Survival analysis: up from Kaplan–Meier–Greenwood

In the type of survival analysis that now is routine, only the points of follow-up at which deaths from the cause at issue occurred make contributions to the Greenwood standard error (SE) of the survival rate’s Kaplan–Meier (KM) point estimate. An equivalent of this ‘KMG’ analysis draws from defined subintervals of the survival period being addressed. The data on each subinterval consist of the number of deaths from the cause at issue and the amount of population–time of follow-up, d _j and T _j , together with the duration of the interval, t _j . The KM point estimate is replicated by \( \exp [-{\sum \nolimits_{j} ({{{d_{j} }/ \mathord{{\vphantom {{d_{j}}{T_{j}}}} \kern-0em} {T_{j} }}})t_{j}}], \) and the KMG interval estimate is replicated by treating the {d _j } as a set of point estimates of Poisson parameters {λ _j }, thus taking the SE of \( \sum \nolimits_{j}({{{d_{j} } \mathord{{\vphantom {{d_{j} } {T_{j} }}} \kern-0em} {T_{j}}}})t_{j}\) to be \([{\sum \nolimits _{j} d_{j} ({{{t_{j} } / \mathord{{\vphantom {{t_{j} } {T_{j} }}} \kern-0em} {T_{j} }}})^{2} }]^{{{1/ \mathord{{\vphantom {1 2}} \kern-0em} 2}}}.\) In both the KMG analysis and this equivalent of it, the SE used to derive the survival rate’s lower confidence limit needs to be augmented by a factor that accounts for the loss of information due to censorings subsequent to the last ‘failure’ in the survival period at issue. But, SE-based interval estimation of survival rate actually needs to be replaced by a first-principles counterpart of it. A suitable point of departure in this is first-principles asymptotic interval estimation of the Poisson parameter \( \lambda =\sum \nolimits _{j}{\lambda_{j}}, \) if not the exact counterpart of this. A confidence limit for the survival rate can then be based on suitable augmentation or contraction of the {d _j } set to \( \{ d_{j}^{*} \} \) consistent with a given limit for λ, the corresponding survival-rate limit being \( \exp [-{\sum \nolimits _{j} ({{{d_{j}^{ *} } \mathord{/ {\vphantom {{d_{j}^{ *} } {T_{j} }}} \kern-0em} {T_{j}}}})t_{j}}]. \) Suitable augmentation is constituted by an identical addition to each \(d_{j}^{1/2},\) suitable contraction by an identical subtraction from each \(d_{j}^{1/2} \ge 1.\)