Bias for the Weibull distribution.

Introduction
Survival time models are commonly employed in medicine and health sciences when analysing data. In these time-to-event analyses, it is often necessary to dichotomise variables that are metrically measured. One example could be to assign patients to different risk groups based on an occurring event. Besides univariable methods, multivariable approaches also exist for establishing cutpoints. Up to now, these multivariable approaches have hardly been investigated.
Methods
Using a Monte Carlo simulation study, we analysed eight multivariable methods from the literature to establish a cutpoint of a biomarker in the context of a semiparametric Cox regression model. The methods are the following: maximising the chi-square statistic, maximising the chi-square statistic with a split-sample approach, maximising the c-index using either the AddFor- or Genetic algorithm, maximising the concordance probability estimator (CPE) with the AddFor- or Genetic algorithm, and minimising the Akaike information criterion (AIC). We compared these methods with each other and in addition with the univariable log-rank minimum p-value approach. The simulation parameters analysed included the cutpoint’s distance from the biomarker’s median, sample size, total censoring, censoring before the end of the follow-up time (drop-outs), and the survival time distribution. Bias and empirical standard error were used as the primary performance measures. Furthermore, each method is illustrated using two practical data examples.
Results
All analysed methods are biased towards the biomarker’s median. Multivariable methods that estimate the cutpoint by using the lowest AIC or the maximum of the chi-square statistic have the lowest bias and empirical standard error in most simulation scenarios. The difference in bias between the methods based on maximising the c-index or maximising the CPE is minimal. Regardless of the distribution used (Weibull, Gompertz, or exponential), the respective bias shows similar dependencies on the simulation parameters.
Conclusions
Multivariable methods to estimate a biomarker’s cutpoint in survival time analyses using the Cox regression model may represent a good alternative to univariable methods. Our simulation has shown that methods maximising the chi-square statistic or minimising the AIC, respectively, perform better than the univariable method using the minimum p-value approach and outperform multivariable methods based on the c-index or CPE.