Extending induced ROC methodology to the functional context

The receiver operating characteristic (ROC) curve is the most widely used measure for evaluating the discriminatory performance of a continuous marker. Often, covariate information is also available and several regression methods have been proposed to incorporate covariate information in the ROC framework. Until now, these methods are only developed for the case where the covariate is univariate or multivariate. We extend ROC regression methodology for the case where the covariate is functional rather than univariate or multivariate. To this end, semiparametric- and nonparametric-induced ROC regression estimators are proposed. A simulation study is performed to assess the performance of the proposed estimators. The methods are applied to and motivated by a metabolic syndrome study in Galicia (NW Spain).     

An interpretation for the ROC curve and inference using GLM procedures

The accuracy of a medical diagnostic test is often summarized in a receiver operating characteristic (ROC) curve. This paper puts forth an interpretation for each point on the ROC curve as being a conditional probability of a test result from a random diseased subject exceeding that from a random nondiseased subject. This interpretation gives rise to new methods for making inference about ROC curves. It is shown that inference can be achieved with binary regression techniques applied to indicator variables constructed from pairs of test results, one component of the pair being from a diseased subject and the other from a nondiseased subject. Within the generalized linear model (GLM) binary regression framework, ROC curves can be estimated, and we highlight a new semiparametric estimator. Covariate effects can also be evaluated with the GLM models. The methodology is applied to a pancreatic cancer dataset where we use the regression framework to compare two different serum biomarkers. Asymptotic distribution theory is developed to facilitate inference and to provide insight into factors influencing variability of estimated model parameters.     

A Parametric ROC Model‐Based Approach for Evaluating the Predictiveness of Continuous Markers in Case–Control Studies

Summary The predictiveness curve shows the population distribution of risk endowed by a marker or risk prediction model. It provides a means for assessing the model's capacity for stratifying the population according to risk. Methods for making inference about the predictiveness curve have been developed using cross‐sectional or cohort data. Here we consider inference based on case–control studies, which are far more common in practice. We investigate the relationship between the ROC curve and the predictiveness curve. Insights about their relationship provide alternative ROC interpretations for the predictiveness curve and for a previously proposed summary index of it. Next the relationship motivates ROC based methods for estimating the predictiveness curve. An important advantage of these methods over previously proposed methods is that they are rank invariant. In addition they provide a way of combining information across populations that have similar ROC curves but varying prevalence of the outcome. We apply the methods to prostate‐specific antigen (PSA), a marker for predicting risk of prostate cancer.     

Estimation of the ROC Curve under Verification Bias

The ROC (receiver operating characteristic) curve is the most commonly used statistical tool for describing the discriminatory accuracy of a diagnostic test. Classical estimation of the ROC curve relies on data from a simple random sample from the target population. In practice, estimation is often complicated due to not all subjects undergoing a definitive assessment of disease status (verification). Estimation of the ROC curve based on data only from subjects with verified disease status may be badly biased. In this work we investigate the properties of the doubly robust (DR) method for estimating the ROC curve under verification bias originally developed by Rotnitzky, Faraggi and Schisterman (2006) for estimating the area under the ROC curve. The DR method can be applied for continuous scaled tests and allows for a non‐ignorable process of selection to verification. We develop the estimator's asymptotic distribution and examine its finite sample properties via a simulation study. We exemplify the DR procedure for estimation of ROC curves with data collected on patients undergoing electron beam computer tomography, a diagnostic test for calcification of the arteries.     

A model free approach to combining biomarkers

For most diseases, single biomarkers do not have adequate sensitivity or specificity for practical purposes. We present an approach to combine several biomarkers into a composite marker score without assuming a model for the distribution of the predictors. Using sufficient dimension reduction techniques, we replace the original markers with a lower-dimensional version, obtained through linear transformations of markers that contain sufficient information for regression of the predictors on the outcome. We combine the linear transformations using their asymptotic properties into a scalar diagnostic score via the likelihood ratio statistic. The performance of this score is assessed by the area under the receiver-operator characteristics curve (ROC), a popular summary measure of the discriminatory ability of a single continuous diagnostic marker for binary disease outcomes. An asymptotic chi-squared test for assessing individual biomarker contribution to the diagnostic score is also derived.     

