Comparison of the accuracy of multiple binary tests in the presence of partial disease verification

In the presence of partial disease verification, the comparison of the accuracy of binary diagnostic tests cannot be carried out through the paired comparison of the diagnostic tests applying McNemar's test, since for a subsample of patients the disease status is unknown. In this study, we have deduced the maximum likelihood estimators for the sensitivities and specificities of multiple binary diagnostic tests and we have studied various joint hypothesis tests based on the chi-square distribution to compare simultaneously the accuracy of these binary diagnostic tests when for some patients in the sample the disease status is unknown. Simulation experiments were carried out to study the type I error and the power of each hypothesis test deduced. The results obtained were applied to the diagnosis of coronary stenosis.     

Risk of Error and the Kappa Coefficient of a Binary Diagnostic Test in the Presence of Partial Verification

The accuracy of a binary diagnostic test is usually measured in terms of its sensitivity and its specificity, or through positive and negative predictive values. Another way to describe the validity of a binary diagnostic test is the risk of error and the kappa coefficient of the risk of error. The risk of error is the average loss that is caused when incorrectly classifying a non-diseased or a diseased patient, and the kappa coefficient of the risk of error is a measure of the agreement between the diagnostic test and the gold standard. In the presence of partial verification of the disease, the disease status of some patients is unknown, and therefore the evaluation of a diagnostic test cannot be carried out through the traditional method. In this paper, we have deduced the maximum likelihood estimators and variances of the risk of error and of the kappa coefficient of the risk of error in the presence of partial verification of the disease. Simulation experiments have been carried out to study the effect of the verification probabilities on the coverage of the confidence interval of the kappa coefficient.     

The effect of verification bias on the comparison of predictive values of two binary diagnostic tests

The comparison of the accuracy of two binary diagnostic tests has traditionally required knowledge of the disease status in all of the patients in the sample via the application of a gold standard. In practice, the gold standard is not always applied to all patients in a sample, and the problem of partial verification of the disease arises. The accuracy of a binary diagnostic test can be measured in terms of positive and negative predictive values, which represent the accuracy of a diagnostic test when it is applied to a cohort of patients. In this paper, we deduce the maximum likelihood estimators of predictive values (PVs) of two binary diagnostic tests, and the hypothesis tests to compare these measures when, in the presence of partial disease verification, the verification process only depends on the results of the two diagnostic tests. The effect of verification bias on the naïve estimators of PVs of two diagnostic tests is studied, and simulation experiments are performed in order to investigate the small sample behaviour of hypothesis tests. The hypothesis tests which we have deduced can be applied when all of the patients are verified with the gold standard. The results obtained have been applied to the diagnosis of coronary stenosis.     

Average kappa coefficient: a new measure to assess a binary test considering the losses associated with an erroneous classification

The weighted kappa coefficient of a binary diagnostic test (BDT) is a measure of performance of a BDT, and is a function of the sensitivity and the specificity of the diagnostic test, of the disease prevalence and the weighting index. Weighting index represents the relative loss between the false positives and the false negatives. In this study, we propose a new measure of performance of a BDT: the average kappa coefficient. This parameter is the average function of the weighted kappa coefficients and does not depend on the weighting index. We have studied three asymptotic confidence intervals (CIs) for the average kappa coefficient, Wald, logit and bias-corrected bootstrap, and we carried out some simulation experiments to study the asymptotic coverage of each of the three CIs. We have written a program in R, called ‘akcbdt’, to estimate the average kappa coefficient of a BDT. This program is available as supplementary material. The results were applied to two examples.     

Asymptotic hypothesis test to compare likelihood ratios of multiple diagnostic tests in unpaired designs

The accuracy of a binary diagnostic test is usually measured in terms of its sensitivity and its specificity. Other measures of the performance of a diagnostic test are the positive and negative likelihood ratios, which quantify the increase in knowledge about the presence of the disease through the application of a diagnostic test, and which depend on the sensitivity and specificity of the diagnostic test. In this article, we construct an asymptotic hypothesis test to simultaneously compare the positive and negative likelihood ratios of two or more diagnostic tests in unpaired designs. The hypothesis test is based on the logarithmic transformation of the likelihood ratios and on the chi-square distribution. Simulation experiments have been carried out to study the type I error and the power of the constructed hypothesis test when comparing two and three binary diagnostic tests. The method has been extended to the case of multiple multi-level diagnostic tests.     

Confidence Intervals of Weighted Kappa Coefficient of a Binary Diagnostic Test

Sensitivity and specificity are classic parameters to assess the performance of a binary diagnostic test. Another useful parameter to measure the performance of a binary test is the weighted kappa coefficient, which is a measure of the classificatory agreement between the binary test and the gold standard. Various confidence intervals are proposed for the weighted kappa coefficient when the binary test and the gold standard are applied to all of the patients in a random sample. The results have been applied to the diagnosis of coronary artery disease.     

