Addressing the End-Cut Preference Problem in Survival Trees with a Smooth Sigmoid Surrogate
- 1Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX 79968, USA
- 2Department of Statistics and Data Science, University of Cincinnati, Cincinnati, OH 45221, USA
- 3Biostatistics and Computational Biology Core, Center for Population Health and Outcomes Research, Children’s Mercy Kansas City, Kansas City, MO 64108, USA
Abstract
The end-cut preference (ECP), the tendency of greedy tree algorithms to select split points near the extremes of a predictor’s range, can produce unstable and highly imbalanced partitions. For survival trees based on the maximized log-rank statistic, we prove that, under the null and an expanding admissible cutpoint range, every maximizing cutpoint leaves any fixed interior quantile interval with probability tending to one. The proof compares the tight interior maximum of a studentized empirical bridge with its diverging maximum over cutpoints approaching an endpoint. An exact variance decomposition of a quadratic Taylor polynomial also identifies a non-negative hard-split term with boundary scale (1−2c)2Λn/{nc(1−c)}. For the smooth sigmoid surrogate (SSS), the corresponding Cox-score term is uniformly Op(a/n), where a is the sigmoid shape parameter. This bound has no endpoint factor. Numerical experiments across censoring levels, signal strengths, split restrictions, and recursive splitting show that SSS generally selects fewer end cuts and yields lower cutpoint RMSE under the alternatives studied, with the clearest gains for weak signals, while retaining the interpretation of ordinary hard splits.
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BibTeX
@article{tenkorang2026addressing,
title = {{Addressing the End-Cut Preference Problem in Survival Trees with a Smooth Sigmoid Surrogate}},
author = {Nicholas Tenkorang and Xiangya Wang and George Ekow Quaye and Xiaogang Su},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203833},
url = {https://www.mdpi.com/2227-7390/14/20/3833},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Addressing the End-Cut Preference Problem in Survival Trees with a Smooth Sigmoid Surrogate
AU - Nicholas Tenkorang
AU - Xiangya Wang
AU - George Ekow Quaye
AU - Xiaogang Su
PY - 2026
DA - 2026/10/21
JO - Mathematics
DO - 10.3390/math14203833
UR - https://www.mdpi.com/2227-7390/14/20/3833
AB - The end-cut preference (ECP), the tendency of greedy tree algorithms to select split points near the extremes of a predictor’s range, can produce unstable and highly imbalanced partitions. For survival trees based on the maximized log-rank statistic, we prove that, under the null and an expanding admissible cutpoint range, every maximizing cutpoint leaves any fixed interior quantile interval with probability tending to one. The proof compares the tight interior maximum of a studentized empirical bridge with its diverging maximum over cutpoints approaching an endpoint. An exact variance decomposition of a quadratic Taylor polynomial also identifies a non-negative hard-split term with boundary scale (1−2c)2Λn/{nc(1−c)}. For the smooth sigmoid surrogate (SSS), the corresponding Cox-score term is uniformly Op(a/n), where a is the sigmoid shape parameter. This bound has no endpoint factor. Numerical experiments across censoring levels, signal strengths, split restrictions, and recursive splitting show that SSS generally selects fewer end cuts and yields lower cutpoint RMSE under the alternatives studied, with the clearest gains for weak signals, while retaining the interpretation of ordinary hard splits.
SN - 2227-7390
ER -