Empirical likelihood for quantile regression models with longitudinal data

We develop two empirical likelihood-based inference procedures for longitudinal data under the framework of quantile regression. The proposed methods avoid estimating the unknown error density function and the intra-subject correlation involved in the asymptotic covariance matrix of the quantile estimators. By appropriately smoothing the quantile score function, the empirical likelihood approach is shown to have a higher-order accuracy through the Bartlett correction. The proposed methods exhibit finite-sample advantages over the normal approximation-based and bootstrap methods in a simulation study and the analysis of a longitudinal ophthalmology data set.     

Nonparametric and semiparametric estimation of the three way receiver operating characteristic surface

In many situations the diagnostic decision is not limited to a binary choice. Binary statistical tools such as receiver operating characteristic (ROC) curve and area under the ROC curve (AUC) need to be expanded to address three-category classification problem. Previous authors have suggest various ways to model the extension of AUC but not the ROC surface. Only simple parametric approaches are proposed for modeling the ROC measure under the assumption that test results all follow normal distributions. We study the estimation methods of three-dimensional ROC surfaces with nonparametric and semiparametric estimators. Asymptotical results are provided as a basis for statistical inference. Simulation studies are performed to assess the validity of our proposed methods in finite samples. We consider an Alzheimer's disease example from a clinical study in the US as an illustration. The nonparametric and semiparametric modelling approaches for the three way ROC analysis can be readily generalized to diagnostic problems with more than three classes.     

COMBINING MULTIPLE MARKERS FOR MULTI‐CATEGORY CLASSIFICATION: AN ROC SURFACE APPROACH

The Receiver Operating Characteristic (ROC) curve and the Area Under the ROC Curve (AUC) are effective statistical tools for evaluating the accuracy of diagnostic tests for binary‐class medical data. However, many real‐world biomedical problems involve more than two categories. The Volume Under the ROC Surface (VUS) and Hypervolume Under the ROC Manifold (HUM) measures are extensions for the AUC under three‐class and multiple‐class models. Inference methods for such measures have been proposed recently. We develop a method of constructing a linear combination of markers for which the VUS or HUM of the combined markers is maximized. Asymptotic validity of the estimator is justified by extending the results for maximum rank correlation estimation that are well known in econometrics. A bootstrap resampling method is then applied to estimate the sampling variability. Simulations and examples are provided to demonstrate our methods.     

Fast and robust bootstrap

In this paper we review recent developments on a bootstrap method for robust estimators which is computationally faster and more resistant to outliers than the classical bootstrap. This fast and robust bootstrap method is, under reasonable regularity conditions, asymptotically consistent. We describe the method in general and then consider its application to perform inference based on robust estimators for the linear regression and multivariate location-scatter models. In particular, we study confidence and prediction intervals and tests of hypotheses for linear regression models, inference for location-scatter parameters and principal components, and classification error estimation for discriminant analysis.     

Exact Bootstrap Variances of the Area Under ROC Curve

The area under the Receiver Operating Characteristic (ROC) curve (AUC) and related summary indices are widely used for assessment of accuracy of an individual and comparison of performances of several diagnostic systems in many areas including studies of human perception, decision making, and the regulatory approval process for new diagnostic technologies. Many investigators have suggested implementing the bootstrap approach to estimate variability of AUC-based indices. Corresponding bootstrap quantities are typically estimated by sampling a bootstrap distribution. Such a process, frequently termed Monte Carlo bootstrap, is often computationally burdensome and imposes an additional sampling error on the resulting estimates. In this article, we demonstrate that the exact or ideal (sampling error free) bootstrap variances of the nonparametric estimator of AUC can be computed directly, i.e., avoiding resampling of the original data, and we develop easy-to-use formulas to compute them. We derive the formulas for the variances of the AUC corresponding to a single given or random reader, and to the average over several given or randomly selected readers. The derived formulas provide an algorithm for computing the ideal bootstrap variances exactly and hence improve many bootstrap methods proposed earlier for analyzing AUCs by eliminating the sampling error and sometimes burdensome computations associated with a Monte Carlo (MC) approximation. In addition, the availability of closed-form solutions provides the potential for an analytical assessment of the properties of bootstrap variance estimators. Applications of the proposed method are shown on two experimentally ascertained datasets that illustrate settings commonly encountered in diagnostic imaging. In the context of the two examples we also demonstrate the magnitude of the effect of the sampling error of the MC estimators on the resulting inferences.     

Consistency of non parametric estimators of the area under the ROC curve

ABSTRACT

The area under the receiver operating characteristic (ROC) curve is a popular summary index that measures the accuracy of a continuous-scale diagnostic test to measure its accuracy. Under certain conditions on estimators of distribution functions, we prove a theorem on strong consistency of the non parametric “plugin” estimators of the area under the ROC curve. Based on this theorem, we construct some new “plugin” consistent estimators. The performance of the non parametric estimators considered is illustrated numerically and the estimators are compared in terms of bias, variance, and mean square error.     