Four Correlation Coefficients with a Third Blocking Variable: Their Efficacy,Relative Efficiency,and Test Statistics

Abstract

The efficacy and the asymptotic relative efficiency (ARE) of a weighted sum of Kendall's taus, a weighted sum of Spearman's rhos, a weighted sum of Pearson's r's, and a weighted sum of z-transformation of the Fisher–Yates correlation coefficients, in the presence of a blocking variable, are discussed. The method of selecting the weighting constants that maximize the efficacy of these four correlation coefficients is proposed. The estimate, test statistics and confidence interval of the four correlation coefficients with weights are also developed. To compare the small-sample properties of the four tests, a simulation study is performed. The theoretical and simulated results all prefer the weighted sum of the Pearson correlation coefficients with the optimal weights, as well as the weighted sum of z-transformation of the Fisher–Yates correlation coefficients with the optimal weights.     

A bootstrap method for comparing correlated kappa coefficients

Cohen's kappa coefficient is traditionally used to quantify the degree of agreement between two raters on a nominal scale. Correlated kappas occur in many settings (e.g., repeated agreement by raters on the same individuals, concordance between diagnostic tests and a gold standard) and often need to be compared. While different techniques are now available to model correlated κ coefficients, they are generally not easy to implement in practice. The present paper describes a simple alternative method based on the bootstrap for comparing correlated kappa coefficients. The method is illustrated by examples and its type I error studied using simulations. The method is also compared with the generalized estimating equations of the second order and the weighted least-squares methods.     

Weighted kappa as a function of unweighted kappas

The kappa coefficient is a widely used measure for assessing agreement on a nominal scale. Weighted kappa is an extension of Cohen's kappa that is commonly used for measuring agreement on an ordinal scale. In this article, it is shown that weighted kappa can be computed as a function of unweighted kappas. The latter coefficients are kappa coefficients that correspond to smaller contingency tables that are obtained by merging categories.     

The Effect of Verification Bias in the Naïve Estimators of Accuracy of a Binary Diagnostic Test

The assessment of a binary diagnostic test requires a knowledge of the disease status of all the patients in the sample through the application of a gold standard. In practice, the gold standard is not always applied to all of the patients, which leads to the problem of partial verification of the disease. When the accuracy of the diagnostic test is assessed using only those patients whose disease status has been verified using the gold standard, the estimators obtained in this way, known as Naïve estimators, may be biased. In this study, we obtain the explicit expressions of the bias of the Naïve estimators of sensitivity and specificity of a binary diagnostic test. We also carry out simulation experiments in order to study the effect of the verification probabilities on the Naïve estimators of sensitivity and specificity.     

Approximate confidence intervals for the weighted kappa coefficient of a binary diagnostic test subject to a case–control design

Case–control design to assess the accuracy of a binary diagnostic test (BDT) is very frequent in clinical practice. This design consists of applying the diagnostic test to all of the individuals in a sample of those who have the disease and in another sample of those who do not have the disease. The sensitivity of the diagnostic test is estimated from the case sample and the specificity is estimated from the control sample. Another parameter which is used to assess the performance of a BDT is the weighted kappa coefficient. The weighted kappa coefficient depends on the sensitivity and specificity of the diagnostic test, on the disease prevalence and on the weighting index. In this article, confidence intervals are studied for the weighted kappa coefficient subject to a case–control design and a method is proposed to calculate the sample sizes to estimate this parameter. The results obtained were applied to a real example.     

On population‐based measures of agreement for binary classifications

The authors describe a model‐based kappa statistic for binary classifications which is interpretable in the same manner as Scott's pi and Cohen's kappa, yet does not suffer from the same flaws. They compare this statistic with the data‐driven and population‐based forms of Scott's pi in a population‐based setting where many raters and subjects are involved, and inference regarding the underlying diagnostic procedure is of interest. The authors show that Cohen's kappa and Scott's pi seriously underestimate agreement between experts classifying subjects for a rare disease; in contrast, the new statistic is robust to changes in prevalence. The performance of the three statistics is illustrated with simulations and prostate cancer data.     

Dependence Between Two Diagnostic Tests with Copula Function Approach: A Simulation Study

The study of the dependence between two medical diagnostic tests is an important issue in health research since it can modify the diagnosis and, therefore, the decision regarding a therapeutic treatment for an individual. In many practical situations, the diagnostic procedure includes the use of two tests, with outcomes on a continuous scale. For final classification, usually there is an additional “gold standard” or reference test. Considering binary test responses, we usually assume independence between tests or a joint binary structure for dependence. In this article, we introduce a simulation study assuming two dependent dichotomized tests using two copula function dependence structures in the presence or absence of verification bias. We compare the test parameter estimators obtained under copula structure dependence with those obtained assuming binary dependence or assuming independent tests.     