梗阻性黄疸MRCP 的循证和临床研究

目的：通过meta、ROC分析以及按病变部位、性质进行的亚组分析分析对目前诊断梗阻性黄疸的非侵入性影像诊断方法（US,Cr和MRCP）进行对比研究。方法：1、采用medline检索。纳入标准为：（a）US、CT和MRCP诊断梗阻性黄疸性疾病的文献（b）病理检查、术中所见或临床、实验室检查结果作为诊断金标准。（c）能够直接或间接获得每个影像方法的真、假阳性数,真、假阴性数。提取数据、文献质量评估通过kappa分析进行一致性检验。统计分析采用漏斗图、SROC分析方法以及协变量分析。2、疑胆胰系疾患接受MRCP检查患者105例,其中同时做US检查者65例。另有同期Cr资料59例,其中同时做US检查者31例。盲法与金标准对比,计算出各诊断方法的真阳性率和假阳性率,ROC分析其诊断效能。同时按病变部位、性质分别计算MR-CP、US及Cr的敏感度、特异度和似然比等指标进行比较分析。结果：1、漏斗图US相关文献分布形状略不规则,CT、MRCP相关文献分布形状类似漏斗形。SROC曲线图MRCP线最靠近左上角,诊断效能高于US和CT、MRCP的Q^＊值（0．9256）高于US（0.8765）和CT（0.8606）。三者间经检验无显著性差别,MRCP和cT问检验Z=0、33,双侧P〉0、25。协变量分析未见对诊断效能有显著性影响因素。2、ROC分析显示,MRCP的曲线最靠近左上角,US次之,Cr在最下面,三者的曲线下面积（Az）分别为0．985,0．981．0、901,均大于0、9,MRCP与Cr间离均差（Z）为0．75,双侧P〉0、25。MRCP、US和Cr诊断胆胰系恶性占位、结石的敏感度分别为100％、83％、82％;92％、71％、76％。经检验,MRCP与US和CT间有显著性差异,P〈0．05。结论：经meta、ROC分析,认为MRCP在诊断梗阻性黄疸疾病中具有优势,诊断效能高于US和Cr。     

Variability of palmprint ridge density in a North Indian population and its use in inference of sex in forensic examinations

Fingerprints and palmprints are unique to an individual, and these biometric characters are used in the identification of individuals. In the recent past, ridge density (ridge count in a defined area) has been explored for its applicability in inference of sex from the fingerprints and palmprints recovered at the crime scene. The present research aims to study the variability of palmprint ridge density in a North Indian population, and its significance in inference of sex in forensic examinations. The sample consisted of 157 healthy young adults (110 females and 47 males) from Shimla city in North India. Bilateral palmprints were taken from all the participants following standard methods. The palmprints were manually analyzed in four defined areas of each palmprint that included the central prominent part of the thenar eminence (P1), the mount distal to the axial triradius on the hypothenar region (P2), the mount proximal to the triradius of the second digit (P3) and the mount proximal to the triradius of the fifth digit (P4). The ridge density was calculated diagonally using a square measuring 5 mm × 5 mm. The sex differences in palmprint ridge densities were statistically analyzed for each of the designated areas using statistical considerations. Receiver operating characteristic (ROC) curve analysis was done to test the overall ability of the palmprint ridge densities obtained from each area in inference of sex. The mean palmprint ridge density was found to be significantly greater in females than in males in all the four defined areas of the palmprint. Ridge densities in P3 and P4 areas of the palmprint showed statistically significant bilateral differences in both males and females. The study observed variations in the ridge density between the four designated areas of the palmprint. Based on the area under the ROC curve (AUC), maximum sexing potential for the palmprint ridge density was observed in the P4 area, followed by P3 area on both right and left sides. ROC analysis of the total palmprint ridge density indicated that the sexing accuracy from the right and left palmprint ridge densities was 70.2% and 71.8% respectively. The study shows variability of palmprint ridge density among sexes and in different regions of the palm on both sides. In view of the considerable overlapping in male and female values and lower levels of accuracy obtained in ROC analysis, the present research concludes that the palmprint ridge density cannot be used as an effective tool in inference of sex. However, in absence of other more reliable means/evidence, it still can be considered as a supportive trait in sex inference.     

