, conditional on covariates, the event time has density \(f(t)\) and survival function \(S(t)=P(T>t)\). For the moment, assume the entry time \(L\) is fixed or is independent of the event time conditional on the covariates in the model. Since a sampled subject is known to satisfy \(T\ge L\), the event-time distribution relevant to that subject is the distribution conditional on surviving to \(L\).

f(t\mid T\ge L)=\frac{f(t)}{S(L)},\qquad t\ge L,

For an exact event at \(y\), the conditional likelihood contribution is \(f(y)/S(L)\). For a subject censored at \(y\), the event time is only known to exceed \(y\), so the corresponding contribution is \(P(T>y\mid T\ge L)=S(y)/S(L)\). These expressions use a continuous event-time model; for continuous times, \(P(T=L)=0\), so whether survival is written with \(>\) or \(\ge\) at the boundary does not change the probability.

L_T= \left\{\frac{f(y)}{S(L)}\right\}^{\delta} \left\{\frac{S(y)}{S(L)}\right\}^{1-\delta}, \qquad y\ge L.

This event-time likelihood is conditional on entry and uses the usual assumption that right censoring is independent of the event time, given the modeled covariates (and any other conditioning variables used in the analysis). If the censoring mechanism is modeled explicitly, its contribution must also be included. When censoring is independent and its distribution is not the target of inference, it can be omitted from the event-time likelihood as in the usual right-censoring treatment.

A useful justification comes directly from conditional probability. For an event at \(y\ge L\), the conditional density is the joint density of an event at \(y\) and survival to \(L\), divided by the probability of survival to \(L\). Since an event at \(y\ge L\) already implies survival to \(L\), the numerator is \(f(y)\). For censoring at \(y\ge L\), the event-time information is \(T>y\), which also implies \(T\ge L\). Thus the conditional tail probability is the ratio \(S(y)/S(L)\).

P(T>y\mid T\ge L) =\frac{P(T>y,\ T\ge L)}{P(T\ge L)} =\frac{S(y)}{S(L)}.

Worked Example: An Event and a Censoring Time After Entry

Suppose a hypothetical subject enters at \(L=2\) years. Conditional on the modeled covariates, assume an exponential event-time distribution with hazard \(\lambda=0.2\) per year, so \(S(t)=e^{-0.2t}\) and \(f(t)=0.2e^{-0.2t}\). First suppose the subject has an event at \(y=5\) years.

The event contribution conditional on reaching entry is \(f(5)/S(2)\). Substituting the values gives \[ \frac{0.2e^{-0.2(5)}}{e^{-0.2(2)}} =0.2e^{-0.2(5-2)} =0.2e^{-0.6} =0.1098 \] per year. This is a conditional density contribution, so it has units of inverse years.

If instead the subject is censored at \(y=5\), the event-time contribution is \[ \frac{S(5)}{S(2)} =\frac{e^{-1}}{e^{-0.4}} =e^{-0.6} =0.5488. \] This is the model probability of remaining event-free from year 2 through year 5, conditional on being event-free at entry. It is not the unconditional probability of surviving from the time origin to year 5.

The conditional calculation is appropriate if the subject’s event-time distribution given the modeled covariates does not otherwise depend on the entry mechanism. If entry timing carries additional information about the event time after conditioning on those covariates, the model must represent that dependence rather than use these ratios uncritically.

Risk-Set Membership With Different Entry Times

At a time \(t\), a subject can contribute to the risk set only after entering and before experiencing the event or being censored. With the convention that subjects entering exactly at \(t\) join immediately after any event at \(t\), define the risk-set indicator just before \(t\) by

R_i(t)=I(L_i<t\le Y_i).

Thus the risk set just before \(t\) is the set of observed subjects for whom \(R_i(t)=1\). A subject who has not yet entered is not at risk in the observed study at that time, even if that person would have been event-free. A subject who experienced an event or was censored earlier is no longer in the risk set. Under continuous event times, exact ties between entry and event are generally negligible; when ties occur, the analysis must use a clearly specified convention.

