ROC graphs for assessing the ability of a diagnostic marker to detect three disease classes with an umbrella ordering

Cutoff estimation and construction of their confidence intervals for continuous biomarkers under ternary umbrella and tree stochastic ordering settings

Tuberculosis (TB) studies often involve four different states under consideration, namely, "healthy," "latent infection," "pulmonary active disease," and "extra-pulmonary active disease." While highly accurate clinical diagnosis tests do exist, they are expensive and generally not accessible in regions where they are most needed; thus, there is an interest in assessing the accuracy of new and easily obtainable biomarkers. For some such biomarkers, the typical stochastic ordering assumption might not be justified for all disease classes under study, and usual ROC methodologies that involve ROC surfaces and hypersurfaces are inadequate. Different types of orderings may be appropriate depending on the setting, and these may involve a number of ambiguously ordered groups that stochastically exhibit larger (or lower) marker scores than the remaining groups. Recently, there has been scientific interest on ROC methods that can accommodate these so-called "tree" or "umbrella" orderings. However, there is limited work discussing the estimation of cutoffs in such settings. In this article, we discuss the estimation and inference around optimized cutoffs when accounting for such configurations. We explore different cutoff alternatives and provide parametric, flexible parametric, and non-parametric kernel-based approaches for estimation and inference. We evaluate our approaches using simulations and illustrate them through a real data set that involves TB patients.

Post-test diagnostic accuracy measures under tree ordering of disease classes

The medical field commonly employs post-test measures such as predictive values and likelihood ratios to assess diagnostic accuracy. Predictive values, including positive and negative values (PPV and NPV), indicate the probability that individuals have a target health condition based on test results. On the other hand, likelihood ratios, including positive and negative ratios (LR+ and LR- respectively), compare the probability of a particular test result between the diseased and non-diseased groups. While predictive values are useful in evaluating diagnostic test accuracy in populations with varying disease prevalence, likelihood ratios provide a direct link between pre-test and post-test probabilities in specific patients. In this study, we introduce and analyze a new approach called generalized predictive values and likelihood ratios, using a tree ordering of disease classes. We evaluate the effectiveness of these methods through simulation studies and illustrate their use with real data on lung cancer.

Receiver operating characteristic estimation and threshold selection criteria in three-class classification problems for clustered data

Statistical evaluation of diagnostic tests, and, more generally, of biomarkers, is a constantly developing field, in which complexity of the assessment increases with the complexity of the design under which data are collected. One particularly prevalent type of data is clustered data, where individual units are naturally nested into clusters. In these cases, Bias can arise from omission, in the evaluation process, of cluster-level effects and/or individual covariates. Focusing on the three-class case and for continuous-valued diagnostic tests, we investigate how to exploit the clustered structure of data within a linear-mixed model approach, both when the assumption of normality holds and when it does not. We provide a method for the estimation of covariate-specific receiver operating characteristic surfaces and discuss methods for the choice of optimal thresholds, proposing three possible estimators. A proof of consistency and asymptotic normality of the proposed threshold estimators is given. All considered methods are evaluated by extensive simulation experiments. As an application, we study the use of the Lysosomal Associated Membrane Protein Family Member 5 gene expression as a biomarker to distinguish among three types of glutamatergic neurons.

Flexible diagnostic measures and new cut-point selection methods under multiple ordered classes

Medical diagnosis is essentially a classification problem and usually it is done with multiple ordered classes. For example, cancer diagnosis might be "non-malignant," "early stage," or "late stage." Therefore, appropriate measures are needed to assess the accuracy of diagnostic markers under multiple ordered classes. However, all existing measures fail to differentiate among some distinctly different biomarkers. This paper presents a multi-step procedure for evaluating biomarker accuracy under multiple ordered classes. This procedure leads to two new flexible overall measures as well as three new cut-point selection methods with great computational ease. The performance of proposed measures and cut-point selection methods are numerically explored via a simulation study. In the end, an ovarian cancer dataset from the Prostate, Lung, Colorectal, and Ovarian cancer study is analyzed. The proposed accuracy measures were estimated for markers CA125 and HE4, and cut-points were estimated for the risk of ovarian malignancy algorithm score.