Partial AUC estimation and regression

Accurate diagnosis of disease is a critical part of health care. New diagnostic and screening tests must be evaluated based on their abilities to discriminate diseased from nondiseased states. The partial area under the receiver operating characteristic (ROC) curve is a measure of diagnostic test accuracy. We present an interpretation of the partial area under the curve (AUC), which gives rise to a nonparametric estimator. This estimator is more robust than existing estimators, which make parametric assumptions. We show that the robustness is gained with only a moderate loss in efficiency. We describe a regression modeling framework for making inference about covariate effects on the partial AUC. Such models can refine knowledge about test accuracy. Model parameters can be estimated using binary regression methods. We use the regression framework to compare two prostate-specific antigen biomarkers and to evaluate the dependence of biomarker accuracy on the time prior to clinical diagnosis of prostate cancer.     

Semiparametric transformation models for multiple continuous biomarkers in ROC analysis

Recent technological advances continue to provide noninvasive and more accurate biomarkers for evaluating disease status. One standard tool for assessing the accuracy of diagnostic tests is the receiver operating characteristic (ROC) curve. Few statistical methods exist to accommodate multiple continuous‐scale biomarkers in the framework of ROC analysis. In this paper, we propose a method to integrate continuous‐scale biomarkers to optimize classification accuracy. Specifically, we develop semiparametric transformation models for multiple biomarkers. We assume that unknown and marker‐specific transformations of biomarkers follow a multivariate normal distribution. Our models accommodate biomarkers subject to limits of detection and account for the dependence among biomarkers by including a subject‐specific random effect. We also propose a diagnostic measure using an optimal linear combination of the transformed biomarkers. Our diagnostic rule does not depend on any monotone transformation of biomarkers and is not sensitive to extreme biomarker values. Nonparametric maximum likelihood estimation (NPMLE) is used for inference. We show that the parameter estimators are asymptotically normal and efficient. We illustrate our semiparametric approach using data from the Endometriosis, Natural History, Diagnosis, and Outcomes (ENDO) study.     

Application of elliptical cylinder as a novel model of microvessel shape for analysis of thyroid capillaries in hypercalcemia

Previously designed stereological method for estimating axial ratios (X0/Y0) of microvessels (MVs) whose shape is approximated by elliptical cylinders was applied for analysis in the rat thyroid perifollicular hemocapillaries in hypercalcemia (Ca gluconate--10% i.m., 0.7 ml/day, for 3 days). Electron microscopy revealed the size of major (X) and minor (Y) radii of ectioning profiles of the capillaries. The X/Y distribution corresponded to use of characteristics foe elliptical cylinder model's stochastically geometric 3D/2D relation of distribution of two--or three-dimensional (3D), and observed, or two-dimensional (2D), axial ratio values. The X0/Y0 values of the MVs under study were estimated as X0/Y0 approximately 1.3 for 79% and X0/Y0 approximately 2.7 for 21% of capillaries. The estimates obtained are of comparable value with the X0/Y0 calculated previously for the thyroid capillaries of the normal rat. They can be employed in physiological studies of the MVs.     

