Left truncation and delayed entry
Not everybody was under observation from day 0. How left truncation differs from left censoring, the two-time Surv(entry, exit, event) form, how far and in which direction survival is wrong without it, and why the whole bias lands on the people who entered late.
Left truncation and left censoring differ over one thing: the denominator
The two terms look alike and get used interchangeably, but they describe two different situations, and the fixes have nothing in common.
| Left censoring | Left truncation | |
|---|---|---|
| What you know | The event has already happened; you only know its time was earlier than some point | Before some point, this person could not have been seen to have the event at all |
| Typical setting | The first serology drawn is already positive; the infection date is unknown | A registry whose enrolment date follows the diagnosis date; age as the time scale |
| What is missing | A time | A stretch of observation |
| The fix | A left- or interval-censored likelihood (Surv(time, time2, type = “interval2”)) | Put the entry time inside Surv() so the person is not in the denominator before it |
Censoring and the risk set already draws the line between “in the data with incomplete time” and “not in the data at all”, and this page does not repeat it. What this page handles is the milder and far more common form of left truncation: people do get in, they just get in late — and the stretch they were late by is exactly the stretch during which they could not have been seen to have the event.
The worked example: a registry whose enrolment follows diagnosis
survival::myeloma holds Mayo Clinic patients with multiple myeloma: 3882 people, 2769 deaths observed, 1113 (28.7%) still alive when follow-up ended.
The point that matters is the time origin: day 0 in this dataset is diagnosis, not enrolment. For someone diagnosed at Mayo the two coincide. For someone diagnosed elsewhere and referred later, they do not, and the gap is recorded in the entry column.
| n | Share | |
|---|---|---|
entry of 0 — under observation from the day of diagnosis | 2194 | 56.5% |
entry above 0 — under observation only later | 1688 | 43.5% |
The median entry over the whole sample is 0, because more than half were enrolled on the spot, and that number tells you nothing. The distribution to look at is the one among the people who entered late: a median delay of 166 days, an interquartile range of 36 to 575 days, and one person who entered 5414 days after diagnosis.
The syntax: entry time goes in the first slot of Surv()
Two times inside Surv() say exactly what needs saying: when this person came under observation, and when they left it.
library(survival)
data(cancer, package = "survival") # myeloma lives inside the `cancer` bundle
# note: data(myeloma, package = "survival") warns that no such dataset exists
# Ignoring delayed entry: everyone is treated as observed from day 0 (diagnosis)
fit_naive <- survfit(Surv(futime, death) ~ 1, data = myeloma)
# Correct: entry is the day this person actually joined the risk set
fit_corr <- survfit(Surv(entry, futime, death) ~ 1, data = myeloma)
summary(fit_naive)$table[c("median", "0.95LCL", "0.95UCL")]
summary(fit_corr)$table[c("median", "0.95LCL", "0.95UCL")]
# Cox takes the identical change: put both times inside Surv()
coxph(Surv(entry, futime, death) ~ year, data = myeloma)Verified with R 4.6.0 and survival 3.8.6. myeloma ships with the survival package; nothing else to install.
import pandas as pd
from lifelines import KaplanMeierFitter, CoxPHFitter
# statsmodels' get_rdataset("myeloma", "survival") fails: the Rdatasets index
# registers it as "myeloma (cancer)" -- the same fact as R needing the cancer
# bundle. Reading the CSV directly is the least trouble.
URL = "https://vincentarelbundock.github.io/Rdatasets/csv/survival/myeloma.csv"
mye = pd.read_csv(URL)
km_naive = KaplanMeierFitter().fit(mye["futime"], mye["death"])
km_corr = KaplanMeierFitter().fit(mye["futime"], mye["death"], entry=mye["entry"])
print(km_naive.median_survival_time_, km_corr.median_survival_time_)
# Cox: entry_col is the first time in Surv()
cph = CoxPHFitter().fit(
mye[["entry", "futime", "death", "year"]],
duration_col="futime", event_col="death", entry_col="entry",
)
print(cph.summary[["coef", "se(coef)", "p"]])lifelines takes entry= in KaplanMeierFitter and entry_col= in CoxPHFitter.fit(); both were run against the R above and reproduce it to the digit (median survival and the Cox coefficient alike). The only difference is getting the data — statsmodels' get_rdataset cannot find myeloma, for the same reason R needs the cancer bundle. See the comment below.