Confidence interval estimation for sensitivity and difference between two sensitivities at a given specificity under tree ordering

This article considers a setting in diagnostic studies (or biomarker study) which involves a healthy class and a diseased class and the latter consists of several subclasses. The problem of interest is to evaluate the accuracy of a biomarker (or a diagnostic test) measured on a continuous scale correctly identifying healthy subjects from diseased subjects without requiring specification of an ordering in terms of marker values for subclasses relative to each other within the diseased class. Such setting is quite common in practice and it falls in the framework of tree ordering or umbrella ordering. This article explores several parametric and nonparametric approaches for estimating confidence intervals of sensitivity of single biomarker and difference between sensitivities of two correlated biomarkers under tree ordering at a given specificity. The performances of all the methods are evaluated and compared by a comprehensive simulation study. A published microarray data set is analyzed using the proposed methods.

Issues and solutions in biomarker evaluation when subclasses are involved under binary classification

In practice, it is common to evaluate biomarkers in binary classification settings (e.g. non-cancer vs. cancer) where one or both main classes involve multiple subclasses. For example, non-cancer class might consist of healthy subjects and benign cases, while cancer class might consist of subjects at early and late stages. The standard practice is pooling within each main class, i.e. all non-cancer subclasses are pooled together to create a control group, and all cancer subclasses are pooled together to create a case group. Based on the pooled data, the area under ROC curve (AUC) and other characteristics are estimated under binary classification for the purpose of biomarker evaluation. Despite the popularity of this pooling strategy in practice, its validity and implication in biomarker evaluation have never been carefully inspected. This paper aims to demonstrate that pooling strategy can be seriously misleading in biomarker evaluation. Furthermore, we present a new diagnostic framework as well as new accuracy measures appropriate for biomaker evaluation under such settings. In the end, an ovarian cancer data set is analyzed.

Measuring diagnostic accuracy for biomarkers under tree-ordering

In the field of diagnostic studies for tree or umbrella ordering, under which the marker measurement for one class is lower or higher than those for the rest unordered classes, there exist a few diagnostic measures such as the naive AUC ( NAUC), the umbrella volume ( UV), and the recently proposed TAUC, i.e. area under a ROC curve for tree or umbrella ordering (TROC). However, an important characteristic about tree or umbrella ordering has been neglected. This paper mainly focuses on promoting the use of the integrated false negative rate under tree ordering ( ITFNR) as an additional diagnostic measure besides TAUC, and proposing the idea of using ( TAUC, ITFNR) instead of TAUC to evaluate the diagnostic accuracy of a biomarker under tree or umbrella ordering. Parametric and non-parametric approaches for constructing joint confidence region of ( TAUC, ITFNR) are proposed. Simulation studies under a variety of settings are carried out to assess and compare the performance of these methods. In the end, a published microarray data set is analyzed.

Inverse probability weighting estimation of the volume under the ROC surface in the presence of verification bias

In diagnostic medicine, the volume under the receiver operating characteristic (ROC) surface (VUS) is a commonly used index to quantify the ability of a continuous diagnostic test to discriminate between three disease states. In practice, verification of the true disease status may be performed only for a subset of subjects under study since the verification procedure is invasive, risky, or expensive. The selection for disease examination might depend on the results of the diagnostic test and other clinical characteristics of the patients, which in turn can cause bias in estimates of the VUS. This bias is referred to as verification bias. Existing verification bias correction in three-way ROC analysis focuses on ordinal tests. We propose verification bias-correction methods to construct ROC surface and estimate the VUS for a continuous diagnostic test, based on inverse probability weighting. By applying U-statistics theory, we develop asymptotic properties for the estimator. A Jackknife estimator of variance is also derived. Extensive simulation studies are performed to evaluate the performance of the new estimators in terms of bias correction and variance. The proposed methods are used to assess the ability of a biomarker to accurately identify stages of Alzheimer's disease.

Assessing calibration of multinomial risk prediction models

Calibration, that is, whether observed outcomes agree with predicted risks, is important when evaluating risk prediction models. For dichotomous outcomes, several tools exist to assess different aspects of model calibration, such as calibration-in-the-large, logistic recalibration, and (non-)parametric calibration plots. We aim to extend these tools to prediction models for polytomous outcomes. We focus on models developed using multinomial logistic regression (MLR): outcome Y with k categories is predicted using k - 1 equations comparing each category i (i = 2, … ,k) with reference category 1 using a set of predictors, resulting in k - 1 linear predictors. We propose a multinomial logistic recalibration framework that involves an MLR fit where Y is predicted using the k - 1 linear predictors from the prediction model. A non-parametric alternative may use vector splines for the effects of the linear predictors. The parametric and non-parametric frameworks can be used to generate multinomial calibration plots. Further, the parametric framework can be used for the estimation and statistical testing of calibration intercepts and slopes. Two illustrative case studies are presented, one on the diagnosis of malignancy of ovarian tumors and one on residual mass diagnosis in testicular cancer patients treated with cisplatin-based chemotherapy. The risk prediction models were developed on data from 2037 and 544 patients and externally validated on 1107 and 550 patients, respectively. We conclude that calibration tools can be extended to polytomous outcomes. The polytomous calibration plots are particularly informative through the visual summary of the calibration performance.

