February 2007
Volume 48, Issue 2
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Eye Movements, Strabismus, Amblyopia and Neuro-ophthalmology  |   February 2007
A Bayesian Analysis of the True Sensitivity of a Temporal Artery Biopsy
Author Affiliations
  • Ryan D. Niederkohr
    From the Division of Nuclear Medicine, Stanford University Hospital and Clinics, Stanford, California; the
  • Leonard A. Levin
    Department of Ophthalmology and Visual Sciences, University of Wisconsin Medical School, Madison, Wisconsin; and the
    Department of Ophthalmology, University of Montreal, Montreal, Quebec, Canada.
Investigative Ophthalmology & Visual Science February 2007, Vol.48, 675-680. doi:10.1167/iovs.06-1106
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      Ryan D. Niederkohr, Leonard A. Levin; A Bayesian Analysis of the True Sensitivity of a Temporal Artery Biopsy. Invest. Ophthalmol. Vis. Sci. 2007;48(2):675-680. doi: 10.1167/iovs.06-1106.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

purpose. The temporal artery biopsy (TAB) has long been the standard for diagnosing temporal arteritis (TA), but in practice this test is less than 100% sensitive; false-negative biopsy results are known to occur. The true sensitivity of a single TAB cannot be directly observed, because there is no true gold standard for comparison. The authors propose a mathematical method for calculating the true sensitivity of the TAB, using data from published bilateral TAB results.

methods. Based on Bayesian methodology, this statistical technique can be used to calculate the true sensitivity of a single TAB with data from studies reporting the results of bilateral simultaneous TABs. This technique also allows for calculation of the true prevalence of TA in a study population. Bootstrap techniques are used to provide confidence intervals. This technique is applied to data derived from four studies in the literature.

results. With this methodology, the sensitivity of a single TAB is calculated to be 87.1% (95% confidence interval, 81.8%–91.7%).

conclusions. Knowledge of the true sensitivity of any imperfect test is necessary for an accurate decision analysis, because it can affect the optimal diagnostic–therapeutic pathway. Although few studies report results of bilateral simultaneous TABs, such data are important because they permit the calculation of the true TAB sensitivity. The authors believe that this mathematical method is superior to observational methods (e.g., clinical criteria) for estimating the true sensitivity of a TAB.

Diagnostic tests with 100% sensitivity and specificity are referred to as gold standard tests. 1 These tests are rarely used in clinical practice, either because they do not exist or because their use is not economically feasible. As a result, diagnostic tests with intrinsic error are frequently used instead. 2 Knowing the sensitivity of an imperfect test is important for decision analysis, because it is a critical parameter that can affect the optimal pathway in a diagnostic or therapeutic decision model. 
The temporal artery biopsy (TAB) has long been considered the gold standard for diagnosing temporal arteritis (TA), but the TAB is less than 100% sensitive. False-negative results occur, mainly due to the presence of “skip lesions” (isolated foci of arteritis) or involvement of sites other than the sampled artery, both of which can be missed on a single biopsy. 3 We recently pooled data from 17 studies reporting the sensitivity of a unilateral TAB in clinical series, revealing a sensitivity of approximately 87% for a unilateral TAB. 4 In these studies, the ultimate standard used most often for comparison was clinical criteria (i.e., the patient was judged clinically to have TA despite a negative biopsy). However, clinical diagnosis of TA also has its own false-negative rate, failing to detect accurately all patients who truly have TA. Since no true gold standard confirmatory test exists, the sensitivity of the TAB cannot be directly measured, and thus reported values represent best estimates. 
TABs are sometimes performed bilaterally in an attempt to increase diagnostic accuracy, and bilateral TABs are sometimes thought to identify the presence or absence of TA with certainty. However, since each TAB has its own intrinsic false-negative rate, even bilateral biopsies are less than 100% sensitive in the diagnosis of TA. 5  
The literature describes several methods of calculating true sensitivity and specificity in the absence of a gold standard diagnostic test, using data from studies reporting the results of repeated (e.g., bilateral) diagnostic tests with binary responses. 1 2 6 7 8 9 All such approaches seek to estimate at least three parameters: test sensitivity, test specificity, and prevalence of disease in the study population. Because these parameters are not directly measurable from experimental data, they are called latent; such methods are often referred to collectively as latent class analysis. Some techniques allow for generalization to multiple test repetitions (e.g., the same test performed n times) and multiple cutoff values (e.g., low/moderate/high risk of disease instead of simply absent/present). Although powerful, such methods can be mathematically complex. Many of these methods require data from four or more tests to derive meaningful results, 1 whereas temporal artery biopsies are almost never performed more than twice. 
A more straightforward method would be preferable in the case of the bilateral TAB: a simple binary test (positive or negative results only), performed only twice, which we assume does not produce false-positive results (i.e., 100% specific). We present a method to calculate the true sensitivity of a single TAB by using Bayesian analysis based on data from published studies reporting the results of bilateral simultaneous biopsies. 
Methods
Data from two binary tests are easily represented in 2 × 2 contingency tables. If the true disease status (i.e., whether a person truly has or does not have TA) of each patient were known, each cell in the 2 × 2 table could be divided into N = A ij + B ij , where A is the number of patients in that cell who are truly disease positive and B is the number of patients who are truly disease negative. In this expression, i and j represent the results of the first and second TABs, respectively. Figure 1depicts the contingency tables demonstrating these definitions. 
If the true disease status of each patient were known, then sensitivity, specificity, and disease prevalence in the study population, would be simple to calculate:  
\[\mathrm{Sensitivity}{=}\ \frac{A_{({+}{-})}{+}A_{({-}{+})}{+}A_{({+}{+})}}{A_{\mathrm{total}}}\]
 
