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.     

ROC curve estimation when covariates affect the verification process

A receiver operating characteristic (ROC) curve is commonly used to measure the accuracy of a medical test. It is a plot of the true positive fraction (sensitivity) against the false positive fraction (1-specificity) for increasingly stringent positivity criterion. Bias can occur in estimation of an ROC curve if only some of the tested patients are selected for disease verification and if analysis is restricted only to the verified cases. This bias is known as verification bias. In this paper, we address the problem of correcting for verification bias in estimation of an ROC curve when the verification process and efficacy of the diagnostic test depend on covariates. Our method applies the EM algorithm to ordinal regression models to derive ML estimates for ROC curves as a function of covariates, adjusted for covariates affecting the likelihood of being verified. Asymptotic variance estimates are obtained using the observed information matrix of the observed data. These estimates are derived under the missing-at-random assumption, which means that selection for disease verification depends only on the observed data, i.e., the test result and the observed covariates. We also address the issues of model selection and model checking. Finally, we illustrate the proposed method on data from a two-phase study of dementia disorders, where selection for verification depends on the screening test result and age.     

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.     

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.     

Order-restricted inference for clustered ROC data with application to fingerprint matching accuracy

Receiver operating characteristic (ROC) curve is commonly used to evaluate and compare the accuracy of classification methods or markers. Estimating ROC curves has been an important problem in various fields including biometric recognition and diagnostic medicine. In real applications, classification markers are often developed under two or more ordered conditions, such that a natural stochastic ordering exists among the observations. Incorporating such a stochastic ordering into estimation can improve statistical efficiency (Davidov and Herman, 2012). In addition, clustered and correlated data arise when multiple measurements are gleaned from the same subject, making estimation of ROC curves complicated due to within-cluster correlations. In this article, we propose to model the ROC curve using a weighted empirical process to jointly account for the order constraint and within-cluster correlation structure. The algebraic properties of resulting summary statistics of the ROC curve such as its area and partial area are also studied. The algebraic expressions reduce to the ones by Davidov and Herman (2012) for independent observations. We derive asymptotic properties of the proposed order-restricted estimators and show that they have smaller mean-squared errors than the existing estimators. Simulation studies also demonstrate better performance of the newly proposed estimators over existing methods for finite samples. The proposed method is further exemplified with the fingerprint matching data from the National Institute of Standards and Technology Special Database 4.     

Semiparametric estimation of the covariate-specific ROC curve in presence of ignorable verification bias

Covariate-specific receiver operating characteristic (ROC) curves are often used to evaluate the classification accuracy of a medical diagnostic test or a biomarker, when the accuracy of the test is associated with certain covariates. In many large-scale screening tests, the gold standard is subject to missingness due to high cost or harmfulness to the patient. In this article, we propose a semiparametric estimation of the covariate-specific ROC curves with a partial missing gold standard. A location-scale model is constructed for the test result to model the covariates' effect, but the residual distributions are left unspecified. Thus the baseline and link functions of the ROC curve both have flexible shapes. With the gold standard missing at random (MAR) assumption, we consider weighted estimating equations for the location-scale parameters, and weighted kernel estimating equations for the residual distributions. Three ROC curve estimators are proposed and compared, namely, imputation-based, inverse probability weighted, and doubly robust estimators. We derive the asymptotic normality of the estimated ROC curve, as well as the analytical form of the standard error estimator. The proposed method is motivated and applied to the data in an Alzheimer's disease research.     

Minimum‐Norm Estimation for Binormal Receiver Operating Characteristic (ROC) Curves

The receiver operating characteristic (ROC) curve is often used to assess the usefulness of a diagnostic test. We present a new method to estimate the parameters of a popular semi‐parametric ROC model, called the binormal model. Our method is based on minimization of the functional distance between two estimators of an unknown transformation postulated by the model, and has a simple, closed‐form solution. We study the asymptotics of our estimators, show via simulation that they compare favorably with existing estimators, and illustrate how covariates may be incorporated into the norm minimization framework.     

