Estimation of AUC or Partial AUC under Test-Result-Dependent Sampling

A novel estimator for the two-way partial AUC

BACKGROUND

The two-way partial AUC has been recently proposed as a way to directly quantify partial area under the ROC curve with simultaneous restrictions on the sensitivity and specificity ranges of diagnostic tests or classifiers. The metric, as originally implemented in the tpAUC R package, is estimated using a nonparametric estimator based on a trimmed Mann-Whitney U-statistic, which becomes computationally expensive in large sample sizes. (Its computational complexity is of order [Formula: see text], where [Formula: see text] and [Formula: see text] represent the number of positive and negative cases, respectively). This is problematic since the statistical methodology for comparing estimates generated from alternative diagnostic tests/classifiers relies on bootstrapping resampling and requires repeated computations of the estimator on a large number of bootstrap samples.

METHODS

By leveraging the graphical and probabilistic representations of the AUC, partial AUCs, and two-way partial AUC, we derive a novel estimator for the two-way partial AUC, which can be directly computed from the output of any software able to compute AUC and partial AUCs. We implemented our estimator using the computationally efficient pROC R package, which leverages a nonparametric approach using the trapezoidal rule for the computation of AUC and partial AUC scores. (Its computational complexity is of order [Formula: see text], where [Formula: see text].). We compare the empirical bias and computation time of the proposed estimator against the original estimator provided in the tpAUC package in a series of simulation studies and on two real datasets.

RESULTS

Our estimator tended to be less biased than the original estimator based on the trimmed Mann-Whitney U-statistic across all experiments (and showed considerably less bias in the experiments based on small sample sizes). But, most importantly, because the computational complexity of the proposed estimator is of order [Formula: see text], rather than [Formula: see text], it is much faster to compute when sample sizes are large.

CONCLUSIONS

The proposed estimator provides an improvement for the computation of two-way partial AUC, and allows the comparison of diagnostic tests/machine learning classifiers in large datasets where repeated computations of the original estimator on bootstrap samples become too expensive to compute.

Considerations on the region of interest in the ROC space

Receiver Operating Characteristic curves have been widely used to represent the performance of diagnostic tests. The corresponding area under the curve, widely used to evaluate their performance quantitatively, has been criticized in several respects. Several proposals have been introduced to improve area under the curve by taking into account only specific regions of the Receiver Operating Characteristic space, that is, the plane to which Receiver Operating Characteristic curves belong. For instance, a region of interest can be delimited by setting specific thresholds for the true positive rate or the false positive rate. Different ways of setting the borders of the region of interest may result in completely different, even opposing, evaluations. In this paper, we present a method to define a region of interest in a rigorous and objective way, and compute a partial area under the curve that can be used to evaluate the performance of diagnostic tests. The method was originally conceived in the Software Engineering domain to evaluate the performance of methods that estimate the defectiveness of software modules. We compare this method with previous proposals. Our method allows the definition of regions of interest by setting acceptability thresholds on any kind of performance metric, and not just false positive rate and true positive rate: for instance, the region of interest can be determined by imposing that ϕ (also known as the Matthews Correlation Coefficient) is above a given threshold. We also show how to delimit the region of interest corresponding to acceptable costs, whenever the individual cost of false positives and false negatives is known. Finally, we demonstrate the effectiveness of the method by applying it to the Wisconsin Breast Cancer Data. We provide Python and R packages supporting the presented method.

High-throughput analysis of cell-cell crosstalk in ad hoc designed microfluidic chips for oncoimmunology applications

Understanding the interactions between immune and cancer cells occurring within the tumor microenvironment is a prerequisite for successful and personalized anti-cancer therapies. Microfluidic devices, coupled to advanced microscopy systems and automated analytical tools, can represent an innovative approach for high-throughput investigations on immune cell-cancer interactions. In order to study such interactions and to evaluate how therapeutic agents can affect this crosstalk, we employed two ad hoc fabricated microfluidic platforms reproducing advanced 2D or 3D tumor immune microenvironments. In the first type of chip, we confronted the capacity of tumor cells embedded in Matrigel containing one drug or Matrigel containing a combination of two drugs to attract differentially immune cells, by fluorescence microscopy analyses. In the second chip, we investigated the migratory/interaction response of naïve immune cells to danger signals emanated from tumor cells treated with an immunogenic drug, by time-lapse microscopy and automated tracking analysis. We demonstrate that microfluidic platforms and their associated high-throughput computed analyses can represent versatile and smart systems to: (i) monitor and quantify the recruitment and interactions of the immune cells with cancer in a controlled environment, (ii) evaluate the immunogenic effects of anti-cancer therapeutic agents and (iii) evaluate the immunogenic efficacy of combinatorial regimens with respect to single agents.

Auxiliary variable-enriched biomarker-stratified design

Clinical trials in the era of precision medicine require assessment of biomarkers to identify appropriate subgroups of patients for targeted therapy. In a biomarker-stratified design (BSD), biomarkers are measured on all patients and used as stratification variables. However, such a trial can be both inefficient and costly, especially when the prevalence of the subgroup of primary interest is low and the cost of assessing the biomarkers is high. Efficiency can be improved and costs reduced by using enriched biomarker-stratified designs, in which patients of primary interest, typically the biomarker-positive patients, are oversampled. We consider a special type of enrichment design, an auxiliary variable-enriched design (AEBSD), in which enrichment is based on some inexpensive auxiliary variable that is positively correlated with the true biomarker. The proposed AEBSD reduces the total cost of the trial compared with a standard BSD when the prevalence rate of true biomarker positivity is small and the positive predictive value (PPV) of the auxiliary biomarker is larger than the prevalence rate. In addition, for an AEBSD, we can immediately randomize the patients selected in the screening process without waiting for the result of the true biomarker test, reducing the treatment waiting time. We propose an adaptive Bayesian method to adjust the assumed PPV while the trial is ongoing. Numerical studies and an example illustrate the approach. An R package is available.

