Two simple algorithms on linear combination of multiple biomarkers to maximize partial area under the ROC curve

Conditional concordance-assisted learning under matched case-control design for combining biomarkers for population screening

Incorporating promising biomarkers into cancer screening practices for early-detection is increasingly appealing because of the unsatisfactory performance of current cancer screening strategies. The matched case-control design is commonly adopted in biomarker development studies to evaluate the discriminative power of biomarker candidates, with an intention to eliminate confounding effects. Data from matched case-control studies have been routinely analyzed by the conditional logistic regression, although the assumed logit link between biomarker combinations and disease risk may not always hold. We propose a conditional concordance-assisted learning method, which is distribution-free, for identifying an optimal combination of biomarkers to discriminate cases and controls. We are particularly interested in combinations with a clinically and practically meaningful specificity to prevent disease-free subjects from unnecessary and possibly intrusive diagnostic procedures, which is a top priority for cancer population screening. We establish asymptotic properties for the derived combination and confirm its favorable finite sample performance in simulations. We apply the proposed method to the prostate cancer data from the carotene and retinol efficacy trial (CARET).

A Stepwise Algorithm for Linearly Combining Biomarkers under Youden Index Maximization

Combining multiple biomarkers to provide predictive models with a greater discriminatory ability is a discipline that has received attention in recent years. Choosing the probability threshold that corresponds to the highest combined marker accuracy is key in disease diagnosis. The Youden index is a statistical metric that provides an appropriate synthetic index for diagnostic accuracy and a good criterion for choosing a cut-off point to dichotomize a biomarker. In this study, we present a new stepwise algorithm for linearly combining continuous biomarkers to maximize the Youden index. To investigate the performance of our algorithm, we analyzed a wide range of simulated scenarios and compared its performance with that of five other linear combination methods in the literature (a stepwise approach introduced by Yin and Tian, the min-max approach, logistic regression, a parametric approach under multivariate normality and a non-parametric kernel smoothing approach). The obtained results show that our proposed stepwise approach showed similar results to other algorithms in normal simulated scenarios and outperforms all other algorithms in non-normal simulated scenarios. In scenarios of biomarkers with the same means and a different covariance matrix for the diseased and non-diseased population, the min-max approach outperforms the rest. The methods were also applied on two real datasets (to discriminate Duchenne muscular dystrophy and prostate cancer), whose results also showed a higher predictive ability in our algorithm in the prostate cancer database.

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

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).

Combining biomarkers by maximizing the true positive rate for a fixed false positive rate

Biomarkers abound in many areas of clinical research, and often investigators are interested in combining them for diagnosis, prognosis, or screening. In many applications, the true positive rate (TPR) for a biomarker combination at a prespecified, clinically acceptable false positive rate (FPR) is the most relevant measure of predictive capacity. We propose a distribution-free method for constructing biomarker combinations by maximizing the TPR while constraining the FPR. Theoretical results demonstrate desirable properties of biomarker combinations produced by the new method. In simulations, the biomarker combination provided by our method demonstrated improved operating characteristics in a variety of scenarios when compared with alternative methods for constructing biomarker combinations. Thus, use of our method could lead to the development of better biomarker combinations, increasing the likelihood of clinical adoption.

On the use of min-max combination of biomarkers to maximize the partial area under the ROC curve

Background

Evaluation of diagnostic assays and predictive performance of biomarkers based on the receiver operating characteristic (ROC) curve and the area under the ROC curve (AUC) are vital in diagnostic and targeted medicine. The partial area under the curve (pAUC) is an alternative metric focusing on a range of practical and clinical relevance of the diagnostic assay. In this article, we adopt and extend the min-max method to the estimation of the pAUC when multiple continuous scaled biomarkers are available and compare the performances of our proposed approach with existing approaches via simulations.

Methods

We conducted extensive simulation studies to investigate the performance of different methods for the combination of biomarkers based on their abilities to produce the largest pAUC estimates. Data were generated from different multivariate distributions with equal and unequal variance-covariance matrices. Different shapes of the ROC curves, false positive fraction ranges, and sample size configurations were considered. We obtained the mean and standard deviation of the pAUC estimates through re-substitution and leave-one-pair-out cross validation.

Results

Our results demonstrate that the proposed method provides the largest pAUC estimates under the following three important practical scenarios: (1) multivariate normally distributed data for non-diseased and diseased participants have unequal variance-covariance matrices; or (2) the ROC curves generated from individual biomarker are relative close regardless of the latent normality distributional assumption; or (3) the ROC curves generated from individual biomarker have straight-line shapes.

Conclusions

The proposed method is robust and investigators are encouraged to use this approach in the estimation of the pAUC for many practical scenarios.

Weighted volume under the three-way receiver operating characteristic surface

It is often necessary to differentiate subjects from multiple categories using medical tests. We may then adopt statistical measures to characterize the performance of these tests. The three-way ROC analysis has been proposed to evaluate the diagnostic accuracy of medical tests with three categories, reflecting the correct classification probabilities across all possible decision thresholds. The geometry of the ROC surface is carefully studied, leading to numerical summary measures such as the volume under the surface. This paper generalizes the global volume under the surface of three-way ROC analysis to the weighted volume under the surface (WVUS) by introducing a weight function emphasizing particular regions of correct classification probabilities. This generalization practically allows researchers to calculate the diagnostic accuracy for a medical or clinical biomarker while satisfactorily high probabilities of correct classification for one or two classes are conditionally ensured. We provide the asymptotic properties of the proposed nonparametric and parametric estimators of WVUS, which could easily lend support to statistical inferences. Some simulations have been conducted to assess the proposed estimators and also to demonstrate the necessity of WVUS. A real data analysis about liver cancer illustrates our methodology.