Local linear smoothing of receiver operating characteristic (ROC) curves

For measuring the accuracy of a continuous diagnostic test, the receiver operating characteristic (ROC) curve is often used. The empirical ROC curve is the most commonly used non-parametric estimator for the ROC curve. Recently, Lloyd (J. Amer. Statist. Assoc. 93(1998) 1356) proposed a kernel smoothing estimator for the ROC curve and showed his estimator has better mean square error than the empirical ROC curve estimator. However, Lloyd's estimator involves two bandwidths and has a boundary problem. In addition, his choice of bandwidths is ad hoc. In this paper we propose another kernel smoothing estimator which involves only one bandwidth and does not have the boundary problem. Furthermore, our choice of the bandwidth is asymptotically optimal.     

Compare diagnostic tests using transformation-invariant smoothed ROC curves()

Receiver operating characteristic (ROC) curve, plotting true positive rates against false positive rates as threshold varies, is an important tool for evaluating biomarkers in diagnostic medicine studies. By definition, ROC curve is monotone increasing from 0 to 1 and is invariant to any monotone transformation of test results. And it is often a curve with certain level of smoothness when test results from the diseased and non-diseased subjects follow continuous distributions. Most existing ROC curve estimation methods do not guarantee all of these properties. One of the exceptions is Du and Tang (2009) which applies certain monotone spline regression procedure to empirical ROC estimates. However, their method does not consider the inherent correlations between empirical ROC estimates. This makes the derivation of the asymptotic properties very difficult. In this paper we propose a penalized weighted least square estimation method, which incorporates the covariance between empirical ROC estimates as a weight matrix. The resulting estimator satisfies all the aforementioned properties, and we show that it is also consistent. Then a resampling approach is used to extend our method for comparisons of two or more diagnostic tests. Our simulations show a significantly improved performance over the existing method, especially for steep ROC curves. We then apply the proposed method to a cancer diagnostic study that compares several newly developed diagnostic biomarkers to a traditional one.     

Nonparametric estimation of the ROC curve based on smoothed empirical distribution functions

The receiver operating characteristic (ROC) curve is a graphical representation of the relationship between false positive and true positive rates. It is a widely used statistical tool for describing the accuracy of a diagnostic test. In this paper we propose a new nonparametric ROC curve estimator based on the smoothed empirical distribution functions. We prove its strong consistency and perform a simulation study to compare it with some other popular nonparametric estimators of the ROC curve. We also apply the proposed method to a real data set.     

A Flexible Method for Estimating the ROC Curve

In this paper we propose a flexible method for estimating a receiver operating characteristic (ROC) curve that is based on a continuous-scale test. The approach is easily understood and efficiently computed, and robust to the smooth parameter selection, which needs intensive computation when using local polynomial and smoothing spline techniques. The results from our simulation experiment indicate that the moderate-sample numerical performance of our estimator is better than the empirical ROC curve estimator and comparable to the local linear estimator. The availability of easy implementation is also illustrated by our simulation. We apply the proposed method to two real data sets.     

ROC Curves in Non‐Parametric Location‐Scale Regression Models

Abstract. The receiver operating characteristic (ROC) curve is a tool of extensive use to analyse the discrimination capability of a diagnostic variable in medical studies. In certain situations, the presence of a covariate related to the diagnostic variable can increase the discriminating power of the ROC curve. In this article, we model the effect of the covariate over the diagnostic variable by means of non‐parametric location‐scale regression models. We propose a new non‐parametric estimator of the conditional ROC curve and study its asymptotic properties. We also present some simulations and an illustration to a data set concerning diagnosis of diabetes.     

The relative performances of improved ridge estimators and an empirical bayes estimator: some monte carlo results

The relative 'performances of improved ridge estimators and an empirical Bayes estimator are studied by means of Monte Carlo simulations. The empirical Bayes method is seen to perform consistently better in terms of smaller MSE and more accurate empirical coverage than any of the estimators considered here. A bootstrap method is proposed to obtain more reliable estimates of the MSE of ridge esimators. Some theorems on the bootstrap for the ridge estimators are also given and they are used to provide an analytical understanding of the proposed bootstrap procedure. Empirical coverages of the ridge estimators based on the proposed procedure are generally closer to the nominal coverage when compared to their earlier counterparts. In general, except for a few cases, these coverages are still less accurate than the empirical coverages of the empirical Bayes estimator.     

