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With left truncation, a subject is observed only if it survives to its entry time $E$: subjects with $T\le E$ are absent from the sample. A business already operating when observation begins has this delayed-entry mechanism. It joins a risk set only after its recorded entry age. Left truncation excludes earlier failures entirely; left censoring instead includes subjects whose event is known to have happened before a specified time. These mechanisms must not be conflated.

Statistician B uses the more appropriate analysis. Its start–stop representation includes left truncation as well as right censoring. Both entry and exit are ages since opening, so a business belongs to the risk set at age $t$ only when $\mathrm{ageatentry}<t\le \mathrm{obsage}$. The event indicator distinguishes closure from a censored end of observation. The counting-process representation does not mean that multiple closures per business have been observed.

Statistician A records only exit and status and treats every sampled business as at risk from opening. That incorrectly adds already-established businesses to risk sets at ages before they entered observation, ignoring their necessary survival to entry. Its first risk count is $60$, whereas B's is $25$. The extra non-events dilute early estimated closure hazards and here bias the estimated survival upward. Correcting left truncation makes later risk sets capable of increasing as businesses enter at their respective ages.

For the Kaplan–Meier estimator with delayed entry, at event ages $t_j$ with $d_j$ closures and $r_j$ businesses at risk,The standard error follows from the Greenwood formula. Between event times this survival function estimate is constant. At age five the last preceding event is at $4.52$, and at age ten it is at $9.90$. Thus the requested estimates, including their precision, areandThese correspond to estimated proportions 46.2% and 28.0%, not to the probability of surviving five or ten additional years conditional on being sampled at an arbitrary age. A's corresponding estimates $0.650$ and $0.407$ use the inappropriate risk sets.

This interpretation assumes appropriately independent entry and censoring and a suitable common lifetime distribution across entry cohorts. A delayed-entry correction does not remove arbitrary informative sampling or calendar effects. More generally, if no observation covers the earliest ages, absolute survival from opening requires additional information about survival before the first observable age; the reported product-limit normalization is the one used in the supplied analysis.

Place discrete hazards $q_j$ at the ordered distinct event times $t_j$. If $d_j$ events occur among $r_j$ individuals at risk immediately before $t_j$, the nonparametric empirical likelihood isEquivalently, an observed event at $x$ contributes the probability mass at $x$, while a right-censored observation contributes the survivor probability beyond its censoring time. Maximization gives ${\hat{q}}_j=d_j/r_j$ and the Kaplan–Meier estimator

Left truncation means an individual is observed only conditional on surviving beyond an entry time $L$. Ignoring it overrepresents long survivors and creates survivorship bias. A subject with event or censoring time $X>L$ contributes its usual likelihood divided by $S(L)$; in risk-set form, that individual enters each $r_j$ only for event times satisfying $L<t_j\le X$.

At each distinct event time $t_j$, let $d_j$ events occur among $r_j$ people in the risk set. The Kaplan–Meier estimator iscensorings remove people from later risk sets but create no factor.

Left truncation, or delayed entry, means that an individual is observed only after surviving to an entry time. A registry assembled from patients alive when a clinic opens is a practical example. Adapt Kaplan–Meier by admitting each person to the risk set only at their entry time.

Period survival analysis is useful when recent prognosis is desired but complete long-term follow-up of a recent diagnosis cohort is unavailable. For a chosen calendar year, intersect every patient's observed follow-up with that year. Express the surviving pieces on the time-since-diagnosis scale, treat the beginning of the calendar window as delayed entry and its end as right censoring, and apply Kaplan–Meier with those entry and exit times.

Measure time in years after age 70. Residence in the care home begins at time 3, so inclusion is conditional on $T>3$: this is left truncation, also called delayed entry. Exact death at time 13 contributes

Construct the two countries' period-specific risk sets with the same left truncation and right censoring rules, then compare their event counts by a log-rank test. This is a nonparametric comparison because it does not specify the shape of either country's hazard function.

Left truncation means that an individual enters observation only after a delayed entry time $L_i$ and is included only if the event time exceeds $L_i$. Earlier failures are absent from the dataset, so analysis must condition on survival to entry.

Survival data describe the time $T$ from a specified origin to an event, together with covariates and the observation mechanism. Under right censoring, the observed exit time is $Y=\min (T,C)$ and the event indicator is $\delta =1_{\{T\le C\}}$. With left truncation, an entry time $E$ is also recorded and subjects are observed only if $T>E$. Recording the time origin, entry, exit and event indicator is essential for constructing valid risk sets.