Signature
S_cond = S_t / S_l
| Inputs | Definition | Unit |
|---|---|---|
S_t | Probability of remaining event free beyond t, measured from the time origin | probability |
S_l | Probability of remaining event free beyond the entry time l, from the same origin | probability |
S_cond | Probability P(T > t given T > l) for a time t at or after the entry time l | probability |
|---|
Function
Survival estimation from left-truncated records with entry-restricted risk sets
Maps records with an entry time, an exit time and an event indicator to survival from the time origin when people come under observation only after the origin. Each person informs survival only beyond entry, conditional on surviving to entry, so risk sets and likelihood contributions start at entry. The notation follows the Left Truncation article and its six registry patients.
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Implementations
Excel
Conditional survival from named survival probabilities
With SurvT and SurvL named, the formula returns survival beyond t given survival to entry, held in CondSurv.
=SurvT/SurvL
Assumptions
One time origin for the entry time and the later time
Both survival probabilities are measured from the same origin, such as diagnosis, and t is at or after l.
Entrants share the survival function of the cohort from the origin
Event time and entry time are independent given that the event time exceeds entry, so late entrants are representative of everyone still event free at their entry time; this can and should be tested.
Worked examples
Surviving to 33 months after entry at 12 months in the registry example
From the corrected Kaplan-Meier curve, survival is 0.1875 at 33 months and 0.6667 at 12 months, so a patient entering at 12 months has a probability of about 0.2812 of surviving to 33 months (computed here for illustration).
S_t = 0.1875; S_l = 0.6667; S_cond = 0.2812
Conditional survival under a constant hazard
With a constant hazard of 0.1 a year, survival is 0.1353 at 20 years and 0.2231 at 15, so a person entering at 15 years survives 5 more with probability about 0.6065, the same as a new person over 5 years (computed here for illustration).
S_t = 0.1353; S_l = 0.2231; S_cond = 0.6065
Common errors
Reading survival after entry as survival from the origin
Stata's manual gives the example of a person arriving 15 years after exposure: had she died earlier she would never have been known, so her later survival must be treated as conditional on having already survived 15 years.
Using entrants' survival for a cohort that starts at diagnosis
In the GENIE cohort studied by Brown and colleagues, median survival from first-line treatment in 132 patients with KRAS wild-type colorectal cancer was 25.8 months after adjustment for left truncation and 40.9 months without it.
Sources
Late entrants' survival conditional on survival to entry in the Stata stset entry
StataCorp. Stata Survival Analysis Reference Manual, Release 19. College Station, TX: Stata Press; 2025. Entry stset, First entry times: origin() marks when a subject first becomes at risk and enter() when the subject first comes under observation; a person arriving 15 years after exposure would never have been known had she died before arriving, so her subsequent survival time must be treated as conditional on having already survived 15 years.
Median survival with and without adjustment for delayed entry in GENIE
Brown S, Lavery JA, Shen R, et al. Implications of selection bias due to delayed study entry in clinical genomic studies. JAMA Oncology. 2022;8(2):287-291. doi:10.1001/jamaoncol.2021.5153 (author manuscript read). Results: in 132 KRAS wild-type stage IV colorectal cancer patients on first-line FOLFOX or FOLFIRI with bevacizumab, median overall survival was 25.8 months after adjustment for left truncation and 40.9 months without adjusting, against 29.0 months in the trial.
Independence of entry and event times required by the delayed-entry adjustment
Betensky RA, Mandel M. Recognizing the problem of delayed entry in time-to-event studies: better late than never for clinical neuroscientists. Annals of Neurology. 2015;78(6):839-844. doi:10.1002/ana.24538 (author manuscript read). Discussion: the adjustment is valid only under the condition that the event time and the study entry time are independent, given that the former necessarily exceeds the latter; this can and should be tested.
Canonical Identity
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