Background
Screening programs for a specific disease or condition of interest are typically divided into two stages. In the first stage, a population of known size
n is screened by simple and rapid but imperfect tests to classify people as likely or unlikely to have the disease or condition of interest. In the second stage, individuals who appear likely to have the disease or condition are examined further using a gold standard test to confirm the disease or condition. To improve the sensitivity of these screenings, two simple and rapid tests are often used in the first stage [
1]. If either of the tests is positive, then a full evaluation of the correct disease status using a gold standard classification is undertaken in the second stage [
2]. This strategy is widely used in screening for chronic diseases, infectious diseases, and animal diseases [
2‐
4]. One characteristic of this strategy is that negatives in both tests are not verified by a gold standard test because the disease probability in the sub-population is so low that further investigations are costly, unacceptable, and possibly unethical. Another characteristic is that the two tests in the first stage are often conditionally dependent on disease status and/or non-disease status [
5]. If the two screening tests have a similar biological basis, as is often the case, the conditional dependence assumption is suitable. For example, when we use two fecal occult blood tests (occult blood tests and immunohistochemical methods) for colon cancer screening, a positive occult blood test is often accompanied by a positive immunochemical result; thus, the results of the two tests are dependent [
2].
Estimating the disease prevalence and test accuracy (sensitivity and specificity) when the two tests are dependent and when individuals negative on both tests are unverified is a considerable challenge, as the disease rate for individuals who are negative on both tests is not identifiable without additional assumptions [
5‐
9]. Li et al. reviewed several methods for resolving the non-identifiability problem in the framework of classical statistics [
8]. They classified these methods into two types. The methods of the first type are intended to solve the non-identifiability problem by introducing additional assumptions about the association of the two tests. Walter et al. assumed that the two test errors were independent [
9], while van der Merwe also assumed that the two tests were independent under the disease status but dependent under non-disease status [
10]. The methods in the second case resolve the non-identifiability problem directly by adding an assumption that is more likely to hold than the independence assumption [
7]. Bohning and Patilea proposed a capture-recapture approach for screening using two diagnostic tests with information on disease status available for the test positives only [
11]. Chu et al. proposed latent class models for screening studies using two screening tests, with a categorical disease status verified in positives test only [
7]. However, these studies did not provide sufficient insight and details to solve the problem [
8]. Li et al. presented a unified framework for solving the non-identifiability problem by saturating the model using certain homogeneous association assumptions [
8]. In their paper, although the five distinct models provided different estimators, they were all saturated models and provided the same maximum likelihood. Therefore, they are not differentiable in terms of goodness-of-fit [
8].
Recently, Bayesian analysis has been used increasingly to solve the non-identifiability problem using prior information [
12‐
17]. Briefly, the use of this analysis is appropriate for three scenarios. The first involves the use of Bayesian model averaging (BMA) to address this challenging estimation problem, as a different homogeneous-dependent model can provide the same goodness-of-fit for the data but with substantively different estimates [
7]. The second scenario involves the application of BMA to inferences over conditional independence and dependence models for the estimation of disease prevalence in a situation involving two imperfect tests in the absence of a gold standard [
18]. The third scenario involves the administration of a moderately effective but relatively cheap screening test to all subjects in the first stage, after which a gold standard test is performed on a subset of the high screen and low screen groups in the second stage [
19].
This article presents a unified framework to handle the non-identifiability problem by a two-stage Bayesian model using two-step hierarchical informative uniform prior on the sensitivities. We provide detailed Bayesian modeling for stages one and two, the specification of the prior distribution, and the calculation of the posteriors of the parameter distributions. The strategies for model evaluation and checking are proposed. We illustrate our approach using an example and simulation and conclude this article with a discussion and some remarks.
Discussion
This article presents additional methods for the estimation problem for screening studies using two screening tests with the disease statuses being verified for test positives only. The core of the methods is that Bayesian modeling is divided into two stages characterized by a joint testing strategy based on the inherent attributes of this type of screening. The first stage is a joint simultaneous testing consisting of two dependent binary tests. The second stage is a special joint sequential testing that consists of simultaneous testing in the first stage and a gold standard test in the second.