Distribution-free ROC analysis using binary regression techniques

Receiver operating characteristic (ROC) regression methodology is used to identify factors that affect the accuracy of medical diagnostic tests. In this paper, we consider a ROC model for which the ROC curve is a parametric function of covariates but distributions of the diagnostic test results are not specified. Covariates can be either common to all subjects or specific to those with disease. We propose a new estimation procedure based on binary indicators defined by the test result for a diseased subject exceeding various specified quantiles of the distribution of test results from non-diseased subjects with the same covariate values. This procedure is conceptually and computationally simplified relative to existing procedures. Simulation study results indicate that the approach has fairly high statistical efficiency. The new ROC regression methodology is used to evaluate childhood measurements of body mass index as a predictive marker of adult obesity.     

Homeostatic model assessment (HOMA) index cut-off values to identify the metabolic syndrome in children

The aim of the study was to establish the best cut-off value for the homeostatic model assessment (HOMA) index in identifying children and adolescents with the metabolic syndrome. The study included 72 non-obese and 68 obese children aged 7 to 16 years. Obesity is defined using the criteria proposed by Cole et al., being included as metabolic syndrome variables waist circumference, systolic blood pressure, diastolic blood pressure and seric values of glucose, uric acid, fasting insulin, leptin, triglycerides and HDL-cholesterol. Children were considered as having the metabolic syndrome when four or more characteristics showed abnormal values. The HOMA index was calculated as the product of the fasting plasma insulin level (microU/mL) and the fasting plasma glucose level (mmol/L), divided by 22.5. HOMA index cut-offs from the 5th to the 95th percentile were used. A receiver operating characteristic (ROC) curve was generated using the different HOMA cut-offs for the screening of the metabolic syndrome. The areas under the ROC curve, 95% confidence intervals, and the point to the ROC curve closest to 1, were calculated. The area under the ROC curve was 0.863 (95% C.I.: 0.797, 0.930). The point closest to 1 corresponds to the 60th percentile of the HOMA index distribution in our sample. HOMA index value at the 60th percentile was 2.28. Cut-off values corresponding to a range of HOMA index from the 50 to the 75 percentile, showed similar distances to 1. HOMA index values for percentiles 50 to 75 ranged from 2.07 to 2.83. In conclusion, HOMA index could be a useful tool to detect children and adolescents with the metabolic syndrome. HOMA cut-off values need to be defined in the paediatric population; however, values near to 3 seem to be adequate.     

Approximate Bayesian computation in population genetics

We propose a new method for approximate Bayesian statistical inference on the basis of summary statistics. The method is suited to complex problems that arise in population genetics, extending ideas developed in this setting by earlier authors. Properties of the posterior distribution of a parameter, such as its mean or density curve, are approximated without explicit likelihood calculations. This is achieved by fitting a local-linear regression of simulated parameter values on simulated summary statistics, and then substituting the observed summary statistics into the regression equation. The method combines many of the advantages of Bayesian statistical inference with the computational efficiency of methods based on summary statistics. A key advantage of the method is that the nuisance parameters are automatically integrated out in the simulation step, so that the large numbers of nuisance parameters that arise in population genetics problems can be handled without difficulty. Simulation results indicate computational and statistical efficiency that compares favorably with those of alternative methods previously proposed in the literature. We also compare the relative efficiency of inferences obtained using methods based on summary statistics with those obtained directly from the data using MCMC.     