Robust Estimation of Area Under ROC Curve Using Auxiliary Variables in the Presence of Missing Biomarker Values

Summary In medical research, the receiver operating characteristic (ROC) curves can be used to evaluate the performance of biomarkers for diagnosing diseases or predicting the risk of developing a disease in the future. The area under the ROC curve (ROC AUC), as a summary measure of ROC curves, is widely utilized, especially when comparing multiple ROC curves. In observational studies, the estimation of the AUC is often complicated by the presence of missing biomarker values, which means that the existing estimators of the AUC are potentially biased. In this article, we develop robust statistical methods for estimating the ROC AUC and the proposed methods use information from auxiliary variables that are potentially predictive of the missingness of the biomarkers or the missing biomarker values. We are particularly interested in auxiliary variables that are predictive of the missing biomarker values. In the case of missing at random (MAR), that is, missingness of biomarker values only depends on the observed data, our estimators have the attractive feature of being consistent if one correctly specifies, conditional on auxiliary variables and disease status, either the model for the probabilities of being missing or the model for the biomarker values. In the case of missing not at random (MNAR), that is, missingness may depend on the unobserved biomarker values, we propose a sensitivity analysis to assess the impact of MNAR on the estimation of the ROC AUC. The asymptotic properties of the proposed estimators are studied and their finite‐sample behaviors are evaluated in simulation studies. The methods are further illustrated using data from a study of maternal depression during pregnancy.     

ROC curve inference for best linear combination of two biomarkers subject to limits of detection

The receiver operating characteristic (ROC) curve is a tool commonly used to evaluate biomarker utility in clinical diagnosis of disease. Often, multiple biomarkers are developed to evaluate the discrimination for the same outcome. Levels of multiple biomarkers can be combined via best linear combination (BLC) such that their overall discriminatory ability is greater than any of them individually. Biomarker measurements frequently have undetectable levels below a detection limit sometimes denoted as limit of detection (LOD). Ignoring observations below the LOD or substituting some replacement value as a method of correction has been shown to lead to negatively biased estimates of the area under the ROC curve for some distributions of single biomarkers. In this paper, we develop asymptotically unbiased estimators, via the maximum likelihood technique, of the area under the ROC curve of BLC of two bivariate normally distributed biomarkers affected by LODs. We also propose confidence intervals for this area under curve. Point and confidence interval estimates are scrutinized by simulation study, recording bias and root mean square error and coverage probability, respectively. An example using polychlorinated biphenyl (PCB) levels to classify women with and without endometriosis illustrates the potential benefits of our methods.     

Estimating Incidence Rates from Population‐Based Case‐Control Studies in the Presence of Nonrespondents

In population‐based case‐control studies, it is of great public‐health importance to estimate the disease incidence rates associated with different levels of risk factors. This estimation is complicated by the fact that in such studies the selection probabilities for the cases and controls are unequal. A further complication arises when the subjects who are selected into the study do not participate (i.e. become nonrespondents) and nonrespondents differ systematically from respondents. In this paper, we show how to account for unequal selection probabilities as well as differential nonresponses in the incidence estimation. We use two logistic models, one relating the disease incidence rate to the risk factors, and one modelling the predictors that affect the nonresponse probability. After estimating the regression parameters in the nonresponse model, we estimate the regression parameters in the disease incidence model by a weighted estimating function that weights a respondent's contribution to the likelihood score function by the inverse of the product of his/her selection probability and his/her model‐predicted response probability. The resulting estimators of the regression parameters and the corresponding estimators of the incidence rates are shown to be consistent and asymptotically normal with easily estimated variances. Simulation results demonstrate that the asymptotic approximations are adequate for practical use and that failure to adjust for nonresponses could result in severe biases. An illustration with data from a cardiovascular study that motivated this work is presented.     

Youden Index and optimal cut-point estimated from observations affected by a lower limit of detection

The receiver operating characteristic (ROC) curve is used to evaluate a biomarker's ability for classifying disease status. The Youden Index (J), the maximum potential effectiveness of a biomarker, is a common summary measure of the ROC curve. In biomarker development, levels may be unquantifiable below a limit of detection (LOD) and missing from the overall dataset. Disregarding these observations may negatively bias the ROC curve and thus J. Several correction methods have been suggested for mean estimation and testing; however, little has been written about the ROC curve or its summary measures. We adapt non-parametric (empirical) and semi-parametric (ROC-GLM [generalized linear model]) methods and propose parametric methods (maximum likelihood (ML)) to estimate J and the optimal cut-point (c *) for a biomarker affected by a LOD. We develop unbiased estimators of J and c * via ML for normally and gamma distributed biomarkers. Alpha level confidence intervals are proposed using delta and bootstrap methods for the ML, semi-parametric, and non-parametric approaches respectively. Simulation studies are conducted over a range of distributional scenarios and sample sizes evaluating estimators' bias, root-mean square error, and coverage probability; the average bias was less than one percent for ML and GLM methods across scenarios and decreases with increased sample size. An example using polychlorinated biphenyl levels to classify women with and without endometriosis illustrates the potential benefits of these methods. We address the limitations and usefulness of each method in order to give researchers guidance in constructing appropriate estimates of biomarkers' true discriminating capabilities.     