\[\mathrm{Specificity}{=}\ \frac{B_{({-}{-})}}{B_{\mathrm{total}}}\]
 
\[\mathrm{Prevalence}{=}\ \frac{A}{A_{\mathrm{total}}{+}B_{\mathrm{total}}}{=}\ \frac{A}{N}.\]
However, since the true disease status of each patient is not known, sensitivity, specificity, and disease prevalence are all latent variables; they are not directly obtainable from experimental values, and thus must be derived mathematically. 
We modified a Bayesian approach to estimate sensitivity and prevalence of disease based on the results of two simultaneous binary diagnostic tests. This method is equivalent to finding the distribution of a mixture of binomial distributions. Our method is based on those described by Black and Craig 1 and Su et al. 6 These methods, in turn, represent modifications of the original Hui and Walter 8 algorithm for estimating such parameters from correlated binary tests. 
There are two possible protocols for performing bilateral TABs. 6 7 “Simultaneous” bilateral biopsies are performed when the surgeon plans from the start to sample both sides regardless of the result of the first biopsy. With “sequential” bilateral biopsies, the decision to perform a second biopsy depends on the result of the first. For example, if the first side is positive for disease (on frozen or permanent section), the surgeon may choose not to perform the contralateral biopsy. If the first side is negative, the surgeon will then proceed with the contralateral biopsy if there is a high suspicion of disease. Note that the definitions of these protocols are not dependent on the actual timing of the two biopsies; whether the procedures take place on the same or different days is irrelevant. Our method is applicable only for studies reporting results of bilateral simultaneous biopsies. 
When a diagnostic test is performed multiple times, the results of each test may be conditionally dependent or conditionally independent. In all cases, the results of the tests are dependent on the true disease state (i.e., the condition). However, given a value of the disease state, the tests themselves may be independent (meaning the result of one test has no impact on the result of the other) or dependent (meaning the result of one test correlates to some degree with the result of the other test). Dependence can occur via either positive or negative correlation. When using Bayesian algorithms, conditional independence is typically assumed. We believe that it is a valid assumption that TAB results are conditionally independent, given the simultaneous nature of the TABs: The decision to perform each biopsy and the interpretation of each biopsy occur independently of one another. 
Consider a study in which a population of N patients undergo bilateral simultaneous TABs. There are four possible outcomes when two simultaneous biopsies are performed: (− −), (− +), (+ −), and (+ +). Of these four possibilities, (− +) and (+ −) represent discordant pairs, while (− −) and (+ +) represent concordant pairs. 
For each disease state, we define the probability of being classified into each of the four cells of each contingency table by the following expression:  
\[P_{ij\ k}{=}P(\mathrm{TAB#1}{=}i,\mathrm{TAB}#2{=}j{\vert}D{=}k),\]
where D represents true disease status and i,j are the results of TAB #1 and TAB #2, respectively. These probabilities are directly related to sensitivity (S n) and specificity (S p) of a single TAB, as well as prevalence of disease (π) in the study population, by the following expressions:  
\[P_{{+}\ {+}{\vert}{+}}{=}S_{\mathrm{n}}{\times}S_{\mathrm{n}}{\times}{\pi}\ P_{{+}{+}{\vert}{-}}{=}(\mathrm{1}{-}S_{\mathrm{p}})(1{-}S_{\mathrm{p}}){\times}(\mathrm{1\ {-}\ {\pi})}\]
 