中国红树林北缘区秋茄树种群空间分布格局和结构特征分析

对位于中国红树林北缘区福鼎市的秋茄树〔Kandelia candel(Linn.)Druce〕天然林和人工林的高度结构和径级结构进行了分析,并以"空间替代时间"的方法研究了秋茄树天然林和人工林的空间分布格局以及天然林的种群结构特征。结果表明:不同滩位秋茄树天然林和人工林的高度结构及径级结构明显不一致。天然林中株高0.0~0.5和1.5~2.0 m的个体占多数;而人工林以株高1.5~2.0 m的个体为主,无株高0.0~0.5 m的幼苗,株高超过2.0 m的植株也极少,表明天然林的高度结构均匀而人工林的高度结构极不合理。根据基径(D)可将秋茄树种群分为13个径级,其中天然林不同径级个体数量随径级增大依次减少,而人工林中径级Ⅲ(6 cm≤D<10 cm)的植株数量最多,且没有径级Ⅸ(30 cm≤D<34 cm)以上的个体,表明天然林的径级结构良好。从分布格局看,天然林幼苗群呈现聚集分布,而其小树群、大树群和老树群均为随机分布;人工林无幼苗群和老树群,小树群呈均匀分布,大树群呈聚集分布。种群静态生命表分析结果表明:秋茄树天然林种群的期间死亡率和消失率在龄级1(D<2 cm)和龄级7(22 cm≤D<26 cm)达到最大,在其他龄级均较小且处于稳定状态;其个体生存率单调下降、累计死亡率单调上升,仅在龄级1时个体生存率高于累计死亡率,从龄级2(2 cm≤D<6 cm)开始均表现为累计死亡率高于生存率;其存活曲线属于典型的Deevey-Ⅲ型,表明秋茄树天然林种群中幼苗丰富但期间死亡率较高,成树的期间死亡率相对较低且个体数量稳定。     

Empirical likelihood inference for the area under the ROC curve

For a continuous-scale diagnostic test, the most commonly used summary index of the receiver operating characteristic curve (ROC) is the area under the curve (AUC) that measures the accuracy of the diagnostic test. In this article, we propose an empirical likelihood (EL) approach for the inference on the AUC. First we define an EL ratio for the AUC and show that its limiting distribution is a scaled chi-square distribution. We then obtain an EL-based confidence interval for the AUC using the scaled chi-square distribution. This EL inference for the AUC can be extended to stratified samples, and the resulting limiting distribution is a weighted sum of independent chi-square distributions. Additionally we conduct simulation studies to compare the relative performance of the proposed EL-based interval with the existing normal approximation-based intervals and bootstrap intervals for the AUC.     

Insights into the area under the receiver operating characteristic curve (AUC) as a discrimination measure in species distribution modelling

Aim The area under the receiver operating characteristic (ROC) curve (AUC) is a widely used statistic for assessing the discriminatory capacity of species distribution models. Here, I used simulated data to examine the interdependence of the AUC and classical discrimination measures (sensitivity and specificity) derived for the application of a threshold. I shall further exemplify with simulated data the implications of using the AUC to evaluate potential versus realized distribution models. Innovation After applying the threshold that makes sensitivity and specificity equal, a strong relationship between the AUC and these two measures was found. This result is corroborated with real data. On the other hand, the AUC penalizes the models that estimate potential distributions (the regions where the species could survive and reproduce due to the existence of suitable environmental conditions), and favours those that estimate realized distributions (the regions where the species actually lives). Main conclusions Firstly, the independence of the AUC from the threshold selection may be irrelevant in practice. This result also emphasizes the fact that the AUC assumes nothing about the relative costs of errors of omission and commission. However, in most real situations this premise may not be optimal. Measures derived from a contingency table for different cost ratio scenarios, together with the ROC curve, may be more informative than reporting just a single AUC value. Secondly, the AUC is only truly informative when there are true instances of absence available and the objective is the estimation of the realized distribution. When the potential distribution is the goal of the research, the AUC is not an appropriate performance measure because the weight of commission errors is much lower than that of omission errors.     

