Coronavirus disease 2019 (COVID-19) rapid antigen tests generate intrinsically fast, inherently spatial, and immediately actionable results. They quickly confirm COVID-19, but weakly rule out infection. Test performance depends on prevalence and testing protocol. Both affect predictive values.
To use original mathematics and visual logistics for interpreting COVID-19 rapid antigen test performance patterns, gauge the influence of prevalence, and evaluate repeated testing.
Mathematica and open access software helped graph relationships, perform recursive computations, and compare performance patterns. PubMed retrieved articles addressing endemic COVID-19 were reviewed.
Tiered sensitivity/specificity comprise the following: T1 (90%/95%), T2 (95%/97.5%), and T3 (100%/≥99%). Performance of self-tests and home antigen tests with US Food and Drug Administration Emergency Use Authorization peaks in low prevalence. Fall-off in performance appears with increasing prevalence because suboptimal sensitivity creates false negatives. The rate of false omissions limits clinical use because of prevalence boundaries based on tolerance for risk. Mathematical analysis supports testing twice to improve predictive values and extend prevalence boundaries nearly to levels of herd immunity.
COVID-19 is quickly becoming endemic. Suboptimal sensitivity of rapid antigen tests limits performance in high prevalence. Risk of contagion in packed spaces (eg, airplanes) might be avoided with dual testing 36 hours apart, allowing time for viral load to increase. Awareness of community prevalence and proof of improved performance with repeated testing will help manage COVID-19 risk, while meeting rapid decision-making needs for highly contagious and new variants (eg, Delta). New COVID-19 variants call for high-quality, low cost, readily accessible, fast, user friendly, and ubiquitous point-of-care testing.
The goals of this research are to apply mathematical relationships and visual logistics to reveal patterns of coronavirus disease 19 (COVID-19) rapid antigen test (RAgT) performance and to facilitate understanding of predictive values, false omission rates, prevalence boundaries, risk tolerance, and repeated (recursive) testing.
COVID-19 variants, such as the India Delta (B.1.617.2), for which Pfizer (BNT162b2) and Johnson & Johnson vaccines are not as effective compared to effectiveness against the wild-type severe acute respiratory syndrome cornavirus-2 (SARS-CoV-2) first detected in Wuhan, China, crop up rapidly and frequently, propelling dynamic surges in contagion that demand early and rapid detection.
In the United States, Delta infections constitute 98.9% of new sequenced cases, according to a Centers for Disease Control and Prevention (CDC) Nowcast projection on September 4, 2021.1 Delta binds strongly to lung cell receptors and must be treated with monoclonal antibodies early on. The virus that caused the first US COVID-19 cases in January 2020 is no longer detected among variants circulating in the country.
About 176.7 million people, or 53.2% of the total US population, have been fully vaccinated as of September 7, 2021.2 Less forgiving and 8 to 10 times more transmissible, Delta is outrunning other variants and now accounts for nearly all hospitalizations and deaths in the unvaccinated. This pandemic of the unvaccinated is accelerating as new waves spread widely. The speed of new contagion warrants rapid testing. On September 4, 2021, a total of 536.9 million tests were reported, 41.7 million were positive, and the 7-day positivity rate was 9.77%.3
The CDC recommends that, “…fully vaccinated people who have come in close contact with someone with suspected or confirmed COVID-19 to be tested three to five days after exposure, and to wear a mask in public indoor settings for fourteen days or until they receive a negative test result.”4
As Delta COVID-19 plays out against the background spectra of testing access, vaccination rates, age groups, and community prevalence worldwide,5 it is placing vulnerable people, such as children, at high risk in the second half of 2021 and early 2022, when a challenging influenza A/B season is expected. In the United States alone, pediatric cases have skyrocketed recently. Nearly 500 children have died of COVID-19 complications.6
Key factors going forward comprise testing frequency, vaccination rates, variant trends, and importantly, prevailing prevalence. This research investigates the benefits of RAgTs using repeated testing, which potentially can make up for low sensitivity and enhance SARS-CoV-2 detection. Access, speed, convenience, and the low cost of RAgTs fulfill important needs in the unfolding era of Delta, which is making COVID-19 a dreaded endemic global disease.
The objectives are (1) to use visual logistics to interpret COVID-19 RAgT performance patterns, (2) to calibrate performance relative to 3 sensitivity and specificity tiers, (3) to assess performance for home use and self-testing with a repeated (recursive) testing protocol, (4) to understand the significance of false omission rates and prevalence boundaries, and as highly infectious variants become entrenched worldwide, (5) to recommend strategies for the use of RAgTs in risk reduction and management.
