Left Censoring

Left Censoring in Survival Data and Economic Models

A left-censored record shows that a patient's event, such as cancer recurrence or the onset of dementia, had already taken place by the time anyone first looked, so the data give only an upper limit for the event time. The time origin is known, for example the date of surgery or of birth, but the event falls somewhere between that origin and the first assessment. Such records arise in trial follow-up, screening, cohorts that enrol people already past a milestone and claims databases that start in a given year. Dropping these patients, or dating their events at the first visit, shifts estimated incidence and time to event, and those estimates feed the transition probabilities, state occupancy and QALYs of an economic model.

Where left-censored records arise in health data

Clark and colleagues, in an introduction to survival analysis, give the standard clinical case. In a study of time to cancer recurrence after surgery, patients first examined 3 months after the operation who already have a recurrence have a left-censored survival time, because the recurrence happened less than 3 months after surgery.

Cain and colleagues describe the same structure in longitudinal studies that enrol people over a range of ages, some already past the milestone of interest. An age at a milestone that occurred before study entry is left censored if the person is enrolled and left truncated if the person is not. Their example is the menopausal transition, in which women already in the late stage at recruitment are enrolled with a left-censored age at onset of that stage.

Screening produces left-censored records when a test finds a condition already present. Yefenof and colleagues give melanoma as an example, observed exactly if the patient detects it, right censored after a negative screen and left censored if it goes undetected until screening.

Administrative data produce left censoring at the start of the data window. Weißbach and colleagues sampled 250,000 people insured with Germany's largest health insurer in 2004 and followed them in the claims data to 2012. Insurants whose dementia had been diagnosed before 2004 were left censored, about 11,000 people beside 36,000 observed incident cases. A biomarker value below a laboratory's limit of detection is left censored in the same way, and Gillespie and colleagues note that survival methods for left-censored data can analyse such values.

How a left-censored record enters the likelihood

Each patient contributes the probability of what was actually seen. For a left-censored record that is the probability that the event had happened by the first observation, so the record enters through the cumulative distribution rather than the density. Under non-informative censoring, standard texts such as Klein and Moeschberger write the likelihood as

$$L(\theta) = \prod_{i \in E} f(t_i; \theta) \prod_{i \in R} S(t_i; \theta) \prod_{i \in C} \left[ 1 - S(t_i; \theta) \right]$$

where $\theta$ is the vector of model parameters, $E$ is the set of patients whose event time $t_i$ is observed exactly, $R$ is the set of right-censored patients with $t_i$ the last event-free observation, $C$ is the set of left-censored patients with $t_i$ the first observation at which the event was already present, $f(t; \theta)$ is the probability density of the event time and $S(t; \theta)$ is the survival function, so that $F(t; \theta) = 1 - S(t; \theta)$ is the cumulative probability that the event has occurred by time $t$. Parametric curves are fitted to this function by maximum likelihood estimation.

A left-censored record is the window $(0, t_i]$, so Yefenof and colleagues treat data made up of exact, right-censored and left-censored records as a special case of interval-censored data.

Nonparametric estimates with left-censored records

The Kaplan-Meier estimator handles right-censored records through the risk set and cannot place an event that happened before observation began. When all the censoring is on the left, as with values below a detection limit, Gillespie and colleagues recommend the reverse Kaplan-Meier estimator, the nonparametric maximum likelihood estimator, which ordinary Kaplan-Meier software can calculate. When left and right censoring occur together the estimate is found iteratively, and Fay and Shaw explain that Turnbull showed how to find it for interval-censored data by the self-consistent algorithm, a special case of the EM algorithm.

Assumptions behind a left-censored analysis

The method needs a time origin, known for every patient, that precedes the first observation. Status at entry must also be ascertained correctly. Cain and colleagues note that their designs assume one can reliably determine whether the milestone occurred before entry, which is not always so, and they call it unwise to treat the first observed occurrence as the actual first one.

The likelihood also treats the timing of the first observation as carrying no information about the event time beyond the bound itself. A fixed assessment schedule meets that condition, while earlier visits for patients with symptoms break it.

Left-censored records say little about the shape of the curve close to the origin, so a parametric fit there leans on its assumed form, and Cain and colleagues warn that such results could be sensitive to distributional assumptions.

