Incorporating a New Summary Statistic into the Min–Max Approach: A Min–Max–Median, Min–Max–IQR Combination of Biomarkers for Maximising the Youden Index

Abstract

Linearly combining multiple biomarkers is a common practice that can provide a better diagnostic performance. When the number of biomarkers is sufficiently high, a computational burden problem arises. Liu et al. proposed a distribution-free approach (min–max approach) that linearly combines the minimum and maximum values of the biomarkers, involving only a single coefficient search. However, the combination of minimum and maximum biomarkers alone may not be sufficient in terms of discrimination. In this paper, we propose a new approach that extends that of Liu et al. by incorporating a new summary statistic, specifically, the median or interquartile range (min–max–median and min–max–IQR approaches) in order to find the optimal combination that maximises the Youden index. Although this approach is more computationally intensive than the one proposed by Liu et al, it includes more information and the number of parameters to be estimated remains reasonable. We compare the performance of the proposed approaches (min–max–median and min–max–IQR) with the min–max approach and logistic regression. For this purpose, a wide range of different simulated data scenarios were explored. We also apply the approaches to two real datasets (Duchenne Muscular Dystrophy and Small for Gestational Age).

Comparing the Min–Max–Median/IQR Approach with the Min–Max Approach, Logistic Regression and XGBoost, Maximising the Youden Index

“…Aznar-Gimeno et al proposed new non-parametric approaches, so-called min-maxmedian (MMM) and min-max-IQR (MMIQR) [27], which extend the idea of the min-max approach by applying our proposed stepwise algorithm [25], following the suggestion of Pepe et al [14,15]. The aim was to include more information in the model while remaining computationally affordable, although more intensive.…”

“…Maintaining the advantage of not being subject to any distributional assumptions, being computationally tractable regardless of the number of original biomarkers while incorporating more information through a new summary statistic (the median or the interquartile range), Aznar-Gimeno et al [27] proposed the so-called min-max-median and min-max-IQR approaches. These approaches are based on estimating the linear combination of these three variables using the proposed stepwise algorithm [25].…”

“…In this sense, our proposed approaches (min-max-median/IQR approach) reduce the dimensional problem by capturing the heterogeneity of the information through summary statistics. Although studies comparing the performance of different machine learning techniques have increased in the literature in recent years, so far, there are no studies comparing the performance of our proposed approaches with machine learning models such as XG-Boost, which has been in high demand in recent years and which can capture more complex relationships than the statistical linear methods compared in other studies [25,27]. The aim of our work was to compare the performance of our proposed min-max-median/IQR approaches with the min-max approach and the machine learning algorithms known as logistic regression and XGBoost, maximising the Youden index.…”