On Enrichment Strategies for Biomarker Stratified Clinical Trials

In the era of precision medicine, drugs are increasingly developed to target subgroups of patients with certain biomarkers. In large all-comer trials using a biomarker stratified design, the cost of treating and following patients for clinical outcomes may be prohibitive. With a fixed number of randomized patients, the efficiency of testing certain treatments parameters, including the treatment effect among biomarker-positive patients and the interaction between treatment and biomarker, can be improved by increasing the proportion of biomarker positives on study, especially when the prevalence rate of biomarker positives is low in the underlying patient population. When the cost of assessing the true biomarker is prohibitive, one can further improve the study efficiency by oversampling biomarker positives with a cheaper auxiliary variable or a surrogate biomarker that correlates with the true biomarker. To improve efficiency and reduce cost, we can adopt an enrichment strategy for both scenarios by concentrating on testing and treating patient subgroups that contain more information about specific treatment parameters of primary interest to the investigators. In the first scenario, an enriched biomarker stratified design enriches the cohort of randomized patients by directly oversampling the relevant patients with the true biomarker, while in the second scenario, an auxiliary-variable-enriched biomarker stratified design enriches the randomized cohort based on an inexpensive auxiliary variable, thereby avoiding testing the true biomarker on all screened patients and reducing treatment waiting time. For both designs, we discuss how to choose the optimal enrichment proportion when testing a single hypothesis or two hypotheses simultaneously. At a requisite power, we compare the two new designs with the BSD design in terms of the number of randomized patients and the cost of trial under scenarios mimicking real biomarker stratified trials. The new designs are illustrated with hypothetical examples for designing biomarker-driven cancer trials.

Two-way partial AUC and its properties

Simultaneous control on true positive rate and false positive rate is of significant importance in the performance evaluation of diagnostic tests. Most of the established literature utilizes partial area under receiver operating characteristic (ROC) curve with restrictions only on false positive rate (FPR), called FPR pAUC, as a performance measure. However, its indirect control on true positive rate (TPR) is conceptually and practically misleading. In this paper, a novel and intuitive performance measure, named as two-way pAUC, is proposed, which directly quantifies partial area under ROC curve with explicit restrictions on both TPR and FPR. To estimate two-way pAUC, we devise a nonparametric estimator. Based on the estimator, a bootstrap-assisted testing method for two-way pAUC comparison is established. Moreover, to evaluate possible covariate effects on two-way pAUC, a regression analysis framework is constructed. Asymptotic normalities of the methods are provided. Advantages of the proposed methods are illustrated by simulation and Wisconsin Breast Cancer Data. We encode the methods as a publicly available R package tpAUC.

Time-dependent classification accuracy curve under marker-dependent sampling

Evaluating the classification accuracy of a candidate biomarker signaling the onset of disease or disease status is essential for medical decision making. A good biomarker would accurately identify the patients who are likely to progress or die at a particular time in the future or who are in urgent need for active treatments. To assess the performance of a candidate biomarker, the receiver operating characteristic (ROC) curve and the area under the ROC curve (AUC) are commonly used. In many cases, the standard simple random sampling (SRS) design used for biomarker validation studies is costly and inefficient. In order to improve the efficiency and reduce the cost of biomarker validation, marker-dependent sampling (MDS) may be used. In a MDS design, the selection of patients to assess true survival time is dependent on the result of a biomarker assay. In this article, we introduce a nonparametric estimator for time-dependent AUC under a MDS design. The consistency and the asymptotic normality of the proposed estimator is established. Simulation shows the unbiasedness of the proposed estimator and a significant efficiency gain of the MDS design over the SRS design.

ROC curve estimation under test-result-dependent sampling

The receiver operating characteristic (ROC) curve is often used to evaluate the performance of a biomarker measured on continuous scale to predict the disease status or a clinical condition. Motivated by the need for novel study designs with better estimation efficiency and reduced study cost, we consider a biased sampling scheme that consists of a SRC and a supplemental TDC. Using this approach, investigators can oversample or undersample subjects falling into certain regions of the biomarker measure, yielding improved precision for the estimation of the ROC curve with a fixed sample size. Test-result-dependent sampling will introduce bias in estimating the predictive accuracy of the biomarker if standard ROC estimation methods are used. In this article, we discuss three approaches for analyzing data of a test-result-dependent structure with a special focus on the empirical likelihood method. We establish asymptotic properties of the empirical likelihood estimators for covariate-specific ROC curves and covariate-independent ROC curves and give their corresponding variance estimators. Simulation studies show that the empirical likelihood method yields good properties and is more efficient than alternative methods. Recommendations on number of regions, cutoff points, and subject allocation is made based on the simulation results. The proposed methods are illustrated with a data example based on an ongoing lung cancer clinical trial.