Left Truncation

Left Truncation in Survival Evidence for Economic Models

Registries, claims databases and clinico-genomic cohorts rarely start observing people at the moment that matters for the analysis. A patient diagnosed in January may only reach a registry when a test is done in June, so the survival clock and the observation clock start on different days. Left truncation, also called delayed entry or late entry, is the name for this structure. Correct analysis restricts risk sets and likelihoods to the time after entry, as an illustrative six-patient example shows, and ignoring it creates a form of immortal time that can distort the real-world survival evidence used in cost-effectiveness models, a risk that NICE's real-world evidence framework addresses.

Why late entrants are a selected group of survivors

Every survival analysis needs a time origin, such as diagnosis, and a rule for when each person is under observation. Stata keeps the two apart. Its origin option marks when a subject becomes at risk, and its enter option when the subject first comes under observation. Its manual notes that a person arriving 15 years after exposure has later survival conditional on having survived those 15 years. In terms of the survival function, an entrant at time $l$ informs only the conditional probability below.

$$P(T > t \mid T > l) = \frac{S(t)}{S(l)}, \qquad t \ge l$$

where $T$ is the event time from the origin, $l$ is the entry time on the same scale, $S(t)$ is the probability of remaining event free beyond $t$, and $S(l)$ is the probability of surviving to entry. People whose event came before their potential entry never appear, so the sample over-represents long survivors. That is the difference from right censoring, where a known person is lost after some follow-up.

Counting-process records and nonparametric estimators

The remedy is to store each person's follow-up as an interval that opens at entry and closes at the event or censoring time. The R survival package calls this the counting process form, Surv(time1, time2, status), with intervals open on the left and closed on the right. A person is at risk only inside that interval, as follows.

$$Y_i(t) = I(L_i < t \le X_i), \qquad n_j = \sum_i Y_i(t_j)$$

where $Y_i(t)$ equals 1 when person $i$ is under observation and at risk at time $t$, $L_i$ is the entry time, $X_i$ is the exit time at the event or censoring, $t_j$ is the $j$th distinct event time and $n_j$ is the size of the risk set then. The Kaplan-Meier estimator page uses the same boundary convention but writes the exit time as $Y_i$.

The estimators are unchanged; only the risk sets differ. The product-limit estimate and the partial likelihood of the Cox proportional hazards model become the following.

$$\hat{S}(t) = \prod_{j:\ t_j \le t} \left(1 - \frac{d_j}{n_j}\right), \qquad L(\beta) = \prod_j \frac{\exp(\beta^{\top} x_{(j)})}{\sum_{i:\ L_i < t_j \le X_i} \exp(\beta^{\top} x_i)}$$

where $\hat{S}(t)$ is estimated survival, $d_j$ is the number of events at $t_j$, $\beta$ is the vector of log hazard ratios, $x_{(j)}$ is the covariate vector of the person with the event at $t_j$ (no ties assumed) and $x_i$ is that of person $i$. Betensky and Mandel describe the adjustment in these terms. The Nelson-Aalen estimator is built from the same $d_j/n_j$ steps, so it needs the same entry-restricted risk sets, as does the Cox-Snell residual plot that applies it to residuals.

In software this is a data step. Stata's stset places each subject in the risk pool from the latest of origin, entry and record start; sts list with the enter option, and survfit in R with entry=TRUE, report how many join the risk set at each time. Those counts belong in numbers-at-risk tables, which with late entry can rise as well as fall.

Fitting parametric curves for extrapolation

Economic models usually extrapolate a fitted parametric survival model, so the likelihood must also condition on survival to entry. Stata's streg manual states that a subject failing at time $t$ contributes the density at $t$ conditional on entry, and a censored subject contributes survival beyond $t$ conditional on entry. On the log scale the contribution is as follows.

$$\ell_i = \delta_i \log f(X_i) + (1 - \delta_i) \log S(X_i) - \log S(L_i)$$

where $\ell_i$ is the log-likelihood contribution of person $i$, $\delta_i$ is 1 for an event and 0 for censoring, and $f$ and $S$ are the density and survival function of the chosen distribution. The final term is what a fit without entry times leaves out. Both streg and the R function flexsurvreg accept start and stop times, flexsurvreg treating counting-type records as left-truncated observations. The fitted curve describes the population from the origin only if the independence assumption below holds.