An application of lomax distributions in receiver operating characteristic(roc)curve analysis

Receiver operating characteristic(ROC)curves are useful for studying the performance of diagnostic tests. ROC curves occur in many fields of applications including psychophysics, quality control and medical diagnostics. In practical situations, often the responses to a diagnostic test are classified into a number of ordered categories. Such data are referred to as ratings data. It is typically assumed that the underlying model is based on a continuous probability distribution. The ROC curve is then constructed from such data using this probability model. Properties of the ROC curve are inherited from the model. Therefore, understanding the role of different probability distributions in ROC modeling is an interesting and important area of research. In this paper the Lomax distribution is considered as a model for ratings data and the corresponding ROC curve is derived. The maximum likelihood estimation procedure for the related parameters is discussed. This procedure is then illustrated in the analysis of a neurological data example.     

Estimating the parameters of Poisson-exponential models

This paper proposes two methods of estimation for the parameters in a Poisson-exponential model. The proposed methods combine the method of moments with a regression method based on the empirical moment generating function. One of the methods is an adaptation of the mixed-moments procedure of Koutrouvelis & Canavos (1999). The asymptotic distribution of the estimator obtained with this method is derived. Finite-sample comparisons are made with the maximum likelihood estimator and the method of moments. The paper concludes with an exploratory-type analysis of real data based on the empirical moment generating function.     

Adaptive estimation of the center of symmetry based on the empirical characteristic function

A procedure based on the empirical characteristic function is proposed for the estimation of the center of symmetric distributions. The method is an adaptive modification of the procedure proposed by Koutrouvelis (1985). The asymptotic behavior of the resulting estimator is investigated and finite sample comparisons are made with the previous nonadaptive estimator and an adaptive trimmed mean proposed by Hogg (1974).     

Robust estimation for ordinal regression

Ordinal regression is used for modelling an ordinal response variable as a function of some explanatory variables. The classical technique for estimating the unknown parameters of this model is Maximum Likelihood (ML). The lack of robustness of this estimator is formally shown by deriving its breakdown point and its influence function. To robustify the procedure, a weighting step is added to the Maximum Likelihood estimator, yielding an estimator with bounded influence function. We also show that the loss in efficiency due to the weighting step remains limited. A diagnostic plot based on the Weighted Maximum Likelihood estimator allows to detect outliers of different types in a single plot.     

Efficiency of a Class of Unbiased Estimators for the Invariant Distribution Function of a Diffusion Process

We consider the problem of the estimation of the invariant distribution function of an ergodic diffusion process when the drift coefficient is unknown. The empirical distribution function is a natural estimator which is unbiased, uniformly consistent and efficient in different metrics. Here we study the properties of optimality for another kind of estimator recently proposed. We consider a class of unbiased estimators and we show that they are also efficient in the sense that their asymptotic risk, defined as the integrated mean square error, attains the same asymptotic minimax lower bound of the empirical distribution function.     

Adjusting ROC curves for covariates in the presence of verification bias

The ROC (receiver operating characteristic) curve is frequently used for describing effectiveness of a diagnostic marker or test. Classical estimation of the ROC curve uses independent identically distributed samples taken randomly from the healthy and diseased populations. Frequently not all subjects undergo a definitive gold standard assessment of disease status (verification). Estimation of the ROC curve based on data only from subjects with verified disease status may be badly biased (verification bias). In this work we investigate the properties of the doubly robust (DR) method for estimating the ROC curve adjusted for covariates (ROC regression) under verification bias. We develop the estimator's asymptotic distribution and examine its finite sample size properties via a simulation study. We apply this procedure to fingerstick postprandial blood glucose measurement data adjusting for age.     

Empirical Bayes rules and Gaussian processes

In this paper we study empirical Bayes (e.B.) rules from a viewpoint which has not yet got any attention in the literature. Since an e.B. estimator can be seen as an estimate of an unknown function, namely the true Bayes estimator, it is natural to consider e.B. estimators as stochastic processes. In this paper we make a first attempt in the direction of this approach. For a certain class of e.B. estimators for the continuous one-parameter exponential family, we investigate the global behaviour on finite intervals. It is shown that the difference between the e.B. and the true Bayes estimator can be represented as a certain type of Gaussian process plus a remainder which is uniformly of smaller order. Several applications of this result are given.     