\[P_{{+}\ {-}{\vert}{+}}{=}S_{\mathrm{n}}(1{-}S_{\mathrm{n}}){\times}{\pi}{\ }{\ }P_{{+}{-}{\vert}{-}}{=}(1{-}S_{\mathrm{p}})S_{\mathrm{p}}{\times}(1\ {-}\ {\pi})\]
 
\[P_{{-}\ {+}{\vert}{+}}{=}(1{-}S_{\mathrm{n}})S_{\mathrm{n}}{\times}{\pi}{\ }{\ }P_{{-}{+}{\vert}{-}}{=}S_{\mathrm{p}}(1{-}S_{\mathrm{p}}){\times}(1\ {-}\ {\pi})\]
 
\[P_{{-}{-}{\vert}{+}}{=}(1{-}S_{\mathrm{n}})(1{-}S_{\mathrm{n}}){\times}{\pi}{\ }{\ }P_{{-}{-}{\vert}{-}}{=}S_{\mathrm{p}}{\times}S_{\mathrm{p}}{\times}(1\ {-}\ {\pi}).\]
 
As stated earlier, we assume that the TAB is 100% specific. The implication of this assumption is that the TAB produces no false-positive results. Although the equations could be solved for both sensitivity and specificity (using mathematical software for iterative techniques), the result would be a specificity less than 100%, which is inconsistent with clinical practice. Few physicians would withhold treatment of TA after a positive biopsy result is obtained. Such a scenario would be contrary to usual practice and would probably have medicolegal implications. In addition, if there were contraindications to steroid therapy (e.g., uncontrolled diabetes), the physician most likely would not have performed the biopsy in the first place. For these reasons, we believe the assumption of 100% specificity is reasonable. 
Given the assumption of 100% specificity, the equations shown in equation 3reduce to:  
\[P_{{+}{+}{\vert}{+}}{=}S_{\mathrm{n}}{\times}S_{\mathrm{n}}{\times}{\pi}\ P_{{+}{+}{\vert}{-}}{=}\mathrm{0}\]
 
\[P_{{+}{-}{\vert}{+}}{=}S_{\mathrm{n}}(1{-}S_{\mathrm{n}}){\times}{\pi}{\ }{\ }P_{{+}{-}{\vert}{-}}{=}0\]
 
\[P_{{-}{+}{\vert}{+}}{=}(1{-}S_{\mathrm{n}})S_{\mathrm{n}}{\times}{\pi}{\ }{\ }P_{{-}{+}{\vert}{-}}{=}0\]
 
\[P_{{-}{-}{\vert}{+}}{=}(1{-}S_{\mathrm{n}})(1{-}S_{\mathrm{n}}){\times}{\pi}{\ }{\ }P_{{-}{-}{\vert}{-}}{=}1\ {-}\ {\pi}.\]
 
