Optimal two-phase sampling for comparing correlated areas under the ROC curves of two screening tests in the presence of verification bias.

The accuracy of a screening test is often measured by the area under the receiver characteristic (ROC) curve (AUC) of a screening test. Two-phase designs have been widely used in diagnostic studies for estimating one single AUC and comparing two AUCs where the screening test results are measured for a large sample (Phase one sample) while the disease status is only verified for a subset of Phase one sample (Phase two sample) by a gold standard. In this paper, we consider the optimal two-phase sampling design for comparing the performance of two ordinal screening tests in classifying disease status. Specifically, we derive an analytical variance formula for the AUC difference estimator and use it to find the optimal sampling probabilities that minimize the variance formula for the AUC difference estimator. According to the proposed optimal two-phase design, the strata with the levels of two tests far apart from each other should be over-sampled while the strata with the levels of two tests close to each other should be under-sampled. Simulation results indicate that two-phase sampling under optimal allocation (OA) achieves a substantial amount of variance reduction, compared with two-phase sampling under proportional allocation (PA). Furthermore, in comparison with a one-phase random sampling, two-phase sampling under OA or PA has a clear advantage in reducing the variance of AUC difference estimator when the variances of the two screening test results in the disease population differ greatly from their counterparts in non-disease population.

Sample size calculation for comparing two screening tests when the gold standard is missing at random.

Sample size formulas have been proposed for comparing two sensitivities (specificities) in the presence of verification bias under a paired design. However, the existing sample size formulas involve lengthy calculations of derivatives and are too complicated to implement. In this paper, we propose alternative sample size formulas for each of three existing tests, two Wald tests and one weighted McNemar's test. The proposed sample size formulas are more intuitive and simpler to implement than their existing counterparts. Furthermore, by comparing the sample sizes calculated based on the three tests, we can show that the three tests have similar sample sizes even though the weighted McNemar's test only use the data from discordant pairs whereas the two Wald tests also use the additional data from accordant pairs.

Simple methods for comparing two predictive values with incomplete data.

Statistical methods have been well developed for comparing the predictive values of two binary diagnostic tests under a paired design. However, existing methods do not make allowance for incomplete data. Although maximum likelihood based method can be used to deal with incomplete data, it requires iterative algorithm for implementation. A simple and easily implemented statistical method is therefore needed. Simple methods exist for comparing two sensitivities or specificities with incomplete data but such simple methods are not available for comparing two predictive values with incomplete data. In this paper, we propose two simple methods for comparing two predictive values with incomplete data. The test statistics derived by these two methods are simple to compute, only involving some minor modification of the existing weighted generalized score statistics with complete data. Simulation results demonstrate that the proposed methods are more efficient than the ad-hoc method that only uses the subjects wit complete data. As an illustration, the proposed methods are applied to an observational study comparing two non-invasive methods in detecting endometriosis.

Joint comparison of the predictive values of multiple binary diagnostic tests: an extension of McNemar's test.

Positive and negative predictive values are important measures of the clinical accuracy of a diagnostic test. Various test statistics have been proposed to compare positive predictive values or negative predictive values of two binary diagnostic tests separately. However, such separate comparisons do not present a complete picture of the relative accuracy of the two diagnostic tests. In this paper, we propose an extension of McNemar's test for the joint comparison of predictive values of multiple diagnostic tests. The proposed extended McNemar's test is intuitive and simple to compute, only involving cell counts of discordant pairs from multiple 2×2 tables. Furthermore, we also propose a re-formulation of an existing Wald test statistic so that it can be implemented more easily than its original form. Simulations demonstrate that the proposed extended McNemar's test statistic preserves type one error much better than the existing Wald test statistic. Thus, we believe that the proposed extended McNemar's test statistic is the preferred statistic to simultaneously compare the predictive values of multiple binary diagnostic tests.

Weighted generalized score test for comparing predictive values in the presence of verification bias.

Positive and negative predictive values of a diagnostic test are two important measures of test accuracy, which are more relevant in clinical settings than sensitivity and specificity. Statistical methods have been well-developed to compare the predictive values of two binary diagnostic tests when test results and disease status fully observed for all study patients. In practice, however, it is common that only a subset of study patients have the disease status verified due to ethical or cost considerations. Methods applied directly to the verified subjects may lead to biased results. A bias-corrected method has been developed to compare two predictive values in the presence of verification bias. However, the complexity of the existing method and the computational difficulty in implementing it has restricted its use. A simple and easily implemented statistical method is therefore needed. In this paper, we propose a weighted generalized score (WGS) test statistic for comparing two predictive values in the presence of verification bias. The proposed WGS test statistic is intuitive and simple to compute, only involving some minor modification of the WGS test statistic when disease status is verified for each study patient. Simulations demonstrate that the proposed WGS test statistic preserves type I error much better than the existing Wald statistic. The method is illustrated with data from a study of methods for the diagnosis of coronary artery disease.

Weighted McNemar's test for the comparison of two screening tests in the presence of verification bias.