Worked example with six registry patients counted two ways

The data are illustrative. Six patients are followed in months since diagnosis, the origin the economic model needs; three were registered at diagnosis and three joined later, when a test was done.

1. Build the correct risk sets. A patient is at risk at month $t$ if entry is before $t$ and exit is at or after $t$. At month 6 the risk set is A, B and C; at month 18, B, C, D and E (F enters at 20); at month 24, C, D, E and F; at month 33, E and F, since C was censored at 30.

2. Estimate survival with entry handled. One death occurs at each event time, which gives the estimate below.

$$\hat{S}(33) = \frac{2}{3} \times \frac{3}{4} \times \frac{3}{4} \times \frac{1}{2} = \frac{18}{96} = 0.1875$$

where each fraction is $(n_j - d_j)/n_j$ at months 6, 18, 24 and 33. The intermediate values are 0.6667, 0.5 and 0.375.

3. Repeat ignoring entry times. Treating everyone as observed from diagnosis puts all six at risk at month 6 and five at month 18 in the calculation below.

$$\hat{S}_{\text{naive}}(33) = \frac{5}{6} \times \frac{4}{5} \times \frac{3}{4} \times \frac{1}{2} = \frac{60}{240} = 0.25$$

where the risk sets are 6, 5, 4 and 2. Survival is overstated at every step, for example 0.8333 against 0.6667 at month 6.

4. See the effect on mean survival. A model turns curves into time lived, and the area under each curve to 36 months is the restricted mean survival time, calculated as follows.

$$\text{RMST}(36) = 1 \times 6 + 0.6667 \times 12 + 0.5 \times 6 + 0.375 \times 9 + 0.1875 \times 3 = 20.94$$

where each term is a survival level times the months it lasts. The naive curve gives 6 + 10 + 4 + 4.5 + 0.75 = 25.25 months, about 21 per cent more time alive.

5. Count the person-time. Follow-up from entry totals 6 + 18 + 30 + 12 + 24 + 13 = 103 months; from diagnosis it is 6 + 18 + 30 + 24 + 36 + 33 = 147 months. The 44 extra months belong to D, E and F before entry, when none could have been recorded as dying. With four deaths, the rate is about 0.47 per person-year on the correct denominator and 0.33 on the inflated one.

The sample is far too small for real estimation, but the direction of each error is general. Guaranteed survivors in early risk sets lift the curve, lengthen mean survival and dilute event rates, including cost and event rates per person-year.

Ignored truncation as a source of immortal time

The 44 months have a name in epidemiology. Suissa defines immortal time as a span of cohort follow-up during which, because of exposure definition, the outcome could not occur, and traces the bias to heart transplantation studies in the 1970s. Hernán and colleagues describe a period in which risk is guaranteed to be exactly zero, arising when follow-up starts before people complete eligibility. Pre-entry time credited to a late entrant is also guaranteed event free, but by selection into the sample rather than by exposure definition, so ignored truncation is related to immortal time bias without being the same thing.

Suissa found that the bias in the rate ratio from misclassified or excluded immortal time grows in proportion to its duration and is larger when the hazard decreases over time. In a simulation by Betensky and Mandel, two groups had identical event-time distributions but one was recruited 1.5 years after the origin; a Cox model ignoring delayed entry estimated a hazard ratio of 5.01 instead of 1, with a type I error of 1, while the adjusted model estimated 1.0 with a type I error of 0.057.

Resetting the clock to entry removes the immortal time but answers a different question, survival after registration rather than after diagnosis, and a model needs the estimand matching its starting cohort. Hernán and colleagues show the design form. When follow-up of prevalent users starts after they began treatment, follow-up is left truncated and the analysis is restricted to those still under follow-up at the reset time zero.