With this theoretical framework defined, we can use experimental data as the basis for calculating the true sensitivity of a TAB. Let X equal the proportion of biopsy pairs that are both negative for disease (concordant negative). Thus, X equals the number of (− −) biopsies divided by the number of all biopsies performed. Let Y equal the proportion of biopsy pairs that are discordant; thus Y equals the sum of (− +) and (+ −) pairs divided by the number of all biopsies performed. Let Z equal the proportion of biopsy pairs that are both positive for disease (concordant positive). Thus, Z equals the number of (+ +) biopsy pairs divided by the number of all biopsies performed. X, Y, and Z are all experimentally derived values that are reported by any study describing the results of bilateral simultaneous biopsies. Note that, by definition, X + Y+ Z = 1 (that is, the sum of the proportions of all biopsy results must equal 100%). 
Now, X, Y, and Z can all be defined by the following expressions that match predicted values with experimental values. Referring back to the equations shown in (4):  
\[X{=}P_{{-}{-}{\vert}{+}}{+}P_{{-}{-}{\vert}{-}}{=}(1{-}S_{\mathrm{n}})(1{-}S_{\mathrm{n}}){\times}{\pi}{+}(1\ {-}\ {\pi}),\]
 
\[Y{=}P_{{+}\ {-}\ or\ {+}\ {-}{\vert}{+}}{=}2S_{\mathrm{n}}(1{-}S_{\mathrm{n}})\ {\times}\ {\pi},\]
and  
\[Z{=}P_{{+}{+}{\vert}{+}}{=}S_{\mathrm{n}}{\times}S_{\mathrm{n}}{\times}{\pi}.\]
Note that, as demonstrated in equation 5 , a concordant negative result can occur in persons who either do not have TA or who have TA and simply have a false-negative TAB result. However, as demonstrated in equations 6 and 7 7 , a single positive TAB defines the patient as having TA, since the TAB is assumed to be 100% specific (i.e., no false positives). 
Given two variables (π and S n) and at least two equations, the expressions can be solved algebraically for each variable. These equations can be solved in any combination. The most simple combination algebraically would be first to solve equation 7for π, yielding:  
\[{\pi}{=}\ \frac{Z}{S_{\mathrm{n}}^{2}}.\]
Substituting this value for π into equation 6 , one can solve for S n, the true sensitivity of a single TAB, in terms of Y and Z:  
\[Y{=}2S_{\mathrm{n}}(1{-}S_{\mathrm{n}}){\times}\ \frac{Z}{S_{\mathrm{n}}^{2}}\]
and  
\[S_{\mathrm{n}}{=}\ \frac{2Z}{Y{+}2Z}.\]
 
Now, substituting this value for S n back into equation 8 , one can solve for π, the true prevalence of disease in the study population:  
\[{\pi}{=}\ \frac{Z}{S_{\mathrm{n}}^{2}}\]
 
\[{\pi}{=}\ \frac{Z}{\left(\frac{2Z}{Y{+}2Z}\right)^{2}}.\]
 
A nonparametric bootstrap technique was used to calculate 95% confidence intervals using “R” (for Windows ver. 2.3.1; Microsoft Inc., Redmond, WA). 10 R” is a free software package for statistical computing and graphics that is available for download at http://www.R-project.org. The procedure is performed by randomly sampling with replacement n data points from each study population of n subjects. Six thousand bootstrap samples were used for each calculation, to ensure stability of interval results, given the random nature of the sampling. 
Though our model assumes that the two biopsies are conditionally independent, it is possible that some degree of dependence is present. If the TAB results of a pair correlate positively with one another, an overestimation of calculated TAB sensitivity would result. 11 To investigate the effect this would have on our technique, we used the technique of Vacek 12 which uses the covariance (e) to parameterize the amount of conditional dependence. Using this method, the following equations for sensitivity and disease prevalence were derived, which can be used to examine the effect of various levels of dependence:  
\[S_{\mathrm{n}e}{=}\ \frac{Z{\pm}\sqrt{Z^{2}{-}(Y{+}2Z)(eZ{+}eY)}}{Y{+}2Z}\]
 