Model selection of generalized estimating equations with multiply imputed longitudinal data

Longitudinal data often encounter missingness with monotone and/or intermittent missing patterns. Multiple imputation (MI) has been popularly employed for analysis of missing longitudinal data. In particular, the MI‐GEE method has been proposed for inference of generalized estimating equations (GEE) when missing data are imputed via MI. However, little is known about how to perform model selection with multiply imputed longitudinal data. In this work, we extend the existing GEE model selection criteria, including the “quasi‐likelihood under the independence model criterion” (QIC) and the “missing longitudinal information criterion” (MLIC), to accommodate multiple imputed datasets for selection of the MI‐GEE mean model. According to real data analyses from a schizophrenia study and an AIDS study, as well as simulations under nonmonotone missingness with moderate proportion of missing observations, we conclude that: (i) more than a few imputed datasets are required for stable and reliable model selection in MI‐GEE analysis; (ii) the MI‐based GEE model selection methods with a suitable number of imputations generally perform well, while the naive application of existing model selection methods by simply ignoring missing observations may lead to very poor performance; (iii) the model selection criteria based on improper (frequentist) multiple imputation generally performs better than their analogies based on proper (Bayesian) multiple imputation.     

Semi-parametric area under the curve regression method for diagnostic studies with ordinal data

The classification accuracy of new diagnostic tests is based on receiver operating characteristic (ROC) curves. The area under the ROC curve (AUC) is one of the well-accepted summary measures for describing the accuracy of diagnostic tests. The AUC summary measure can vary by patient and testing characteristics. Thus, the performance of the test may be different in certain subpopulation of patients and readers. For this purpose, we propose a direct semi-parametric regression model for the non-parametric AUC measure for ordinal data while accounting for discrete and continuous covariates. The proposed method can be used to estimate the AUC value under degenerate data where certain rating categories are not observed. We will discuss the non-standard asymptotic theory, since the estimating functions were based on cross-correlated random variables. Simulation studies based on different classification models showed that the proposed model worked reasonably well with small percent bias and percent mean-squared error. The proposed method was applied to the prostate cancer study to estimate the AUC for four readers, and the carotid vessel study with age, gender, history of previous stroke, and total number of risk factors as covariates, to estimate the accuracy of the diagnostic test in the presence of subject-level covariates.     

Identifiability and estimation of causal effects in randomized trials with noncompliance and completely nonignorable missing data

Summary .  In this article, we first study parameter identifiability in randomized clinical trials with noncompliance and missing outcomes. We show that under certain conditions the parameters of interest are identifiable even under different types of completely nonignorable missing data: that is, the missing mechanism depends on the outcome. We then derive their maximum likelihood and moment estimators and evaluate their finite-sample properties in simulation studies in terms of bias, efficiency, and robustness. Our sensitivity analysis shows that the assumed nonignorable missing-data model has an important impact on the estimated complier average causal effect (CACE) parameter. Our new method provides some new and useful alternative nonignorable missing-data models over the existing latent ignorable model, which guarantees parameter identifiability, for estimating the CACE in a randomized clinical trial with noncompliance and missing data.     

A robust method using propensity score stratification for correcting verification bias for binary tests

Sensitivity and specificity are common measures of the accuracy of a diagnostic test. The usual estimators of these quantities are unbiased if data on the diagnostic test result and the true disease status are obtained from all subjects in an appropriately selected sample. In some studies, verification of the true disease status is performed only for a subset of subjects, possibly depending on the result of the diagnostic test and other characteristics of the subjects. Estimators of sensitivity and specificity based on this subset of subjects are typically biased; this is known as verification bias. Methods have been proposed to correct verification bias under the assumption that the missing data on disease status are missing at random (MAR), that is, the probability of missingness depends on the true (missing) disease status only through the test result and observed covariate information. When some of the covariates are continuous, or the number of covariates is relatively large, the existing methods require parametric models for the probability of disease or the probability of verification (given the test result and covariates), and hence are subject to model misspecification. We propose a new method for correcting verification bias based on the propensity score, defined as the predicted probability of verification given the test result and observed covariates. This is estimated separately for those with positive and negative test results. The new method classifies the verified sample into several subsamples that have homogeneous propensity scores and allows correction for verification bias. Simulation studies demonstrate that the new estimators are more robust to model misspecification than existing methods, but still perform well when the models for the probability of disease and probability of verification are correctly specified.     

