1. Purpose

                
This tool finds the optimal cutpoint for continuous variables in survival analysis.
                      Instead of using the median as a one-size-fits-all solution, it tests all possible
                      cutpoints and selects the one that maximizes the survival difference between groups.

                
2. Methods

                
    
                  
  • 
                    BDS-Cut (Bootstrap Density-Stabilized Cutpoint):
                     Two-stage procedure: (1) brute-force scan all candidate cutpoints for visualization; (2) stabilize via 200 bootstrap replications, reporting the density peak as the final cutpoint. More stable and less prone to overfitting than raw optimum.
                  
    • 
                  
  • 
                    MaxStat (Maximally Selected Rank Statistics):
                     Uses the maxstat package to find the cutpoint that maximizes the log-rank statistic, with a multiplicity-corrected p-value (conditional Monte Carlo). This is the recommended method when a valid p-value is needed for hypothesis testing.
                  
    • 
                  
  • 
                    Median / Tertile / Quartile:
                     Traditional quantile-based splitting.
                  
    • 
                  
  • 
                    Custom:
                     User-specified cutpoint value(s).
                  
    • 
                

                
3. Output

                
    
                  
  • 
                    LogRank_P:
                     P-value from log-rank test comparing survival curves.
                  
    • 
                  
  • 
                    Adjusted_P (MaxStat only):
                     Multiplicity-corrected p-value from maxstat.test (Lau94). This corrects for the multiple testing inherent in scanning all cutpoints.
                  
    • 
                  
  • 
                    HR:
                     Hazard ratio from Cox model (High vs Low for BDS-Cut; Gn vs G1 for quantile methods).
                  
    • 
                  
  • 
                    BS_Stability (BDS-Cut only):
                     Bootstrap stability flag.
                              Stable (>0.8): reproducible across bootstrap samples.
                              Moderate (0.5-0.8): interpret with caution.
                              Unstable (<0.5): driven by chance; do not use.
                  
    • 
                  
  • 
                    BS_CI (BDS-Cut only):
                     95% confidence interval of the bootstrap cutoff distribution.
                  
    • 
                

                
4. Caution

                
Note: The BDS-Cut method searches all possible cutpoints, which inflates
                           Type I error. The reported p-value is not corrected for
                           multiple testing. Always validate the selected cutpoint in an independent
                           cohort. See Altman et al. (1994) JNCI.

                
5. Algorithm Principles (原理说明)

                
This tool implements a two-stage, bootstrap-stabilized cutpoint procedure for continuous predictors in right-censored survival data. It is designed as an exploratory stabilization strategy, not as a significance-testing framework.

                
5.1 Stage 1 - Brute-force scan (exploration only)

                
For a continuous variable X, all interior candidate cutpoints c are examined under the constraint that each side retains at least max(10%, 5) of observations. For each c, subjects are split into G_L = {X ≤ c} and G_H = {X > c}, and a two-sample log-rank statistic is computed:

                
χ²(c) = (O_L - E_L)²/V_L + (O_H - E_H)²/V_H

                
The raw optimum is: ĉ_raw = arg min_c p_logrank(c)

                

                  This raw optimum is known to suffer from selection bias
                  : scanning many c under H_0 inflates Type I error (Altman et al., 1994; Lausen & Schumacher, 1992). Therefore ĉ_raw and its associated p are never used for inference in this tool; they only drive the scan plot.
                

                
5.2 Stage 2 - Bootstrap density-peak stabilization

                
We draw B = 200 bootstrap samples with replacement from (X, T, δ). On each bootstrap sample b, the same brute-force scan is repeated, yielding ĉ_raw^(b). Let C = {ĉ_raw^(b) : b = 1,...,B, valid}.

                
The reported cutpoint is the mode of the bootstrap distribution, approximated by the peak of a kernel density estimate:

                
ĉ_stable = arg max_x f̂_KDE(x; C)

                

                  Rationale: 
                  the raw optimum is a high-variance estimator (sensitive to single observations); the bootstrap distribution reveals which region of X consistently separates groups. The density peak is a post-selection stabilizer, conceptually aligned with bootstrap aggregation of cutpoints (Thiele, 2021; cutpointr vignette).
                