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.     

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.     

Doubly robust point and variance estimation in the presence of imputed survey data

Imputation is often used in surveys to treat item nonresponse. It is well known that treating the imputed values as observed values may lead to substantial underestimation of the variance of the point estimators. To overcome the problem, a number of variance estimation methods have been proposed in the literature, including resampling methods such as the jackknife and the bootstrap. In this paper, we consider the problem of doubly robust inference in the presence of imputed survey data. In the doubly robust literature, point estimation has been the main focus. In this paper, using the reverse framework for variance estimation, we derive doubly robust linearization variance estimators in the case of deterministic and random regression imputation within imputation classes. Also, we study the properties of several jackknife variance estimators under both negligible and nonnegligible sampling fractions. A limited simulation study investigates the performance of various variance estimators in terms of relative bias and relative stability. Finally, the asymptotic normality of imputed estimators is established for stratified multistage designs under both deterministic and random regression imputation. The Canadian Journal of Statistics 40: 259–281; 2012 © 2012 Statistical Society of Canada     

A non-inferiority test for diagnostic accuracy in the absence of the golden standard test based on the paired partial areas under receiver operating characteristic curves

Non-inferiority tests are often measured for the diagnostic accuracy in medical research. The area under the receiver operating characteristic (ROC) curve is a familiar diagnostic measure for the overall diagnostic accuracy. Nevertheless, since it may not differentiate the diverse shapes of the ROC curves with different diagnostic significance, the partial area under the ROC (PAUROC) curve, another summary measure emerges for such diagnostic processes that require the false-positive rate to be in the clinically interested range. Traditionally, to estimate the PAUROC, the golden standard (GS) test on the true disease status is required. Nevertheless, the GS test may sometimes be infeasible. Besides, in a lot of research fields such as the epidemiology field, the true disease status of the patients may not be known or available. Under the normality assumption on diagnostic test results, based on the expectation-maximization algorithm in combination with the bootstrap method, we propose the heuristic method to construct a non-inferiority test for the difference in the paired PAUROCs without the GS test. Through the simulation study, although the proposed method might provide a liberal test, as a whole, the empirical size of the proposed method sufficiently controls the size at the significance level, and the empirical power of the proposed method in the absence of the GS is as good as that of the non-inferiority in the presence of the GS. The proposed method is illustrated with the published data.     

Bootstrap-based inferential improvements in beta autoregressive moving average model

We consider the issue of performing accurate small sample inference in beta autoregressive moving average model, which is useful for modeling and forecasting continuous variables that assume values in the interval (0,?1). The inferences based on conditional maximum likelihood estimation have good asymptotic properties, but their performances in small samples may be poor. This way, we propose bootstrap bias corrections of the point estimators and different bootstrap strategies for confidence interval improvements. Our Monte Carlo simulations show that finite sample inference based on bootstrap corrections is much more reliable than the usual inferences. We also presented an empirical application.     

Nonparametric covariate adjustment for receiver operating characteristic curves

The accuracy of a diagnostic test is typically characterized using the receiver operating characteristic (ROC) curve. Summarizing indexes such as the area under the ROC curve (AUC) are used to compare different tests as well as to measure the difference between two populations. Often additional information is available on some of the covariates which are known to influence the accuracy of such measures. The authors propose nonparametric methods for covariate adjustment of the AUC. Models with normal errors and possibly non‐normal errors are discussed and analyzed separately. Nonparametric regression is used for estimating mean and variance functions in both scenarios. In the model that relaxes the assumption of normality, the authors propose a covariate‐adjusted Mann–Whitney estimator for AUC estimation which effectively uses available data to construct working samples at any covariate value of interest and is computationally efficient for implementation. This provides a generalization of the Mann–Whitney approach for comparing two populations by taking covariate effects into account. The authors derive asymptotic properties for the AUC estimators in both settings, including asymptotic normality, optimal strong uniform convergence rates and mean squared error (MSE) consistency. The MSE of the AUC estimators was also assessed in smaller samples by simulation. Data from an agricultural study were used to illustrate the methods of analysis. The Canadian Journal of Statistics 38:27–46; 2010 © 2009 Statistical Society of Canada     

A Framework for Random-Effects ROC Analysis: Biases with the Bootstrap and Other Variance Estimators

In this article, we analyze the three-way bootstrap estimate of the variance of the reader-averaged nonparametric area under the receiver operating characteristic (ROC) curve. The setting for this work is medical imaging, and the experimental design involves sampling from three distributions: a set of normal and diseased cases (patients), and a set of readers (doctors). The experiment we consider is fully crossed in that each reader reads each case. A reading generates a score that indicates the reader's level of suspicion that the patient is diseased. The distribution of scores for the normal patients is compared to the distribution of scores for the diseased patients via an ROC curve, and the area under the ROC curve (AUC) summarizes the reader's diagnostic ability to separate the normal patients from the diseased ones. We find that the bootstrap estimate of the variance of the reader-averaged AUC is biased, and we represent this bias in terms of moments of success outcomes. This representation helps unify and improve several current methods for multi-reader multi-case (MRMC) ROC analysis.     