\[{\pi}_{\mathit{e}}\mathrm{{=}}\frac{\mathrm{Z}}{\mathrm{S}_{\mathrm{n}_{e}}^{2}{+}e}.\]
The dependence covariance parameter e has a quantifiable maximum value based on the calculated false negative rate of the TAB. 12 We performed a sensitivity analysis with input parameter e ranging from 0% to 100% of maximum e to evaluate the effect of various degrees of dependence on sensitivity and prevalence results derived by our technique. 
We searched the medical literature using PubMed/Medline for studies that report the outcomes of bilateral TABs. Although many such studies exist, few of them explicitly clarify whether their TABs were performed in a simultaneous or sequential fashion. Of the studies that report this information, most report results of sequential biopsies only, presumably because the sequential protocol more closely matches actual clinical practice (i.e., if an initial biopsy is positive for TA, a second biopsy is not likely to be performed). 
Results
Our literature search revealed only four studies that explicitly report the experimental results of bilateral simultaneous TABs. 13 14 15 16 For two studies, 13 16 additional data were provided by the authors via personal communications. Using the experimental data provided, we applied the mathematical technique described herein to each study as well as to the pooled data from all four studies. We specifically excluded equivocal cases (i.e., cases in which the biopsy sample was insufficient in size or nondiagnostic) from our analysis. Data from these studies, as well as the values derived for true sensitivity (S n) of a single TAB and true prevalence of disease (π) using our technique, are shown in Table 1 . Using our method, the true sensitivity of a single TAB in these studies is 87.1% (95% confidence interval, 81.8%–91.7%). The true prevalence of TA in patients undergoing TAB in these studies is 26.4% (95% confidence interval, 22.2%–30.7%). 
To verify the stability of the technique, we also pooled study data using two separate techniques and compared the results to those described herein. First, we pooled the sensitivities calculated by our method from each of the four studies using the meta-analytic techniques used by the Cochrane Collaboration. 17 We estimated the between-study heterogeneity by using a χ2-based Q statistic, with heterogeneity considered significant for P < 0.10. The Q statistic indicated significant heterogeneity between the four included studies; as a result, a random-effects (DerSimonian and Laird) model with the inverse variance-weighting method was indicated and used. The pooled sensitivity obtained with this meta-analytic method was 88.9% (95% confidence interval, 78.0%–99.8%). 
Second, we used a random sampling bootstrap technique to estimate the pooled mean of all four studies. As there were a total of 367 patients in all four studies, random samples of 367 patients (with replacement) were drawn from the entire population and repeated 6000 times to ensure stability of results, given the random nature of the sampling. With this method, the pooled sensitivity obtained was 86.7% (95% confidence interval, 81.3%–91.3%). 
Using the overall data from the four TAB studies in the literature, we performed a sensitivity analysis with input parameter e ranging from 0% to 100% of its maximum possible value, to evaluate the effect of various degrees of dependence on sensitivity and prevalence results derived by our technique (Fig. 2) . At 100% of maximum e (representing peak dependence), calculated sensitivity decreased from 87.1% to 79.2%. From 0% to 70% of maximum e, the calculated sensitivity value falls within the 95% confidence interval calculated by our method. The prevalence value changed very little and remained within the calculated 95% confidence interval over the entire range of dependence levels. These findings suggest that our model is robust over a wide range of dependence levels. 
Discussion
Given any study reporting absolute numbers for the four possible outcomes of two tests performed in a simultaneous fashion, as defined herein, latent class analysis allows for the calculation of true mathematical test sensitivity and true prevalence of disease within the study population. Schulzer et al. 9 have described the application of this general technique for estimating the sensitivity and specificity of visual field testing for optic nerve damage progression in normal-tension glaucoma. A mathematical method is ideal for the calculation of true sensitivity in the absence of a gold standard test. As a reference standard, clinical criteria are subjective and dependent on interobserver variation and thus do not represent true sensitivity. A mathematical derivation from the results of bilateral biopsies is objective and relies on the test itself to calculate the true sensitivity experimentally. 
The calculated sensitivities for each study showed significant variability, ranging from 69% to 98.7% (Table 1) . The differences between studies are probably the result of factors such as variable TAB methodology (e.g., length of specimen) or interpretation (e.g., number of levels examined per specimen, skill of pathologists). However, when the overall mean TAB sensitivity calculated by our method (87.1%) is compared with pooled means calculated by meta-analytic techniques (88.9%) and a random-sampling bootstrap technique (86.7%), the results are quite similar, which suggests that, despite significant heterogeneity between the included studies, our method is robust and capable of providing stable results. In addition, the calculated sensitivity of 87.1% closely matches the value of 86.9%, which was derived from our previously reported literature review of unilateral TAB sensitivity, 4 and further supports this mathematical method as capable of estimating the true sensitivity. 
Our 87.1% calculated sensitivity of a single TAB is slightly lower than previously published estimates which approached 90% or greater, 4 consistent with the fact that using clinical criteria as a reference standard can produce false-negative results. Our calculated true prevalence of TA in these study populations (26.4%) may appear high, especially when compared to published population-based estimates that report a prevalence of ∼1%, even in the oldest age groups. 18 However, the high true prevalence values should not be surprising when one considers that these populations include patients who are undergoing TAB. The patients in these studies probably demonstrated characteristic symptoms or signs that suggested the presence of TA. In this sense, they represent selected populations in which the likelihood of disease is greater than in the general population. 