Statistical methods have been well-developed for comparing two binary screening tests in the presence of verification bias. However, the complexity of existing methods and the computational difficulty in implementing them have restricted their use. A simple and easily implemented statistical method is therefore needed. In this paper, we propose a weighted McNemar's test statistic for comparing two sensitivities(specificities). The proposed test statistics are intuitive and simple to compute, only involving some minor modification of a McNemar's test statistic using the estimated verification probabilities for discordant pairs. Simulations demonstrate that the proposed weighted McNemar's test statistics preserve type I error as well as or better than the existing statistics. Furthermore, unlike the existing methods, the proposed weighted McNemar's test statistics can still be applied even when none of the accordant pairs are verified.

Assess predictive values of a binary diagnostic test under a nested case-control design.

Predictive values of a binary diagnostic test are often evaluated under a random sample design. When the disease is rare, however, such a design might not be as efficient as a nested case-control design where the cases are oversampled from a large existing cohort. Under a nested case-control design, direct proportion estimators of predictive values are biased because cases are oversampled. Consistent estimates of predictive values can be easily obtained by inverse probability weighting (IPW) method. The only difficulty with these IPW estimators has been the absence of expressions for their variances. To fill this gap, in the current paper, we obtain the asymptotic variance formulas for the IPW estimators of predictive values. Unlike their counterparts from weighted logistic regression, our variance formulas take into account the variance of the estimated weights in the IPW estimators of predictive values. We further use the proposed variance formulas to examine the gain in efficiency under a nested case-control design compared with a simple random sampling design. Our results clearly show that when the disease is rare, a nested case-control design can achieve a substantial amount of variance reduction by oversampling cases, compared with a random sample design. Finally, we compare via simulation the accuracy of the proposed variance formulas with the existing methods and illustrate the proposed method by a real data example evaluating the accuracy of D-dimer test.

Nonparametric inference of the area under ROC curve under two-phase cluster sampling.

Nonparametric inference of the area under ROC curve (AUC) has been well developed either in the presence of verification bias or clustering. However, current nonparametric methods are not able to handle cases where both verification bias and clustering are present. Such a case arises when a two-phase study design is applied to a cohort of subjects (verification bias) where each subject might have multiple test results (clustering). In such cases, the inference of AUC must account for both verification bias and intra-cluster correlation. In the present paper, we propose an IPW AUC estimator that corrects for verification bias and derive a variance formula to account for intra-cluster correlations between disease status and test results. Results of a simulation study indicate that the method that assumes independence underestimates the true variance of the IPW AUC estimator in the presence of intra-cluster correlations. The proposed method, on the other hand, provides a consistent variance estimate for the IPW AUC estimator by appropriately accounting for correlations between true disease statuses and between test results.

Simple confidence interval and region formulas for comparing diagnostic likelihood ratios under a paired design.

A population-based paired design is often used for comparing the diagnostic likelihood ratios of two binary diagnostic tests. However, a case-control paired design, which involves the application of both diagnostic tests to two independent samples, is a good alternative study design especially when the disease is rare. Existing methods for comparing two diagnostic likelihood ratios have been mainly focused on the population-based paired design with little attention paid to the case-control paired design. In this paper, we derive a confidence interval formula for the relative diagnostic likelihood ratio (the ratio of two diagnostic likelihood ratios), which can be used for the comparison of two positive or negative diagnostic likelihood ratios separately. We also derive a confidence region formula for the two relative positive and negative diagnostic likelihood ratios, which allows simultaneous comparison of two positive and negative diagnostic likelihood ratios. The proposed confidence interval and region formulas are simple to compute and can be used for both population-based paired design and case-control paired designs. Simulation studies are used to assess the finite sample performance of the confidence interval and region formulas. The proposed methods are applied to a real data set on coronary artery disease and two diagnostic tests.

Optimal two-phase sampling for estimating the area under the receiver operating characteristic curve.

Statistical methods are well developed for estimating the area under the receiver operating characteristic curve (AUC) based on a random sample where the gold standard is available for every subject in the sample, or a two-phase sample where the gold standard is ascertained only at the second phase for a subset of subjects sampled using fixed sampling probabilities. However, the methods based on a two-phase sample do not attempt to optimize the sampling probabilities to minimize the variance of AUC estimator. In this paper, we consider the optimal two-phase sampling design for evaluating the performance of an ordinal test in classifying disease status. We derived the analytic variance formula for the AUC estimator and used it to obtain the optimal sampling probabilities. The efficiency of the two-phase sampling under the optimal sampling probabilities (OA) is evaluated by a simulation study, which indicates that two-phase sampling under OA achieves a substantial amount of variance reduction with an over-sample of subjects with low and high ordinal levels, compared with two-phase sampling under proportional allocation (PA). Furthermore, in comparison with an one-phase random sampling, two-phase sampling under OA or PA have a clear advantage in reducing the variance of AUC estimator when the variance of diagnostic test results in the disease population is small relative to its counterpart in nondisease population. Finally, we applied the optimal two-phase sampling design to a real-world example to evaluate the performance of a questionnaire score in screening for childhood asthma.