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If you have a dataset, sorted into ascending order by survival time (minimum of censoring and event time), and this dataset contains at least one censoring before the first event (so that the start of the censoring indicator looks similar to 0 0 0 1 for example), how are these censored events dealt with? Do they count towards the risk set for the first event? Do they contribute to the estimation of survival rates or model fitting at all? Any information on how different methods deal with censorings before the first event would be appreciated.

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The risk set for the first event only includes cases still at risk at that time, so any cases censored earlier than that time do not enter that (or any) risk set, at least in standard Kaplan-Meier or Cox proportional hazards analyses. You can check this by comparing results on a data set where you either include or exclude such early-censored cases.

It's possible that some methods of fitting defined parametric models to survival data might include information on early-censored cases, but I don't have expertise in that type of modeling. Others on this site are better equipped to address parametric survival modeling.

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To complete @EdM's answer, if an event is right censored before the first observed event time, it will contribute to the log likelihood function, i.e. it will contribute

$\log(S(t_i | \theta))$

where $\theta$ is the parameter of interest.

However, this contribution will be very minimal; for most values of $\theta$ implied by the rest of the data, $S(t_i | \theta) \approx 1$, implying for little impact on the likelihood function, and thus very little impact of the estimation of $\theta$ itself.

To tie back into EdM's answer, the reason these observations do not affect the semi- and non-parametric models is that in those cases, $S(t_i|\hat \theta) = 1$, as assigning probability mass before the first event would only decrease the likelihood function.

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