The true sensitivity of any diagnostic test is useful to know, because it may alter patient management and outcome. The ultimate goal of clinical decision-making is to select a diagnostic and/or therapeutic pathway that maximizes benefits while minimizing risks. The integration of evidence-based data into this process can decrease the level of uncertainty in clinical decision-making. Decision analysis is a technique that employs quantitative methods to delineate elements of the decision-making process and to compare the expected consequences of pursuing different strategies. 19 Regardless of the disease being studied, such models always rely on the use of accurate parameters, such as the sensitivity and specificity of diagnostic tests, to generate meaningful results. For example, we have recently published a decision-making model for the management of patients with suspected TA. 4 The results of this model (e.g., when to perform unilateral TAB, when to perform bilateral TAB, if/when to treat with steroids) are critically dependent on an accurate estimate for the sensitivity of the TAB. 
Discordance rates observed with bilateral biopsies, which represent the proportion of bilateral biopsy pairs with conflicting results, are sometimes used as direct estimates of the potential increase in sensitivity gained by performing two biopsies instead of one. We do not believe that this is true, for several reasons. First, there is a 50% chance that the positive side of a discordant biopsy would have been randomly chosen in the case of a unilateral TAB. If this occurs, a unilateral biopsy would be considered 100% sensitive, compared with a discordant bilateral biopsy pair of 50% sensitivity (i.e., the first biopsy identified the disease correctly, but the second biopsy failed to do so). If the discordance rate alone is used to estimate increased diagnostic yield with two biopsies, one would conclude that the second biopsy actually reduced diagnostic yield by 50%, which is not mathematically consistent. Second, in practice, the selection of which side to sample in a unilateral TAB is not always random. Most surgeons tend to sample the side that is clinically abnormal (e.g., on palpation) or potentially symptomatically involved (e.g., monocular visual loss on that side). As a result, the discordance rate may not accurately estimate the increase in sensitivity with the second biopsy, because the true pretest probability of disease in a purely unilateral biopsy could vary, depending on which side is initially chosen. 
Third, sensitivity cannot be calculated in the same way for both simultaneous and sequential biopsies. To illustrate this concept, consider a study population undergoing sequential biopsies, where only a negative result on the first biopsy triggers the performance of the second biopsy. Because one test result directly influences the decision to perform the second test, the prevalence of disease in the group undergoing the second biopsy is necessarily different from that of the overall group before the first biopsy is performed. The true likelihood of disease after one negative biopsy is likely to be much lower than that in the undifferentiated population. Using the simultaneous approach, each biopsy is performed, regardless (independently) of the result of the other, so there is no differentiation in the study population between the biopsy events. A single method for calculating the sensitivity of both simultaneous and sequential biopsies fails to account for this key difference. Our mathematical method applies only when each testing event is conditionally independent of one another; thus, it is only valid for deriving the sensitivity from studies reporting results of simultaneous biopsies. 
Similarly, the selection of patients undergoing bilateral biopsies versus unilateral biopsy may not be truly random, raising the possibility of selection bias. Patients who undergo bilateral biopsies could have a different pretest probability of disease than patients for whom a unilateral biopsy result is deemed adequate. Cases perceived as diagnostically difficult may be chosen to undergo simultaneous biopsy, whereas clear-cut cases may undergo biopsy of one side only, with a second biopsy only if the first was negative. Also, studies publishing bilateral biopsy results typically report on patients who were treated at tertiary medical centers and may have more aggressive disease than their unselected counterparts. While selection bias is largely unavoidable in our analysis, it may influence generalization of our technique. 
The technique described in our study could be extended to other similar diagnostic situations in which one test is performed multiple times at different anatomic sites, as long as such tests are performed in a simultaneous fashion, are conditionally independent, and there are no false-positive results. Fine-needle aspiration biopsies throughout the body (e.g., thyroid, breast) are common examples of diagnostic studies that may meet these criteria. 
Conclusion
We have presented a mathematical method in which data are used from bilateral simultaneous TABs to calculate the true sensitivity of a single TAB as well as the true prevalence of TA in the study population. This model is based on the Bayesian approach to the binomial distribution, which has been well characterized in previous studies. We applied this technique to four studies in the literature that explicitly report the results of bilateral simultaneous biopsies. We believe that this mathematical method is superior to observational methods (e.g., clinical criteria) for estimating the true sensitivity of a single TAB because it uses the test itself to calculate true sensitivity experimentally. Limitations to our study include the inability to use data from sequential biopsies and the potential for influence by selection bias. Our methodology could be extended to other diagnostic tests that do not have gold standards, as long as the specificity of the test can be assumed to be 100%. 
 