How far it is wrong, and in which direction
Same data, same people, same event definition. The only difference is one extra column inside Surv().
figures/scripts/B3-08-left-truncation.R| Formula | Median survival (95% CI) | S(500 d) | S(1000 d) | S(2000 d) |
|---|---|---|---|---|
Ignoring delayed entrysurvfit(Surv(futime, death) ~ 1, data = myeloma) | 1004 (952–1060) | 71.8% | 50.4% | 26.2% |
Handling delayed entrysurvfit(Surv(entry, futime, death) ~ 1, data = myeloma) | 764 (728–811) | 63.2% | 41.0% | 19.2% |
| Difference (first minus second) | 240 days | 8.5 pp | 9.4 pp | 7.0 pp |
The direction is not a quirk of this dataset — it is fixed: ignoring left truncation overestimates survival. Median survival rises from 764 days to 1004 days, a gain of 240 days, or 1.31 times the correct value. Survival at day 1000 rises from 41.0% to 50.4%, a gap of 9.4 percentage points.
In percentage points, of the three days tabulated above the gap is widest at day 1000 (9.4 pp), narrowing to 7.0 pp by day 2000. That is not the bias shrinking: by then both curves are close to the floor, so there is less room left to differ in.
The mechanism: identical numerator, different denominator
The cleanest way in is this: both formulas count exactly the same events.
Either way the count is 2769 deaths, not one added and not one removed. What changes is the size of the risk set that Kaplan-Meier divides by at each of those deaths. Inflate the denominator and every “fraction who died at this instant” is diluted; multiply the diluted terms together and the whole curve lifts.
figures/scripts/B3-08-left-truncation.R| Day | Denominator, ignoring delayed entry | Denominator, handling delayed entry | Not yet entered | Share of the naive denominator |
|---|---|---|---|---|
| 0 | 3882 | 2194 | 1688 | 43.5% |
| 500 | 2335 | 1859 | 476 | 20.4% |
| 1000 | 1469 | 1228 | 241 | 16.4% |
| 1500 | 929 | 805 | 124 | 13.3% |
| 2000 | 602 | 534 | 68 | 11.3% |
| 2500 | 387 | 352 | 35 | 9.0% |
| 3000 | 235 | 219 | 16 | 6.8% |
The day-0 gap of 1688 is exactly the number of people with entry above 0 — not an approximation of it, the same set. The gap shrinks as referrals arrive, but at day 2000 the naive denominator still contains 11.3% who do not belong in it.
The bias is not spread evenly over the sample
Split the sample and run each half separately, and the most persuasive thing on this page appears.
| Subgroup | n | Events | Median, ignoring delayed entry (95% CI) | Median, handling delayed entry (95% CI) | Difference |
|---|---|---|---|---|---|
| entry == 0 (diagnosed at Mayo) | 2194 | 1889 | 777 (729–823) | 777 (729–823) | 0 days |
| entry > 0 (referred later) | 1688 | 880 | 1462 (1339–1581) | 795 (734–885) | 667 days |
For the patients enrolled on the spot the two formulas give identical median survival (777 days), down to both ends of the confidence interval. That is no surprise: their entry is 0, so the two expressions are saying the same thing about them.
For the referred patients the two formulas differ by 667 days. The entire bias in the whole analysis lands on those 1688 people.
Two settings you will meet this term
One: a cohort with age as the time scale
In chronic disease epidemiology the time axis is often not “years of follow-up” but age — because for cardiovascular events, dementia or fracture, age is the dominant time scale, and using follow-up time while adjusting for age as a covariate demotes the most important variable in the problem.
Once the axis is age, everybody is left truncated: a participant enrolled at 62 was not observed from age 0 to 62, and the reason they could appear on your enrolment list at all is that they reached 62.
# Time scale = age in years. Age at enrolment is entry, age at the end of
# follow-up is exit. lung, already loaded above, stands in for the shape:
# age is age at enrolment and time is follow-up in days.
cohort <- data.frame(
age_entry = lung$age,
futime_years = lung$time / 365.25,
event = as.integer(lung$status == 2),
sex = factor(lung$sex, levels = c(1, 2), labels = c("Male", "Female")),
ecog = lung$ph.ecog
)
cohort$age_exit <- cohort$age_entry + cohort$futime_years
coxph(Surv(age_entry, age_exit, event) ~ ecog + sex, data = cohort)Two: registry and biobank studies, where enrolment follows disease onset
This is the first trap a student meets working with claims data or a hospital registry. Whenever the time origin and the date of entry into the database are not the same day, the stretch between them is left truncated:
- A hospital cancer registry uses the date of diagnosis as the origin, but the patient was registered only on referral
- A biobank uses the date of onset, but the specimen arrived, and entered the analysis, later
- A database that begins in a given year, used to study patients diagnosed before that year — their origin predates the database, and appearing in it at all means they survived to the day it opened
- The first prescription as the origin, in a database whose pharmacy records only start in some year
The code is the same every time: entry is days from the origin to database entry, exit is days from the origin to the event or the end of follow-up. The other traps in database studies and real-world data — claims exist for reimbursement, not research; what RECORD asks you to state — stack on top of this one rather than replacing it.
