= Kaplan–Meier estimator with delayed entry
{c}
{title2=$\widehat S(t)$}
For independently <left-truncated> and <right-censored> <survival data>, let $E_i$ and $Y_i$ be entry and exit times on the same clock. At each distinct event time $t_j$, the <risk set> is $R_j=\{i:E_i<t_j\le Y_i\}$, with size $r_j$, and $d_j$ events occur. The <Kaplan–Meier estimator> is
$$
\widehat S(t)=\prod_{t_j\le t}\left(1-\frac{d_j}{r_j}\right).
$$
Before entry a subject contributes neither an event nor time at risk. Thus adding observed entrants can increase successive <risk sets>; pretending that all subjects were observed from time zero distorts the estimated <hazard function>. The estimator identifies the <survival function> over ages covered by observation; normalization from birth requires observation arbitrarily close to time zero or additional knowledge of earlier survival.
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