Worked example with recurrence found at the first scan

All numbers in this example are illustrative. Ten patients are followed from surgery, with the first planned scan at month 3. Three already have a recurrence at that scan and are left censored at 3 months. After the scan, recurrence is diagnosed when symptoms bring patients back, and the dates are recorded at months 5, 7, 9 and 11. Three patients remain recurrence-free at month 12 and are right censored there. The analysis fits an exponential distribution with constant monthly hazard $\lambda$.

Step 1. Write the log-likelihood. Each left-censored patient contributes $\ln(1 - e^{-3\lambda})$, the log probability of recurrence by month 3. Each exact recurrence at time $t$ contributes $\ln \lambda - \lambda t$, and each right-censored patient contributes $-12\lambda$. The exact times sum to 5 + 7 + 9 + 11 = 32 months and the right-censored patients add 3 × 12 = 36 months, so

$$\ell(\lambda) = 3 \ln\left(1 - e^{-3\lambda}\right) + 4 \ln \lambda - 68 \lambda$$

where $\ell(\lambda)$ is the log-likelihood, $\lambda$ is the monthly hazard, the leading 3 counts the left-censored patients, 4 counts the exact recurrences and 68 is total follow-up outside the left-censored records.

Step 2. Solve for the hazard. Setting the derivative to zero gives

$$\frac{9 e^{-3\lambda}}{1 - e^{-3\lambda}} + \frac{4}{\lambda} = 68$$

where the first term is the contribution of the three left-censored patients and the second that of the four exact recurrences. The equation has no closed form and is solved numerically, giving $\hat{\lambda} = 0.09684$ per month. At that value $e^{-3\hat{\lambda}} = e^{-0.2905} = 0.7479$, the first term is 9 × 0.7479 / 0.2521 = 26.70 and the second is 4 / 0.09684 = 41.30, which sum to 68.00. Mean recurrence-free time is 1 / 0.09684 = 10.33 months, the median is 0.6931 / 0.09684 = 7.16 months and the fitted 12-month recurrence-free probability is $e^{-1.162} = 0.313$. The fitted probability of recurrence by month 3 is one minus 0.7479, or 0.252, against 3 of 10 observed, a gap too small in ten patients to test the constant hazard.

Step 3. Compare three shortcuts. Without left censoring, the exponential estimate is events divided by total time at risk. Dating the three left-censored recurrences at the scan gives 7 events over 3 × 3 + 32 + 36 = 77 months, a hazard of 7 / 77 = 0.0909 and a mean of 11.00 months. Excluding the three patients gives 4 events over 68 months, a hazard of 4 / 68 = 0.0588 and a mean of 17.00 months. Imputing month 1.5 for each gives 7 events over 4.5 + 32 + 36 = 72.5 months, a hazard of 7 / 72.5 = 0.0966 and a mean of 10.36 months.

Step 4. Interpret the differences. Dating events at the scan adds 0.67 months (11.00 less 10.33), or 0.056 years, to mean recurrence-free time. At an illustrative recurrence-free utility of 0.80, that is 0.80 × 0.056 = 0.045 undiscounted QALYs per patient credited to time not spent recurrence-free, before subtracting the utility those months carry after recurrence. Exclusion adds 6.67 months (0.556 years), worth 0.80 × 0.556 = 0.44 recurrence-free QALYs, because the patients with the earliest events vanish from the data. Midpoint imputation lands close here only because the window is short relative to a 10-month mean and the assumed hazard is constant, which is not a general property.

Bias from ignoring or excluding left-censored records

Cain and colleagues simulated menopausal transition studies with left truncation on age at the final menstrual period. Accounting for left censoring of age at onset of the late stage reduced bias by about one half, and the large negative bias in estimated time spent in the late stage was primarily due to failure to account for left censoring. Where people already past a milestone can be identified, they prefer recording current status on all potential participants and analysing those past it as left censored, which gives estimates that are less biased and more precise than excluding them.

In the German claims data, Weißbach and colleagues found that including the left-censored insurants increased estimated dementia incidence at younger ages and attenuated the increase with age. It also reduced the slope of the trend across birth cohorts, because the share of left-censored people was larger in the older cohorts.