                
5.3 What we report, and what we deliberately do NOT report

                

                      
QuantityReported?Notes

                      
                        
ĉ_stable✓Density peak of bootstrap cutpoints

                        
Bootstrap 95% CI of ĉ✓2.5%-97.5% quantiles of C

                        
Stability index 1-SD(C)/|median(C)|✓Flag: Stable (>0.8), Moderate (0.5-0.8), Unstable (<0.5)

                        
KM curve at ĉ_stable✓Descriptive only

                        
Log-rank p at ĉ_stable on full data✓ (labeled "uncorrected")Exploratory, not adjusted for cutpoint search

                        
HR (High vs Low) at ĉ_stable✓Optimistic: conditional on selected cutpoint

                        
Multiplicity-corrected p-value✗Use maxstat::maxstat.test() or exact distribution methods for that

                        
Optimism-corrected p / HR✗Use Harrell-style bootstrap validation externally

                      
                    

                
5.4 Why this is not a 'new test'

                
The density-peak selector reduces estimation instability but does not correct the selection-induced Type I error inflation (Hothorn & Lausen, 2003; Li et al., 2023). Thus:

                
    
                  
  • 
                    For hypothesis testing
                     ('is this biomarker prognostic?'), report maxstat::maxstat.test() alongside.
                  
    • 
                  
  • 
                    For exploratory threshold generation
                     ('which cutoff is worth validating externally?'), this tool's output is sufficient.
                  
    • 
                  
  • Any cutpoint from this tool must be validated in an independent cohort before clinical use.
    • 
                

                
5.5 Minimum data requirements (enforced)

                
    
                  
  • Total n ≥ 10 (soft gate; warning otherwise)
    • 
                  
  • Total events ≥ 30 recommended
    • 
                  
  • Each side of any candidate cutpoint ≥ 5 events ideally (10% rule is sample-size guard, not event guard)
    • 
                

                
6. References

                
    
                  
  1. Altman DG, Lausen B, Sauerbrei W, Schumacher M. Dangers of using 'optimal' cutpoints in the evaluation of prognostic factors. J Natl Cancer Inst. 1994;86(11):829-835.
    2. 
                  
  2. Lausen B, Schumacher M. Maximally selected rank statistics. Biometrics. 1992;48(1):73-85.
    3. 
                  
  3. Hothorn T, Lausen B. On the exact distribution of maximally selected rank statistics. Comput Stat Data Anal. 2003;43(2):121-137.
    4. 
                  
  4. Hollander N, Sauerbrei W, Schumacher M. Confidence intervals for the effect of a prognostic factor after selection of an 'optimal' cutpoint. Stat Med. 2004;23(11):1701-1713.
    5. 
                  
  5. Contal C, O'Quigley J. An application of changepoint methods in studying the effect of age on survival in breast cancer. Comput Stat Data Anal. 1999;30(3):253-270.
    6. 
                  
  6. Thiele C, Hirschfeld G. cutpointr: Improved estimation and validation of optimal cutpoints in R. J Stat Softw. 2021;98(11):1-27.
    7. 
                  
  7. Li N, Song Y, Lin CD, Tu D. Bootstrap adjustment to minimum p-value method for predictive classification. Stat Sin. 2023;33(3):2065-2086.
    8. 
                  
  8. Hothorn T, Lausen B. Maximally Selected Rank Statistics in R (maxstat package vignette). CRAN.
    9. 
                  
  9. Thiele C. Robust cutpoint estimation (cutpointr package vignette). CRAN.
    10. 
                

                
7. Citation

                

                  If you use BDS-Cut in a publication, please cite: 
                  We used the bootstrap density-peak stabilized cutpoint (BDS-Cut) procedure as implemented in KMcutoff (math81.com) for exploratory threshold generation, with external validation planned per Altman 1994.