Assessing the discriminative ability of risk models for more than two outcome categories

The discriminative ability of risk models for dichotomous outcomes is often evaluated with the concordance index (c-index). However, many medical prediction problems are polytomous, meaning that more than two outcome categories need to be predicted. Unfortunately such problems are often dichotomized in prediction research. We present a perspective on the evaluation of discriminative ability of polytomous risk models, which may instigate researchers to consider polytomous prediction models more often. First, we suggest a "discrimination plot" as a tool to visualize the model's discriminative ability. Second, we discuss the use of one overall polytomous c-index versus a set of dichotomous measures to summarize the performance of the model. Third, we address several aspects to consider when constructing a polytomous c-index. These involve the assessment of concordance in pairs versus sets of patients, weighting by outcome prevalence, the value related to models with random performance, the reduction to the dichotomous c-index for dichotomous problems, and interpretation. We illustrate these issues on case studies dealing with ovarian cancer (four outcome categories) and testicular cancer (three categories). We recommend the use of a discrimination plot together with an overall c-index such as the Polytomous Discrimination Index. If the overall c-index suggests that the model has relevant discriminative ability, pairwise c-indexes for each pair of outcome categories are informative. For pairwise c-indexes we recommend the 'conditional-risk' method which is consistent with the analytical approach of the multinomial logistic regression used to develop polytomous risk models.

On ROC analysis with nonbinary reference standard

Statistical methods for the evaluation of the accuracy of diagnostic tests usually assume a binary true disease status. However, this assumption may not be realistic in practical settings in which "disease" is defined by dichotomizing continuous or ordinal categorical measures using a pre-specified threshold value. In this paper, we focus on the analysis of studies in which both the diagnostic test and the reference standard are reported as continuous measures. We propose a semiparametric model for estimating the sensitivity, specificity, and the ROC curve as functions of reference standard thresholds. Under suitable order restrictions on the mean of the test result variable, fitting is done via two alternative approaches: isotonic regression and monotone smoothing splines. The model provides the basis to assess the effect of varying reference standard threshold on the performance of a diagnostic test. An example to evaluate the ability of the maximal SUV-lean (standardized uptake value normalized to lean body mass) in predicting axillary node involvement in women diagnosed with breast cancer is presented.

Two-stage k-sample designs for the ordered alternative problem

In preclinical studies and clinical dose-ranging trials, the Jonckheere-Terpstra test is widely used in the assessment of dose-response relationships. Hewett and Spurrier (1979) presented a two-stage analog of the test in the context of large sample sizes. In this paper, we propose an exact test based on Simon's minimax and optimal design criteria originally used in one-arm phase II designs based on binary endpoints. The convergence rate of the joint distribution of the first and second stage test statistics to the limiting distribution is studied, and design parameters are provided for a variety of assumed alternatives. The behavior of the test is also examined in the presence of ties, and the proposed designs are illustrated through application in the planning of a hypercholesterolemia clinical trial. The minimax and optimal two-stage procedures are shown to be preferable as compared with the one-stage procedure because of the associated reduction in expected sample size for given error constraints.

Parametric and non-parametric confidence intervals of the probability of identifying early disease stage given sensitivity to full disease and specificity with three ordinal diagnostic groups

In practice, there exist many disease processes with three ordinal disease classes, that is, the non-diseased stage, the early disease stage, and the fully diseased stage. Because early disease stage is likely the best time window for treatment interventions, it is important to have diagnostic tests that have good diagnostic ability to discriminate the early disease stage from the other two stages. In this paper, we present both parametric and non-parametric approaches for confidence interval estimation of probability of detecting early disease stage given the true classification rates for non-diseased group and diseased group, namely, the specificity and the sensitivity to full disease. We analyze a data set on the clinical diagnosis of early-stage Alzheimer's disease from the neuropsychological database at the Washington University Alzheimer's Disease Research Center using the proposed approaches.

