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February 28, 2023


The author, Olivier Bouaziz, developed a new method to calculate pseudo-observations for the survival function and also the restricted mean survival time (RMST) that used formulas which were based on the original estimators and not on the jackknife ones.  Prior to this, Andersen et al had developed pseudo-observations using a jackknife method from a survival function and subsequently used as response varaibles in a particular regression model. Bouaziz provided the background on the jackknife pseudo obervations and how they can be estimated through a generalized estimating equation using the geese function in the geepack package in R.  He then goes ont ot show approximating pseudo-observations for right-censored data in Section 3.  They show deriving pseudo-observations from the orginal sample for survival without using the jackknife procedure. This would be then be faster computationally than the jackknife method. They also presented using an infinitesmal jackknife method to estimate survival in the R survival package in the fuction pseudo.

In Section 4 he discusssed the computation of pseudo observations for interval-censored data.  They appeared to dismiss the non-parametric case but showed in the parametric case that the pseudo observations could be obtained without actually computing the required number of jackknife estimators. This led to computational efficiency since computing the pseudo observations was similar to computing a pointwise estimate from the Newton-Raphson.  In Section 5, they discussed their simulations scenarios for right censored data and for interval censored data where for each they based the pseudo observations on a Kaplan-Meier. They found almost identical results between the jackknife and the approximate formula.  In Section 6, a real data example is based on the Cardiovascular Health Study analysis performed in Zhao (2021), which is a based on a deep neural network.  They found the reduction in computational time is a major advantage for their formula, which really assists in implementing a neural network which is computationally intensive. They also looked at a Signal Tandmobiel data, longtindual dental survey, using three different models: standard Cox model, logistic model for conditional survival, and the conditional RMST model that they presented.

In their Discussion, they do mention that a reviewer pointed out there already has been an alternative to their fast formula for right censored data in the pseudo function of the survival package which is based on an infinitesimal jackknife.  This has not yet been extended to the inteval censored data approach. Finally, the asymptotic formulas based on the Von Mises approach developed in this paper for pseudovalues of the survival function and the RMST are available in a package, FastPseudo, avalable at their Github site.


Written by,


Usha Govindarajulu


Keywords: survival, pseudo observations, RMST, neural network, jackknife,Von Mises



Bouaziz, O. (2023). “Fast approximations of pseudo-observations in the context of right censoring and interval censoring” Biometrical Journal.


Andersen, P. K., Borgan, Ø., Gill, R. D., & Keiding, N. (1993). Statistical models based on counting processes. Springer Series in Statistics. Springer-Verlag.