Consequences for decision models

Age-specific incidence from registries or claims data often becomes a transition probability in a Markov model, such as the move from a healthy state into dementia. Dropping people who already have the disease when the data window opens, or dating their onset at entry, alters the age pattern of the hazard that the model applies to every simulated cohort.

For endpoints such as relapse-free survival, dating events found at a first assessment at that date overstates time in the event-free state, as Step 4 shows. In a partitioned survival model the shift moves QALYs and state costs into the pre-event state, and a fitted parametric survival model carries it into the extrapolated tail. When the timing of first assessments differs between arms, the shift can bias the incremental estimate too.

Zero healthcare costs are not left-censored values. Mihaylova and colleagues, reviewing cost methods, treat zeros as genuine observations rather than unobserved values, and describe two-part models as widely used where non-users create excess zeros.

Left censoring beside right censoring, interval censoring and truncation

Left censoring is one form of censoring. Its neighbours differ in which side of the event time is missing and whether the patient appears in the data at all, and interval censoring includes left censoring as the window that starts at the origin. Right censoring places the event after the last event-free observation, and under left truncation a person whose event came before entry is never enrolled.

The contrast with left truncation decides the analysis. A truncated design needs the likelihood conditioned on reaching entry, which Cain and colleagues apply by dividing by the probability of inclusion given age at entry, and they note that collecting current status at recruitment converts a left-truncation problem into a left-censoring problem.

Misreadings of left-censored data

A positive result at a first test records when the condition was found, not when it began, so a curve built on such dates describes time to detection under one assessment schedule. The bound from a left-censored record is informative and is not missing data to be deleted.

The name reverse Kaplan-Meier has two unrelated uses. Clark and colleagues use it to summarise follow-up, with the event indicator reversed so that being censored becomes the outcome, whereas the estimator Gillespie and colleagues recommend estimates the distribution of left-censored values.

A small proportion of left-censored records does not make them safe to ignore when they cluster in one subgroup. In the German claims sample the 11,000 left-censored insurants were under 5% of 250,000, yet their share differed between birth cohorts, and that imbalance changed the estimated cohort trend.

Sources

• Cain KC, Harlow SD, Little RJ, Nan B, Yosef M, Taffe JR, Elliott MR. Bias due to left truncation and left censoring in longitudinal studies of developmental and disease processes. American Journal of Epidemiology. 2011;173(9):1078-1084. https://doi.org/10.1093/aje/kwq481
• Clark TG, Bradburn MJ, Love SB, Altman DG. Survival analysis part I: basic concepts and first analyses. British Journal of Cancer. 2003;89(2):232-238. https://pmc.ncbi.nlm.nih.gov/articles/PMC2394262/
• Fay MP, Shaw PA. Exact and asymptotic weighted logrank tests for interval censored data: the interval R package. Journal of Statistical Software. 2010;36(2):1-34. https://doi.org/10.18637/jss.v036.i02
• Gillespie BW, Chen Q, Reichert H, Franzblau A, Hedgeman E, Lepkowski J, Adriaens P, Demond A, Luksemburg W, Garabrant DH. Estimating population distributions when some data are below a limit of detection by using a reverse Kaplan-Meier estimator. Epidemiology. 2010;21(Suppl 4):S64-S70. https://doi.org/10.1097/EDE.0b013e3181ce9f08 (abstract checked)
• Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data. 2nd ed. New York: Springer; 2003.
• Mihaylova B, Briggs A, O'Hagan A, Thompson SG. Review of statistical methods for analysing healthcare resources and costs. Health Economics. 2011;20(8):897-916. https://doi.org/10.1002/hec.1653
• Weißbach R, Kim Y, Dörre A, Fink A, Doblhammer G. Left-censored dementia incidences in estimating cohort effects. Lifetime Data Analysis. 2021;27(1):38-63. https://doi.org/10.1007/s10985-020-09505-1
• Yefenof J, Goldberg Y, Wiler J, Mandelbaum A, Ritov Y. Self-reporting and screening: data with right-censored, left-censored, and complete observations. Statistics in Medicine. 2022;41(18):3561-3578. https://doi.org/10.1002/sim.9434