Weighted empirical likelihood inference for multiple samples

We propose a weighted empirical likelihood approach to inference with multiple samples, including stratified sampling, the estimation of a common mean using several independent and non-homogeneous samples and inference on a particular population using other related samples. The weighting scheme and the basic result are motivated and established under stratified sampling. We show that the proposed method can ideally be applied to the common mean problem and problems with related samples. The proposed weighted approach not only provides a unified framework for inference with multiple samples, including two-sample problems, but also facilitates asymptotic derivations and computational methods. A bootstrap procedure is also proposed in conjunction with the weighted approach to provide better coverage probabilities for the weighted empirical likelihood ratio confidence intervals. Simulation studies show that the weighted empirical likelihood confidence intervals perform better than existing ones.     

Semi-parametric estimation of ROC curves based on binomial regression modelling

This paper presents a method of estimating a receiver operating characteristic (ROC) curve when the underlying diagnostic variable X is continuous and fully observed. The new method is based on modelling the probability of response given X , rather than the distribution of X given response. The method offers advantages in modelling flexibility and computational simplicity. The resulting ROC curve estimates are semi-parametric and can, in principle, take an infinite variety of shapes. Moreover, model selection can be based on standard methods within the binomial regression framework. Statistical accuracy of the curve estimate is provided by a simply implemented bootstrap approach.     

Nonparametric methodology for the time-dependent partial area under the ROC curve

To assess the classification accuracy of a continuous diagnostic result, the receiver operating characteristic (ROC) curve is commonly used in applications. The partial area under the ROC curve (pAUC) is one of the widely accepted summary measures due to its generality and ease of probability interpretation. In the field of life science, a direct extension of the pAUC into the time-to-event setting can be used to measure the usefulness of a biomarker for disease detection over time. Without using a trapezoidal rule, we propose nonparametric estimators, which are easily computed and have closed-form expressions, for the time-dependent pAUC. The asymptotic Gaussian processes of the estimators are established and the estimated variance-covariance functions are provided, which are essential in the construction of confidence intervals. The finite sample performance of the proposed inference procedures are investigated through a series of simulations. Our method is further applied to evaluate the classification ability of CD4 cell counts on patient's survival time in the AIDS Clinical Trials Group (ACTG) 175 study. In addition, the inferences can be generalized to compare the time-dependent pAUCs between patients received the prior antiretroviral therapy and those without it.     

Bootstrap methods for heteroskedastic regression models: evidence on estimation and testing

This paper uses Monte Carlo simulation analysis to study the finite-sample behavior of bootstrap estimators and tests in the linear heteroskedastic model. We consider four different bootstrapping schemes, three of them specifically tailored to handle heteroskedasticity. Our results show that weighted bootstrap methods can be successfully used to estimate the variances of the least squares estimators of the linear parameters both under normality and under nonnormality. Simulation results are also given comparing the size and power of the bootstrapped Breusch-Pagan test with that of the original test and of Bartlett and Edgeworth-corrected tests. The bootstrap test was found to be robust against unfavorable regression designs.     

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.     

Weighted empirical adaptive variance estimators for correlated data regression

Estimating equations based on marginal generalized linear models are useful for regression modelling of correlated data, but inference and testing require reliable estimates of standard errors. We introduce a class of variance estimators based on the weighted empirical variance of the estimating functions and show that an adaptive choice of weights allows reliable estimation both asymptotically and by simulation in finite samples. Connections with previous bootstrap and jackknife methods are explored. The effect of reliable variance estimation is illustrated in data on health effects of air pollution in King County, Washington.     

CONFIDENCE INTERVALS FOR THE DIFFERENCE BETWEEN TWO PARTIAL AUCS

As new diagnostic tests are developed and marketed, it is very important to be able to compare the accuracy of a given two continuous‐scale diagnostic tests. An effective method to evaluate the difference between the diagnostic accuracy of two tests is to compare partial areas under the receiver operating characteristic curves (AUCs). In this paper, we review existing parametric methods. Then, we propose a new semiparametric method and a new nonparametric method to investigate the difference between two partial AUCs. For the difference between two partial AUCs under each method, we derive a normal approximation, define an empirical log‐likelihood ratio, and show that the empirical log‐likelihood ratio follows a scaled chi‐square distribution. We construct five confidence intervals for the difference based on normal approximation, bootstrap, and empirical likelihood methods. Finally, extensive simulation studies are conducted to compare the finite‐sample performances of these intervals, and a real example is used as an application of our recommended intervals. The simulation results indicate that the proposed hybrid bootstrap and empirical likelihood intervals outperform other existing intervals in most cases.