Accuracy and cut-off point selection in three-class classification problems using a generalization of the Youden index

We study properties of the index J(3), defined as the accuracy, or the maximum correct classification, for a given three-class classification problem. Specifically, using J(3) one can assess the discrimination between the three distributions and obtain an optimal pair of cut-off points c(1)<c(2) in the sense that the sum of the correct classification proportions will be maximized. It also serves as the generalization of the Youden index in three-class problems. Parametric and non-parametric approaches for estimation and testing are considered and methods are applied to data from an MRS study on human immunodeficiency virus (HIV) patients.

Evaluation of diagnostic accuracy in detecting ordered symptom statuses without a gold standard

Our research is motivated by 2 methodological problems in assessing diagnostic accuracy of traditional Chinese medicine (TCM) doctors in detecting a particular symptom whose true status has an ordinal scale and is unknown-imperfect gold standard bias and ordinal scale symptom status. In this paper, we proposed a nonparametric maximum likelihood method for estimating and comparing the accuracy of different doctors in detecting a particular symptom without a gold standard when the true symptom status had an ordered multiple class. In addition, we extended the concept of the area under the receiver operating characteristic curve to a hyper-dimensional overall accuracy for diagnostic accuracy and alternative graphs for displaying a visual result. The simulation studies showed that the proposed method had good performance in terms of bias and mean squared error. Finally, we applied our method to our motivating example on assessing the diagnostic abilities of 5 TCM doctors in detecting symptoms related to Chills disease.

Exact confidence interval estimation for the difference in diagnostic accuracy with three ordinal diagnostic groups

In the cases with three ordinal diagnostic groups, the important measures of diagnostic accuracy are the volume under surface (VUS) and the partial volume under surface (PVUS) which are the extended forms of the area under curve (AUC) and the partial area under curve (PAUC). This article addresses confidence interval estimation of the difference in paired VUS s and the difference in paired PVUS s. To focus especially on studies with small to moderate sample sizes, we propose an approach based on the concepts of generalized inference. A Monte Carlo study demonstrates that the proposed approach generally can provide confidence intervals with reasonable coverage probabilities even at small sample sizes. The proposed approach is compared to a parametric bootstrap approach and a large sample approach through simulation. Finally, the proposed approach is illustrated via an application to a data set of blood test results of anemia patients.

Identification of a panel of sensitive and specific DNA methylation markers for squamous cell lung cancer

Background

Lung cancer is the leading cause of cancer death in men and women in the United States and Western Europe. Over 160,000 Americans die of this disease every year. The five-year survival rate is 15% – significantly lower than that of other major cancers. Early detection is a key factor in increasing lung cancer patient survival. DNA hypermethylation is recognized as an important mechanism for tumor suppressor gene inactivation in cancer and could yield powerful biomarkers for early detection of lung cancer. Here we focused on developing DNA methylation markers for squamous cell carcinoma of the lung. Using the sensitive, high-throughput DNA methylation analysis technique MethyLight, we examined the methylation profile of 42 loci in a collection of 45 squamous cell lung cancer samples and adjacent non-tumor lung tissues from the same patients.

Results

We identified 22 loci showing significantly higher DNA methylation levels in tumor tissue than adjacent non-tumor lung. Of these, eight showed highly significant hypermethylation in tumor tissue (p < 0.0001): GDNF, MTHFR, OPCML, TNFRSF25, TCF21, PAX8, PTPRN2 and PITX2. Used in combination on our specimen collection, this eight-locus panel showed 95.6% sensitivity and specificity.

Conclusion

We have identified 22 DNA methylation markers for squamous cell lung cancer, several of which have not previously been reported to be methylated in any type of human cancer. The top eight markers show great promise as a sensitive and specific DNA methylation marker panel for squamous cell lung cancer.

Comparison of ROC Umbrella Volumes with an Application to the Assessment of Lung Cancer Diagnostic Markers

Receiver operating characteristic (ROC) analysis is widely used to assess the ability of diagnostic markers to correctly classify into one of two disease classes. ROC surfaces and umbrella surfaces generalize the utility of ROC analysis when there are three disease classes. Identification of lung cancer diagnostic markers is an active area of research since prognosis for those diagnosed with lung cancer is so poor and there is not an accurate method for early detection of lung cancer. A study conducted for the assessment of DNA methylation markers motivated the comparison of ROC umbrella surfaces which is developed in this article using U-statistics and bootstrap methodology.