Time-dependent ROC curves for censored survival data and a diagnostic marker

ROC curves are a popular method for displaying sensitivity and specificity of a continuous diagnostic marker, X, for a binary disease variable, D. However, many disease outcomes are time dependent, D(t), and ROC curves that vary as a function of time may be more appropriate. A common example of a time-dependent variable is vital status, where D(t) = 1 if a patient has died prior to time t and zero otherwise. We propose summarizing the discrimination potential of a marker X, measured at baseline (t = 0), by calculating ROC curves for cumulative disease or death incidence by time t, which we denote as ROC(t). A typical complexity with survival data is that observations may be censored. Two ROC curve estimators are proposed that can accommodate censored data. A simple estimator is based on using the Kaplan-Meier estimator for each possible subset X > c. However, this estimator does not guarantee the necessary condition that sensitivity and specificity are monotone in X. An alternative estimator that does guarantee monotonicity is based on a nearest neighbor estimator for the bivariate distribution function of (X, T), where T represents survival time (Akritas, M. J., 1994, Annals of Statistics 22, 1299-1327). We present an example where ROC(t) is used to compare a standard and a modified flow cytometry measurement for predicting survival after detection of breast cancer and an example where the ROC(t) curve displays the impact of modifying eligibility criteria for sample size and power in HIV prevention trials.     

Smoothed empirical likelihood inference for ROC curve in the presence of missing biomarker values

This paper considers statistical inference for the receiver operating characteristic (ROC) curve in the presence of missing biomarker values by utilizing estimating equations (EEs) together with smoothed empirical likelihood (SEL). Three approaches are developed to estimate ROC curve and construct its SEL-based confidence intervals based on the kernel-assisted EE imputation, multiple imputation, and hybrid imputation combining the inverse probability weighted imputation and multiple imputation. Under some regularity conditions, we show asymptotic properties of the proposed maximum SEL estimators for ROC curve. Simulation studies are conducted to investigate the performance of the proposed SEL approaches. An example is illustrated by the proposed methodologies. Empirical results show that the hybrid imputation method behaves better than the kernel-assisted and multiple imputation methods, and the proposed three SEL methods outperform existing nonparametric method.     

Brief communication: Body mass index,body adiposity index,and percent body fat in asians

Human obesity is a growing epidemic throughout the world. Body mass index (BMI) is commonly used as a good indicator of obesity. Body adiposity index (BAI = hip circumference (cm)/stature (m)^1.5 ? 18), as a new surrogate measure, has been proposed recently as an alternative to BMI. This study, for the first time, compares BMI and BAI for predicting percent body fat (PBF; estimated from skinfolds) in a sample of 302 Buryat adults (148 men and 154 women) living in China. The BMI and BAI were strongly correlated with PBF in both men and women. The correlation coefficient between BMI and PBF was higher than that between BAI and PBF for both sexes. For the linear regression analysis, BMI better predicted PBF in both men and women; the variation around the regression lines for each sex was greater for BAI comparisons. For the receiver operating characteristic (ROC) analysis, the area under the ROC curve for BMI was higher than that for BAI for each sex, which suggests that the discriminatory capacity of the BMI is higher than the one of BAI. Taken together, we conclude that BMI is a more reliable indicator of PBF derived from skinfold thickness in adult Buryats. Am J Phys Anthropol 152:294–299, 2013. © 2013 Wiley Periodicals, Inc.     

Combining diagnostic test results to increase accuracy

When multiple diagnostic tests are performed on an individual or multiple disease markers are available it may be possible to combine the information to diagnose disease. We consider how to choose linear combinations of markers in order to optimize diagnostic accuracy. The accuracy index to be maximized is the area or partial area under the receiver operating characteristic (ROC) curve. We propose a distribution-free rank-based approach for optimizing the area under the ROC curve and compare it with logistic regression and with classic linear discriminant analysis (LDA). It has been shown that the latter method optimizes the area under the ROC curve when test results have a multivariate normal distribution for diseased and non-diseased populations. Simulation studies suggest that the proposed non-parametric method is efficient when data are multivariate normal.The distribution-free method is generalized to a smooth distribution-free approach to: (i) accommodate some reasonable smoothness assumptions; (ii) incorporate covariate effects; and (iii) yield optimized partial areas under the ROC curve. This latter feature is particularly important since it allows one to focus on a region of the ROC curve which is of most relevance to clinical practice. Neither logistic regression nor LDA necessarily maximize partial areas. The approaches are illustrated on two cancer datasets, one involving serum antigen markers for pancreatic cancer and the other involving longitudinal prostate specific antigen data.