Worked Example: Constructing Risk Sets

Consider four hypothetical subjects, with all times measured in years from a shared origin:

Subject	Entry \(L_i\)	Observed endpoint \(Y_i\)	Endpoint status
A	0	4	Event
B	2	5	Censored
C	3	3.5	Event
D	4	6	Event

Just before the event time \(t=3.5\), subject A satisfies \(0<3.5\le4\), B satisfies \(2<3.5\le5\), and C satisfies \(3<3.5\le3.5\). Therefore, the risk set is \(\{A,B,C\}\). Subject D is not included because \(L_D=4\) is later than 3.5.

Just before \(t=4\), C has already left observation because of its event at 3.5; A is still included immediately before its event at 4, and B remains under observation until censoring at 5. D has entry time 4 and is excluded at that instant under the stated convention. The risk set is therefore \(\{A,B\}\) immediately before 4. A’s event then removes A; D joins immediately after that time. The definition makes clear why subjects should not be counted as at risk before their actual entry.

Entry Can Change Who Is Observed

When entry is independent of event time conditional on the model variables, conditioning on survival to each known entry time gives the appropriate event-time contribution. But delayed entry also selects the observed sample: subjects with shorter event times are less likely to survive long enough to enter. Consequently, the distribution of covariates or latent types among entrants can differ from its distribution in the source population.

For a discrete type \(B\) with source-population probabilities \(P(B=b)\), and type-specific survival \(S_b(L)\) to a common entry time \(L\), Bayes’ rule gives

P(B=b\mid T\ge L) =\frac{P(B=b)S_b(L)} {\sum_{b'}P(B=b')S_{b'}(L)}.

The survival weighting is not an optional adjustment when the target is the event-time distribution among entrants or the source-population composition. It represents the selection mechanism that determined who could be observed. In a regression or joint model, conditioning variables may explain some of the selection, but the analyst must still state what distribution is being modeled and whether entry is ignorable given those variables.

Worked Example: Selection Changes a Mixture Distribution

Suppose a hypothetical source population has two event-risk types, \(B=0\) and \(B=1\), with source proportions \(0.1\) and \(0.2\) in the subpopulation under consideration. The remainder of the population is outside this two-type comparison. Let the type-specific event times be exponential, with \(\lambda_0=0.1\) and \(\lambda_1=0.3\) per year. All entrants are observed only if event-free at \(L=2\) years.

The survival probabilities at entry are \(S_0(2)=e^{-0.2}=0.8187308\) and \(S_1(2)=e^{-0.6}=0.5488116\). The source-weighted amounts represented among entrants are \[ 0.1(0.8187308)=0.0818731,\qquad 0.2(0.5488116)=0.1097623. \] Therefore the proportion of type 1 among entrants in this two-type subpopulation is \[ \frac{0.1097623}{0.0818731+0.1097623} =0.5728. \] It is larger than the type 1 source proportion among these two types, \(0.2/(0.1+0.2)=0.6667\)? No: \(0.5728\) is smaller than \(0.6667\), as expected because type 1 has lower survival to entry.

Now consider the event-time density at \(t=3\), conditional on entry at 2. For type 0 it is \[ \frac{f_0(3)}{S_0(2)} =0.1e^{-0.1(3-2)} =0.0904837 \] per year. For type 1 it is \[ \frac{f_1(3)}{S_1(2)} =0.3e^{-0.3(3-2)} =0.2222455 \] per year. Weighting by the entrant composition gives \[ (1-0.5728)(0.0904837)+(0.5728)(0.2222455) \approx 0.1660 \] per year, using the unrounded entrant weight in the calculation. This is the mixture density among entrants at year 3.

For comparison, if the two types had equal source proportions, their entrant type-1 proportion would instead be \[ \frac{0.5(0.5488116)} {0.5(0.8187308)+0.5(0.5488116)} =0.4013. \] The observed mixture depends on both source composition and survival to entry. Using source proportions alone as the weights for an entrant sample omits this selection.

