KaplanMeier

Statistical assessments with the Kaplan-Meier survival function (lower/upper limits) to test whether a measured value x0 (typically the mean of a distribution) is associated with some population x, accounting for lower limits in x. If the Kaplan-Meier survival function at x0 is outside the limits of 0.01 to 0.99, one can confidently reject the null hypothesis that the measurement x0 is associated with the measurements x.

I originally developed and implemented this script for Flury et al. 2024 for tests involving a sample of 89 Lyman continuum measurements, 39 of which were upper limits requiring the censoring treatment of the Kaplan-Meier survival curve.

Examples

Kaplan Meier Survival Test

``` python import matplotlib.pyplot as plt from numpy.random import seed,rand,randn from KaplanMeier import *

seed(123) x = randn(100) c = rand(100)<0.3 x0 = array([1.65]) x0_err = array([[0.3],[0.5]])

kmx,kmy = kmcurve(x,c) px,pe = kmeval(x0,x,c,x0err=x0err) ``` which gives the results below

Mantel Log Rank Test with Kaplan Meier Survival

python from numpy.random import seed,rand,randn from KaplanMeier import * seed(123) x1 = rand(50) # all reference measurements c1 = x1<0.1 # where measurements are upper limits x2 = randn(30)*0.1+0.5 # new/test measurements c2 = x2<-0.1 # where measurements are upper limits D,Z,pvalue = km_logrank(x1,c1,x2,c2) # get p-value print(f'p-value = {pvalue:.6f}') which prints the following to the command line python p-value = 0.037192

BibTeX reference

While this code is provided publicly, I request that any use thereof be cited in any publications in which this code is used. BibTeX formatted reference provided below.

bibtex @ARTICLE{Flury2024, author = {{Flury}, Sophia R. and {Jaskot}, Anne E. and {the LzLCS Collaboration}}, title = "{The Low-Redshift Lyman Continuum Survey: The Roles of Stellar Feedback and ISM Geometry in LyC Escape}", journal = {\apjs}, keywords = {Reionization, Galactic and extragalactic astronomy, Ultraviolet astronomy, Hubble Space Telescope, 1383, 563, 1736, 761, Astrophysics - Astrophysics of Galaxies, Astrophysics - Cosmology and Nongalactic Astrophysics}, year = 2024, month = {}, volume = {}, number = {}, eid = {}, pages = {}, doi = {10.48550/arXiv.2409.12118}, archivePrefix = {arXiv}, eprint = {2409.12118}, url = {https://github.com/sflury/KaplanMeier}, primaryClass = {astro-ph.GA}, adsurl = {https://ui.adsabs.harvard.edu/abs/}, adsnote = {Provided by the SAO/NASA Astrophysics Data System} }

An additional reference to consider is the original Kaplan & Meier (1958) paper which first presented the Kaplan-Meier statistic. The BibTeX entry for their paper is listed below.

bibtex @article{KaplanMeier1958, author = {E. L. Kaplan and Paul Meier}, title = {Nonparametric Estimation from Incomplete Observations}, journal = {Journal of the American Statistical Association}, volume = {53}, number = {282}, pages = {457-481}, year = {1958}, publisher = {Taylor & Francis}, doi = {10.1080/01621459.1958.10501452}, }

The related log-rank test is based on the formalism from Mantel (1966). The BibTeX entrey for his paper is listed below.

bibtex @article{Mantel1966, title={Evaluation of survival data and two new rank order statistics arising in its consideration}, author={Nathan Mantel}, journal={Cancer Chemotherapy Reports}, year={1966}, volume={50}, number={3}, pages={163-170} }

DOI

Licensing

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see https://www.gnu.org/licenses/.