Receiver operating characteristic surfaces in the presence of verification bias

Summary.  In diagnostic medicine, the receiver operating characteristic (ROC) surface is one of the established tools for assessing the accuracy of a diagnostic test in discriminating three disease states, and the volume under the ROC surface has served as a summary index for diagnostic accuracy. In practice, the selection for definitive disease examination may be based on initial test measurements and induces verification bias in the assessment. We propose a non-parametric likelihood-based approach to construct the empirical ROC surface in the presence of differential verification, and to estimate the volume under the ROC surface. Estimators of the standard deviation are derived by both the Fisher information and the jackknife method, and their relative accuracy is evaluated in an extensive simulation study. The methodology is further extended to incorporate discrete baseline covariates in the selection process, and to compare the accuracy of a pair of diagnostic tests. We apply the proposed method to compare the diagnostic accuracy between mini-mental state examination and clinical evaluation of dementia, in discriminating between three disease states of Alzheimer's disease.     

Assessing accuracy of a continuous screening test in the presence of verification bias

Summary.  In studies to assess the accuracy of a screening test, often definitive disease assessment is too invasive or expensive to be ascertained on all the study subjects. Although it may be more ethical or cost effective to ascertain the true disease status with a higher rate in study subjects where the screening test or additional information is suggestive of disease, estimates of accuracy can be biased in a study with such a design. This bias is known as verification bias. Verification bias correction methods that accommodate screening tests with binary or ordinal responses have been developed; however, no verification bias correction methods exist for tests with continuous results. We propose and compare imputation and reweighting bias-corrected estimators of true and false positive rates, receiver operating characteristic curves and area under the receiver operating characteristic curve for continuous tests. Distribution theory and simulation studies are used to compare the proposed estimators with respect to bias, relative efficiency and robustness to model misspecification. The bias correction estimators proposed are applied to data from a study of screening tests for neonatal hearing loss.     

Testing the difference between two independent regression models

ABSTRACT

In some situations, for example, in biology or psychology studies, we wish to determine whether the linear relationship between response variable and predictor variables differs in two populations. The analysis of the covariance (ANCOVA) or, equivalently, the partial F-test approaches are the commonly used methods. In this study, the asymptotic distribution for the difference between two independent regression coefficients was established. The proposed method was used to derive the asymptotic confidence set for the difference between coefficients and hypothesis testing for the equality of the two regression models. Then a simulation study was conducted to compare the proposed method with the partial F method. The performance of the new method was comparable with that of the partial F method.     

Behavior of agreement measures in the presence of zero cells and biased marginal distributions

Kappa and B assess agreement between two observers independently classifying N units into k categories. We study their behavior under zero cells in the contingency table and unbalanced asymmetric marginal distributions. Zero cells arise when a cross-classification is never endorsed by both observers; biased marginal distributions occur when some categories are preferred differently between the observers. Simulations studied the distributions of the unweighted and weighted statistics for k=4, under fixed proportions of diagonal agreement and different patterns off-diagonal, with various sample sizes, and under various zero cell count scenarios. Marginal distributions were first uniform and homogeneous, and then unbalanced asymmetric distributions. Results for unweighted kappa and B statistics were comparable to work of Muñoz and Bangdiwala, even with zero cells. A slight increased variation was observed as the sample size decreased. Weighted statistics did show greater variation as the number of zero cells increased, with weighted kappa increasing substantially more than weighted B. Under biased marginal distributions, weighted kappa with Cicchetti weights were higher than with squared weights. Both statistics for observer agreement behaved well under zero cells. The weighted B was less variable than the weighted kappa under similar circumstances and different weights. In general, B's performance and graphical interpretation make it preferable to kappa under the studied scenarios.     

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.     

A comparative study of confidence intervals to assess biosimilarity from analytical data

Assessment of analytical similarity of tier 1 quality attributes is based on a set of hypotheses that tests the mean difference of reference and test products against a margin adjusted for standard deviation of the reference product. Thus, proper assessment of the biosimilarity hypothesis requires statistical tests that account for the uncertainty associated with the estimations of the mean differences and the standard deviation of the reference product. Recently, a linear reformulation of the biosimilarity hypothesis has been proposed, which facilitates development and implementation of statistical tests. These statistical tests account for the uncertainty in the estimation process of all the unknown parameters. In this paper, we survey methods for constructing confidence intervals for testing the linearized reformulation of the biosimilarity hypothesis and also compare the performance of the methods. We discuss test procedures using confidence intervals to make possible comparison among recently developed methods as well as other previously developed methods that have not been applied for demonstrating analytical similarity. A computer simulation study was conducted to compare the performance of the methods based on the ability to maintain the test size and power, as well as computational complexity. We demonstrate the methods using two example applications. At the end, we make recommendations concerning the use of the methods.     

Adaptive Neyman's smooth tests of homogeneity of two samples of survival data

The problem of testing whether two samples of possibly right-censored survival data come from the same distribution is considered. The aim is to develop a test which is capable of detection of a wide spectrum of alternatives. A new class of tests based on Neyman's embedding idea is proposed. The null hypothesis is tested against a model where the hazard ratio of the two survival distributions is expressed by several smooth functions. A data-driven approach to the selection of these functions is studied. Asymptotic properties of the proposed procedures are investigated under fixed and local alternatives. Small-sample performance is explored via simulations which show that the power of the proposed tests appears to be more robust than the power of some versatile tests previously proposed in the literature (such as combinations of weighted logrank tests, or Kolmogorov–Smirnov tests).