Assumptions, Diagnostics, and Alternatives

The conditional likelihood is useful only when its assumptions match how subjects entered and were followed. The principal assumptions are:

Correct time origin and entry time. Event and entry times must be measured on a compatible scale. Resetting time to zero at each subject’s entry changes the estimand and does not in general represent time since the common origin.
Correct event-time model conditional on included variables. The functions \(f(t)\) and \(S(t)\) must describe event times for the modeled covariates or other conditioning variables.
Entry ignorability after conditioning, or explicit modeling of entry. A common sufficient setup treats entry time as fixed by design, or independent of event time conditional on modeled variables, with inclusion determined by survival to entry. If entry depends on unmodeled prognosis, the conditional distribution may not be \(f(y)/S(L)\).
Appropriate handling of right censoring. Censoring must be independent of event time conditional on the modeled information, unless its mechanism is explicitly included.
Correct inclusion mechanism. Individuals who fail before entry are not observed. If the study instead has information on such individuals, or entry is based on additional eligibility rules, the sampling likelihood must reflect that design.

Useful diagnostics start with the study timeline: verify that no recorded endpoint precedes entry, check that the time origin is consistently defined, and inspect whether entry times differ systematically across covariates or clinical groups. Compare entry patterns with known recruitment rules. A strong relationship between entry time and prognosis after conditioning on the model variables is a warning that entry may be informative. If early events could not be captured, an apparent scarcity of early failures in the observed data is a feature of truncation, not evidence that the event hazard was small in the source population.

The main alternative is to ignore delayed entry and use the usual right-censoring contributions \(f(y)^\delta S(y)^{1-\delta}\), or to place every observed person in the risk set from the time origin. Both approaches treat subjects as if they were observable before they entered. They can overstate early risk-set sizes and fail to account for survival selection. Another approach is to model the entry process jointly with the event process; this is useful when entry is informative, but requires a defensible model for that process and sufficient information to estimate it. A simple conditional likelihood is preferable when its assumptions are credible, not merely because it is easier to compute.

Delayed entry does not by itself specify a particular event-time model or an estimation method. It changes the likelihood and risk sets supplied to those methods. The Kaplan–Meier estimator and regression approaches use risk sets in different ways, but both must respect delayed entry when it is present. If truncation is severe, few subjects remain to inform the event-time distribution at early times; conditional likelihood cannot recover information about portions of the population or time range that the design never observes without additional assumptions.

Common Pitfall

Common mistake: “The subject had an event at year 5, so the likelihood contribution is just \(f(5)\), and the subject can be counted at risk from year 0.” This reasoning ignores that the subject was only sampled because they survived until entry. Correction: condition the event-time contribution on survival to the subject’s entry time, \(f(5)/S(L)\), and include the subject in risk sets only after entry. The denominator and the risk-set restriction address the selection created by delayed entry.

Check Your Understanding

Answer each question, then compare with the brief solutions.

A subject enters at year 3 and is censored at year 7. Write the event-time likelihood contribution conditional on entry.
For an event at \(y\) after entry \(L\), explain why the conditional density is \(f(y)/S(L)\), rather than \(f(y)\).
A subject has \(L=4\) and \(Y=8\). Is the subject in the risk set immediately before time \(t=3\), \(t=5\), and \(t=8\)? Use \(R_i(t)=I(L_i<t\le Y_i)\).
With equal source proportions in two types and entry survivals \(0.8187\) and \(0.5488\), calculate the entrant proportion for the type with survival 0.5488.

Solutions. (1) The contribution is \(S(7)/S(3)\), since the subject is known to have survived event-free to entry and remains event-free through censoring. (2) Conditional probability divides the joint density of an event at \(y\) and survival to \(L\), which is \(f(y)\) for \(y\ge L\), by \(P(T\ge L)=S(L)\). (3) At 3 the subject has not entered, so is excluded; at 5 the subject is included because \(4<5\le8\); at 8 the subject is included immediately before the censoring endpoint under the stated risk-set convention. (4) The entrant proportion is \(0.5(0.5488)/\{0.5(0.8187)+0.5(0.5488)\}=0.4013\), to four significant figures.