Real-world and registry survival evidence for economic models

Left truncation matters most when real-world data supply a model's survival curve or comparator arm. Brown and colleagues studied stage IV colorectal and non-small cell lung cancer in the AACR Project GENIE Biopharma Collaborative, where patients entered at genomic profiling, a median of 6.25 and 2.34 months after diagnosis. Adjustment changed median overall survival from diagnosis by 13.0 and 12.0 months.

Their comparisons with trials used survival from the start of first-line treatment. In 132 patients with KRAS wild-type stage IV colorectal cancer on FOLFOX or FOLFIRI with bevacizumab, median survival was 25.8 months adjusted and 40.9 unadjusted, against 29.0 in the trial arm. In 76 patients with EGFR-mutant stage IV lung cancer on erlotinib or gefitinib, it was 31.2 months adjusted and 42.7 unadjusted, against 31.8. The authors noted that these informal comparisons do not adjust for clinical and sociodemographic differences, but judged the unadjusted estimates to strain credulity, and warned that restricting analysis to patients with no delay introduces other selection biases.

An overstated curve flows into life-years, QALYs and time-dependent costs, and so into the ICER. The NICE real-world evidence framework (ECD9, 2022, since updated) asks comparative studies to avoid time-related biases from misaligned eligibility, treatment assignment and start of follow-up, naming immortal time and prevalent-user bias. It prefers new-user designs because prevalent users have, by definition, remained on treatment and survived for a period.

Khudyakov and colleagues developed estimators of the ICER and incremental net benefit from left-truncated, right-censored data with stratified Cox models and restricted mean survival time, for treatment switching among AIDS patients in Tanzania, where truncation came from delayed treatment start. Only 1.4 per cent of eligible patients started the second treatment on becoming eligible. One scenario estimates a conditional restricted mean survival, $E(T - r \mid T > r)$ for a truncation time $r$.

Independent truncation and what to do when it fails

The risk-set correction fixes the denominators but rests on an assumption about why people enter when they do. Betensky and Mandel state that it is valid only when event time and entry time are independent, given that the event time exceeds entry, and that this should be tested; a treatment started at entry that changes the hazard breaks it. Brown and colleagues describe dependent truncation when clinical worsening prompts genomic sequencing, so entry is correlated with prognosis, and point to Kendall's tau statistics or regression methods to detect it.

A weaker assumption is available. Sondhi describes conditionally independent truncation, where event and entry times are independent given measured confounders, likens it to data missing at random, and notes that it can be tested. Under it, adjusted Cox models estimate conditional hazard ratios, and weighting a real-world cohort towards a reference covariate distribution, such as a single-arm trial's, recovers marginal survival in that reference population, or a treatment-versus-external-control hazard ratio for the trial population, provided the reference data represent the target population. In the cohort Betensky and Mandel discuss, which used age from birth as the time scale, a test detected dependence and the analysts added age-dependent terms to their Cox models.

A further limit applies even when entry is independent. If few or no people are observed near the origin, survival there is unstable or not estimable, so curves should only be read where the risk set has support; Khudyakov and colleagues require their truncation time to exceed the shortest delay observed in each group for this reason.

Left truncation, censoring and staggered recruitment

Several nearby ideas are confused with left truncation, and each needs different handling. The table sets out the contrasts.

Staggered recruitment alone does not create left truncation, although some authors use "staggered entry" for the truncated case. A landmark analysis is a deliberate relative. It starts the clock at a landmark among those still event free, making the conditioning explicit.

Errors in analysing and reporting left-truncated cohorts

The errors above share a root. Starting every record at zero lifts survival curves, extrapolated tails and any hazard ratio between groups with different entry delays, while dropping late entrants adds its own selection. Reports should show entrants in numbers-at-risk tables and argue or test independent entry, since the risk-set correction fixes only the denominators.

Sources

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• 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. https://doi.org/10.1001/jamaoncol.2021.5153 (author manuscript checked: https://pmc.ncbi.nlm.nih.gov/articles/PMC9190030/)
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