March 13, 2024
New immunotherapies forcancer have shown to have different treatment effects often with delay compared to other cytotoxic treatments and, therefore, this has called for different
ways of modeling the survival curves. The usual log-rank test has relied on the proportional hazardsassumption for the test but delays in survival would cause violations in that
assumption. Various alternatives have been proposed such as a weighted
log-rank, restricted mean survival time, etc.. The authors proposed a technique that wouldhave allowed for the delay and also crossing in curves, which theoretically has
been a no go with proportional hazards assumption.
In the first stage,one finds conditional estimates of the survival function by maximizing a
nonparametric log-likelihood that is conditional on the value of two crossing
parameters. Then, in the second stage, one estimates these crossing parameters
by maximizing a profile log-likelihood function. For the first part, they created a terms,theta and gamma, to represent crossing time. The authors then estimated survival under a non-parametric paradigm proposed by Park et al (2012). They only considered single crossingpossibilities and non-smooth hazard possibilities. The authors then characterized ways toestimate the survival probabilities between the two curves. The first mentioned was at specific timepoints, which can avoid the proportional hazard assumption. The second was the
proportion surviving up to crossing. Thethird was restricted mean survival time up to a particular defined time, tau. The fourth was restricted residual mean life. The fifth was crossing time conditional survival curves. Finally the last was pre and post crossingaverage hazard ratios.
They then had run simulations where they assumed survivalfollowed a piecewise exponential distribution in both treatment arms and they showed six scenarios for different types of crossing curves. They found their methods of single-crossing constrained estimates of the survival curve in
general performed better than a standard Kaplan-Meier based curve but this was
not always the case in every scenario. They also tested this in a real data
example and then in the discussion, they suggested their method may work better
as a secondary analysis under condition of delayed treatment effects since they
had an issue with robustness of model misspecification. They then discussed all sorts of otherextensions or further possible developments for their method.
Written by,
Usha Govindarajulu
Keywords: survival,delayed treatment effects, Kaplan-Meier, log-rank test, restricted mean
survival time
References
Henderson NC, Nam, K, and Feng D (2024). “Nonparametric analysis of delayed treatment effects usingsingle-crossing constraints” Biometrical Journal. https://doi.org/10.1002/bimj.202200165
Park, Y., Kalbfleisch, J. D., & Taylor,J. M. (2012). Constrained nonparametric maximum likelihood estimation ofstochastically ordered survivor functions. Canadian Journal ofStatistics, 40(1), 22–39.
https://onlinelibrary.wiley.com/cms/asset/941bc966-1829-4bad-bd83-4caf1e3c7434/bimj2560-fig-0001-m.jpg