Figure 1.
 
2 × 2 Contingency tables.
Figure 1.
 
2 × 2 Contingency tables.
Table 1.
 
Application of Mathematical Method to Published Data
Table 1.
 
Application of Mathematical Method to Published Data
Figure 2.
 
Effect of conditional dependence on calculated sensitivity and prevalence results. A sensitivity analysis, with the covariance e, a parameter for dependence, as the input parameter, was performed using overall data from the four TAB studies in the literature. The dependence ranged from 0% to 100% of maximum possible dependence, thereby evaluating the effect of various degrees of dependence on the TAB sensitivity and prevalence results derived by our technique.
Figure 2.
 
Effect of conditional dependence on calculated sensitivity and prevalence results. A sensitivity analysis, with the covariance e, a parameter for dependence, as the input parameter, was performed using overall data from the four TAB studies in the literature. The dependence ranged from 0% to 100% of maximum possible dependence, thereby evaluating the effect of various degrees of dependence on the TAB sensitivity and prevalence results derived by our technique.
The authors thank Linnea Boyev, Neil Miller, and Helen Danesh-Meyer for providing additional data from their published studies. 
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Figure 1.
 
2 × 2 Contingency tables.
Figure 1.
 
2 × 2 Contingency tables.
Figure 2.
 
Effect of conditional dependence on calculated sensitivity and prevalence results. A sensitivity analysis, with the covariance e, a parameter for dependence, as the input parameter, was performed using overall data from the four TAB studies in the literature. The dependence ranged from 0% to 100% of maximum possible dependence, thereby evaluating the effect of various degrees of dependence on the TAB sensitivity and prevalence results derived by our technique.
Figure 2.
 
Effect of conditional dependence on calculated sensitivity and prevalence results. A sensitivity analysis, with the covariance e, a parameter for dependence, as the input parameter, was performed using overall data from the four TAB studies in the literature. The dependence ranged from 0% to 100% of maximum possible dependence, thereby evaluating the effect of various degrees of dependence on the TAB sensitivity and prevalence results derived by our technique.
Table 1.
 
Application of Mathematical Method to Published Data
Table 1.
 
Application of Mathematical Method to Published Data
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