Multiple Imputation Methods for Estimating Regression Coefficients in the Competing Risks Model with Missing Cause of Failure

We propose a method to estimate the regression coefficients in a competing risks model where the cause-specific hazard for the cause of interest is related to covariates through a proportional hazards relationship and when cause of failure is missing for some individuals. We use multiple imputation procedures to impute missing cause of failure, where the probability that a missing cause is the cause of interest may depend on auxiliary covariates, and combine the maximum partial likelihood estimators computed from several imputed data sets into an estimator that is consistent and asymptotically normal. A consistent estimator for the asymptotic variance is also derived. Simulation results suggest the relevance of the theory in finite samples. Results are also illustrated with data from a breast cancer study.     

Estimating the area under the ROC curve when transporting a prediction model to a target population

We propose methods for estimating the area under the receiver operating characteristic (ROC) curve (AUC) of a prediction model in a target population that differs from the source population that provided the data used for original model development. If covariates that are associated with model performance, as measured by the AUC, have a different distribution in the source and target populations, then AUC estimators that only use data from the source population will not reflect model performance in the target population. Here, we provide identification results for the AUC in the target population when outcome and covariate data are available from the sample of the source population, but only covariate data are available from the sample of the target population. In this setting, we propose three estimators for the AUC in the target population and show that they are consistent and asymptotically normal. We evaluate the finite-sample performance of the estimators using simulations and use them to estimate the AUC in a nationally representative target population from the National Health and Nutrition Examination Survey for a lung cancer risk prediction model developed using source population data from the National Lung Screening Trial.     

Assessment of evaluation criteria for survival prediction from genomic data

Survival prediction from high-dimensional genomic data is dependent on a proper regularization method. With an increasing number of such methods proposed in the literature, comparative studies are called for and some have been performed. However, there is currently no consensus on which prediction assessment criterion should be used for time-to-event data. Without a firm knowledge about whether the choice of evaluation criterion may affect the conclusions made as to which regularization method performs best, these comparative studies may be of limited value. In this paper, four evaluation criteria are investigated: the log-rank test for two groups, the area under the time-dependent ROC curve (AUC), an R2-measure based on the Cox partial likelihood, and an R2-measure based on the Brier score. The criteria are compared according to how they rank six widely used regularization methods that are based on the Cox regression model, namely univariate selection, principal components regression (PCR), supervised PCR, partial least squares regression, ridge regression, and the lasso. Based on our application to three microarray gene expression data sets, we find that the results obtained from the widely used log-rank test deviate from the other three criteria studied. For future studies, where one also might want to include non-likelihood or non-model-based regularization methods, we argue in favor of AUC and the R2-measure based on the Brier score, as these do not suffer from the arbitrary splitting into two groups nor depend on the Cox partial likelihood.     

An asymptotic theory for model selection inference in general semiparametric problems

Hjort & Claeskens (2003) developed an asymptotic theoryfor model selection, model averaging and subsequent inferenceusing likelihood methods in parametric models, along with associatedconfidence statements. In this article, we consider a semiparametricversion of this problem, wherein the likelihood depends on parametersand an unknown function, and model selection/averaging is tobe applied to the parametric parts of the model. We show thatall the results of Hjort & Claeskens hold in the semiparametriccontext, if the Fisher information matrix for parametric modelsis replaced by the semiparametric information bound for semiparametricmodels, and if maximum likelihood estimators for parametricmodels are replaced by semiparametric efficient profile estimators.Our methods of proof employ Le Cam's contiguity lemmas, leadingto transparent results. The results also describe the behaviourof semiparametric model estimators when the parametric componentis misspecified, and also have implications for pointwise-consistentmodel selectors.     

Generalized linear mixture models for handling nonignorable dropouts in longitudinal studies

This paper presents a method for analysing longitudinal data when there are dropouts. In particular, we develop a simple method based on generalized linear mixture models for handling nonignorable dropouts for a variety of discrete and continuous outcomes. Statistical inference for the model parameters is based on a generalized estimating equations (GEE) approach (Liang and Zeger, 1986). The proposed method yields estimates of the model parameters that are valid when nonresponse is nonignorable under a variety of assumptions concerning the dropout process. Furthermore, the proposed method can be implemented using widely available statistical software. Finally, an example using data from a clinical trial of contracepting women is used to illustrate the methodology.