How this page relates to the others
- Censoring, truncation and why the risk set is the denominator — back to censoring and the risk set. Its table of the three censoring types and the table at the top of this page are a matched pair.
- How a KM curve is computed, and why a numbers-at-risk table is compulsory — see the Kaplan-Meier curve. The first figure here carries both risk rows precisely because those two rows are the entire reason the curves separate.
- The Cox model — see the Cox proportional hazards model. Left truncation is the identical one-line change in
coxph(), and every risk set in the partial likelihood is filtered byentryautomatically. - Immortal time bias — a relative of delayed entry, but not the same thing, and worth keeping apart:
- Delayed entry: during that stretch nobody was observing the person at all. The error is counting observation that never happened.
- Immortal time bias: during that stretch you can see the person, but you grouped them by a status that was not yet known at the time (transplanted or not, responder or not), so guaranteed-alive time was credited to the wrong arm. The fixes point in opposite directions: the first takes time out of the risk set, the second keeps it in and moves it to the correct arm. B3-01 works through immortal time with the Stanford heart transplant data, and time-varying covariates and landmark analysis is where the fix lives.
- Being eligible at all is itself a filter — see selection bias.
Surv(entry, ...)repairs the denominator; it does not repair a population that has already been screened.
Common misuses
| Misuse | Why it is wrong |
|---|---|
Enrolment follows the origin, but the code says Surv(time, event) | No error is raised and survival is overstated systematically; on this page by 240 days |
Reading the overall median entry of 0 as “this does not affect us” | The bias sits on the people who entered late, and a whole-sample quantile hides them |
Using exit - entry as follow-up time instead of two times | That changes the time origin rather than correcting anything; the clinical question changes with it |
| Age as the time scale, but only follow-up time in the code | Everyone is left truncated in that setting; it is the default there, not the exception |
| Treating left truncation as left censoring and reaching for an interval likelihood | Left censoring is missing a time, left truncation is missing observation; the likelihoods differ |
Claiming selection bias is handled because Surv(entry, exit, event) was used | Only the denominator was repaired; who got the chance to be enrolled is still filtered |
| A Methods section that says only “Kaplan-Meier was used” | The reader cannot tell whether delayed entry was handled, and this is the irreproducible line |
| Assuming the other assumptions hold once delayed entry is handled | Non-informative censoring and proportional hazards are separate assumptions, each needing its own check |
Reproducing every number on this page
/opt/homebrew/bin/Rscript figures/scripts/B3-08-left-truncation.RRead the figure
The answer comes from the same statistical output that produced this page's figures, not from a number typed in beside them.
The same myeloma data give different median survival depending on whether delayed entry is handled. Which figure should be reported?
Show the answer and why
Correct answer: 764 days - putting each patient's entry time into the risk set
A referred patient had to survive until referral to appear in this dataset at all, and during that stretch they could not die by definition. Treating them as observed from diagnosis hands that impossible-to-die time to the survival curve, pushing the median from 764 days up to 1004. 728 is the lower confidence bound on the corrected median: it moves with the sample, so using it as a point estimate is not caution but a different quantity.
There are 3882 patients here. Which number tells you whether delayed entry is a problem worth handling?
Show the answer and why
Correct answer: 1688 have an entry time above zero, so observation of them begins after diagnosis
1688 patients entered late, close to half the cohort - and that share is what governs how much bias ignoring it introduces. 2194 entered at time zero and are unproblematic, but "the rest can be set aside" has it exactly backwards; 1113 is the censored count, and delayed entry is the opposite phenomenon: censoring is not seeing what came after, delayed entry is not seeing what came before. In the data this is a single entry column, and leaving it out of Surv() produces no complaint and a perfectly plausible answer.
Overall, ignoring versus handling delayed entry moves median survival by only 240 days, and someone concludes the problem is negligible. Which number best refutes that?
Show the answer and why
Correct answer: The late-entry group differs by 667 days, so the whole bias sits with them
Patients entering at time zero have no delayed-entry problem, so both methods give them the same median and the difference is 0; the entire bias comes from the late-entry group, where it is 667 days. The overall 240 is what remains after mixing the two together - half the cohort has no bias at all, so averaging them in is bound to look reassuring. "It barely moves overall" is therefore not a reason to ignore left truncation; the questions are how large the affected group is and how far it is off.
Chapters that use this method
Watch next
Censoring and Truncation [Survival Analysis 2/8]Sources and licences
This page is original writing