Another method presented in this study involves assigning a diffuse and less risky two-step hierarchical prior structure for sensitivities. Because reasonably informative prior information is necessary for resolving the non-identifiability problem in this type of study, we emphasized the setting of the sensitivity prior distribution based on the following three points. First, the priors of the sensitivity are more important than the priors of the specificity or prevalence when the prevalence is lower than 50%, as is often found in this type of study. Second, we placed the prior information of the sensitivity on the least possible quantity, i.e., diffuse prior or less risky prior information, such as two-step hierarchical prior structure, to reduce the influence of subjective opinion as much as possible on the results. Third, we set the prior structure to be as easy as possible for use by non-statisticians so that uniform prior distributions could be used in this study.
We found that when the priors of the sensitivities were assigned as appropriate priors, such as two-step hierarchical uniform priors, in the absence of priors for the specificities and prevalence (e.g., assigning a uniform distribution on (0,1)), the parameters involved in this study could still be estimated well. However, changing the prior structure of the sensitivities has a substantive effect on the estimators (Tables
4,
6 and
7). The reasons for these results are presented in the
Simulation studies section. Here, we would like to state that this finding has important practical significance because the sensitivities (related to the diseased population) can often, but not always, be obtained more easily than the specificities (related to the healthy population) in medical practice. This characteristic is partially based on the increased feasibility of applying the gold standard test to the diseased population compared with the healthy population, owing to the frequent invasiveness, expense, and sophistication of many gold standard tests, particularly ones with further ethical implications. Therefore, we suggest that the prior structure of the sensitivities should be thoroughly investigated for scenarios in low-prevalence populations.
To our knowledge, this study is the first to estimate test accuracy and disease prevalence for two dependent dichotomous screening tests with unverified negative individuals on both tests using two-stage Bayesian modeling. Recently, Bohning and Patilea, Chu et al., and Li et al. proposed two-stage methods to address the non-identifiability problem [
7,
8,
11]. Their two-stage estimation methods differ from ours. In the first stage, Bohning and Patilea [
11], Chu et al., and Li et al. [
7,
8] estimated the parameters of known cells; in the second stage, they estimated the parameters of unknown cells by making assumptions regarding the cell probabilities of various homogeneous association models. They circumvented the non-identifiability problem using a capture–recapture approach or constrained maximum likelihood estimation. Because all of the models are saturated, homogeneous association assumptions are not testable. Therefore, the dependence structure modeling considered can only be viewed as a sensitivity analysis [
7]. They did suggest that Bayesian methods that incorporate prior information could be a reasonable alternative to obtain improved estimates, although further research is needed [
8]. Their suggestion was an important factor in initiating our study.
Our two-stage approach has certain advantages over the previous methods. First, our method can be understood more easily and used with greater convenience by non-statisticians because the procedure of constructing the model fully agrees with the practical screening procedure. Additionally, the modeling often only involves the prior distribution for the sensitivities, because the prior distributions of the specificities and prevalence are assigned a non-informative uniform distribution of (0, 1), respectively. Moreover, the method of model checking is simple, intuitive, and convenient, as
pD and
DIC can be obtained directly using WinBUGS. Second, this method improves non-identifiability more reasonably, as probabilistic constraints not deterministic constraints (simplifying the model) are used. Third, the method could obtain more information to evaluate the screening strategy than the capture–recapture approach or constrained maximum likelihood estimation [
7,
8,
11]. For example, the method could conveniently calculate the joint sensitivities and joint specificities and make corresponding statistical inferences, such as 95% credible intervals (Table
3). In the framework of frequency statistics, such as capture–recapture approaches or constrained maximum likelihood estimations, it is often difficult to obtain statistical inferences about the joint sensitivities and joint specificities in these types of studies.
Conclusions
We developed a two-stage Bayesian method for two dependent dichotomous screening tests with unverified individuals who are negative on both tests. We also addressed the ad hoc model evaluation and checking procedures based on empirical information for these types of studies. Although the practical example and simulation studies considering various practical situations showed that these models fit well, three points should be clarified. (1) Due to the lack of full data, it is impossible to check the model completely without further information. (2) Although our Bayesian two-stage modeling improves non-identifiability based on probabilistic constraints, the informative prior information on the sensitivities must be assigned reasonably, even though the prior could be diffuse, as in the two-step hierarchical uniform prior. (3) Using our real example and simulated scenarios related to real-life situations, the priors of the sensitivities are more important than the priors of the specificities. However, when the prevalence is large, especially if it is greater than 50% (rare in medical screening tests), the same attention should be paid to the priors of the specificities.
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
LJ and CF conceived and conducted the study, and drafted the manuscript. YH participated in the design, data generation and interpretation of the analyses. ZP and LL participated in the simulation and interpretation of the analyses. All the authors read and approved the final manuscript.