Unbiased Estimation of the Distribution Function of an Exponential Population Using Order Statistics with Application in Ranked Set Sampling

In this paper we consider the problem of unbiased estimation of the distribution function of an exponential population using order statistics based on a random sample. We present a (unique) unbiased estimator based on a single, say ith, order statistic and study some properties of the estimator for i = 2. We also indicate how this estimator can be utilized to obtain unbiased estimators when a few selected order statistics are available as well as when the sample is selected following an alternative sampling procedure known as ranked set sampling. It is further proved that for a ranked set sample of size two, the proposed estimator is uniformly better than the conventional nonparametric unbiased estimator, further, for a general sample size, a modified ranked set sampling procedure provides an unbiased estimator uniformly better than the conventional nonparametric unbiased estimator based on the usual ranked set sampling procedure.     

An alternative method for global and partial comparison of two diagnostic systems based on ROC curves

In this paper, an alternative method for the comparison of two diagnostic systems based on receiver operating characteristic (ROC) curves is presented. ROC curve analysis is often used as a statistical tool for the evaluation of diagnostic systems. However, in general, the comparison of ROC curves is not straightforward, in particular, when they cross each other. A similar difficulty is also observed in the multi-objective optimization field where sets of solutions defining fronts must be compared with a multi-dimensional space. Thus, the proposed methodology is based on a procedure used to compare the performance of distinct multi-objective optimization algorithms. In general, methods based on the area under the ROC curves are not sensitive to the existence of crossing points between the curves. The new approach can deal with this situation and also allows the comparison of partial portions of ROC curves according to particular values of sensitivity and specificity of practical interest. Simulations results are presented. For illustration purposes, considering real data from newborns with very low birthweight, the new method was applied in order to discriminate the better index for evaluating the risk of death.     

On Robust and Efficient Estimation of the Center of Symmetry

In this article, a class of estimators of the center of symmetry based on the empirical characteristic function is examined. In the spirit of the Hodges–Lehmann estimator, the resulting procedures are shown to be a function of the pairwise averages. The proposed procedures are also shown to have an equivalent representation as the minimizers of certain distances between two corresponding kernel density estimators. An alternative characterization of the Hodges–Lehmann estimator is established upon the use of a particularly simple choice of kernel.     

Untransformed first observation problem in regression model with moving average process

This paper shows the impact of underestimation of variance of an estima-tor when first observation is left untransformed to simplify the computational procedure. In fact, the bias of the variance is not diminishing even for large sample size for the model considered. By partitioning the covariance matrix into two parts, this paper explains why least square estimator with untrans-formed first observation shows such a consequence. To demonstrate this, an exact GLS estimator is developed by modifying an approximate estimator. Nonetheless, the computational simplicity remains same.     

Empirical Likelihood-Based Confidence Intervals for the Sensitivity of a Continuous-Scale Diagnostic Test with Missing Data

In a continuous-scale diagnostic test, the receiver operating characteristic (ROC) curve is useful to evaluate the range of the sensitivity at the cut-off point that yields a desired specificity. Many current studies on inference of the ROC curve focus on the complete data case. In this paper, an imputation-based profile empirical likelihood ratio for the sensitivity, which is free of bandwidth selection, is defined and shown to follow an asymptotically scaled Chi-square distribution. Two new confidence intervals are proposed for the sensitivity with missing data. Simulation studies are conducted to evaluate the finite sample performance of the proposed intervals in terms of coverage probability. A real example is used to illustrate the new methods.     

Estimating mixtures of normal distributions via empirical characteristic function

This paper uses the empirical characteristic function (ECF) procedure to estimate the parameters of mixtures of normal distributions. Since the characteristic function is uniformly bounded, the procedure gives estimates that are numerically stable. It is shown that, using Monte Carlo simulation, the finite sample properties of th ECF estimator are very good, even in the case where the popular maximum likelihood estimator fails to exist. An empirical application is illustrated using the monthl excess return of the Nyse value-weighted index.     

Robust linear discriminant analysis using S‐estimators

The authors consider a robust linear discriminant function based on high breakdown location and covariance matrix estimators. They derive influence functions for the estimators of the parameters of the discriminant function and for the associated classification error. The most B‐robust estimator is determined within the class of multivariate S‐estimators. This estimator, which minimizes the maximal influence that an outlier can have on the classification error, is also the most B‐robust location S‐estimator. A comparison of the most B‐robust estimator with the more familiar biweight S‐estimator is made.