Table 1 presents the mathematical design criteria for the 3 tiers, which are intended to systematically harmonize Bayesian post hoc performance of COVID-19 diagnostics. The design criteria include simultaneously the performance level, sensitivity, specificity, target prevalence boundary, and false omission rate, RFO, which reflects risk for missed diagnosis at user-defined tolerance levels of 5%, 10%, and 20%. Please refer to open access articles by Kost in the Archives of Pathology & Laboratory Medicine7,8 for descriptions of mathematical methods, visual logistics, computational design and software, and human ethics.
Rapid Antigen Tests
Antigen tests considered for inclusion here received US Food and Drug Administration (FDA) Emergency Use Authorization (EUA) status9 in 2020 continuing through May 2021 when the CDC allowed vaccinated people to remove masks in public, creating controversy and broader demand for rapid detection of SARS-CoV-2. Then, subsequently in July 2021, the CDC reversed its masks-off decision4 and recommended more testing when Delta became a dominant threat.
It is not the intent to analyze all FDA EUA or Conformité Européenne (CE)–Marked RAgTs but instead to select popular ones for self-testing and home testing (Table 2), and then graph performance and illustrate the effects of repeated (recursive) testing as a continuous function of prevalence from 0% to 100%, the first such report.
The supplemental digital content (available at https://meridian.allenpress.com/aplm in the January 2022 table of contents) provides an updated equation set. Equations 7 through 14 are used to calculate continuous positive predictive value (PPV) and negative predictive value (NPV), plus associated parameters through rearrangement of variables.
Recursive formulas for PPV (Eq. 22a) and NPV (Eq. 22b) allow calculation of predictive values for repeated testing, which ideally should be performed with different test designs, so-called orthogonal testing. When testing only twice with the same assay, a single equation can be derived to simplify recursive graphing for prevalence 0% to 100%.
Transformation of Pretest Probability (Prevalence) to Posttest Probability
A proof in Kost,8 with intermediate steps detailed, demonstrates how sensitivity and specificity modulate the pretest probability of COVID-19 to generate posttest probability for a negative test result, the false omission rate, or RFO (Eq. 20). The proof is summarized here, where TP is true positive; FP, false positive; TN, true negative; and FN, false negative.
We use this transformation: Pretest probability → Pretest odds → Likelihood ratio → Posttest odds → Posttest probability. Pretest Odds = [Pretest Probability]/[1 – (Pretest Probability)] = p/(1 – p), where p is the prevalence. Therefore, p/(1 − p) = (TP + FN)/(TN + FP), which is equivalent to the ratio [+ COVID-19]:[− COVID-19].
Multiplying the pretest odds by the likelihood ratio for a negative test result generates the posttest odds of a negative test result for COVID-19. For a negative test result, the likelihood ratio is (1 – x)/y = [FN/TN] • [(TN + FP)/(TP + FN)], where x is sensitivity and y, specificity. Next, we calculate [p/(1 − p)] • [(1 – x)/y] = FN/TN.
Since the posttest probability is [Posttest Odds]/[1 + (Posttest Odds)], the posttest probability is [FN/(TN + FN). However, FN/(TN + FN) is the false omission rate. Therefore, the Posttest Probability = RFO, but RFO = 1 − NPV, so the posttest probability is 1− NPV.
The prevalence boundary is defined as the prevalence at which RFO exceeds a specified risk tolerance, such as 5% (1 in 20 diagnoses missed), 10% (1 in 10 missed), or 20% (1 in 5 missed).
The prevalence boundary is calculated by using Equation 26 and apparent where the RFO curve intersects the horizontal line demarcating risk tolerance. Precedent for RFO = 5% is found in a report of reverse transcription–polymerase chain reaction (RT-PCR) stepwise testing for ruling out COVID-19 devised by Raschke et al10 and published in the Archives.
Figure 1 presents the logic of repeated (recursive) testing. First-round testing uses community prevalence (left) to generate the post hoc Bayesian viewpoint of PPV and NPV. Then, for the second round the partitioned new prevalences are the PPV for the COVID-19 TP set and 1 − NPV for the COVID-19 FN set, as shown on the right and derived at the bottom of the figure, respectively. For the populations as a whole, those who have COVID-19 are the sum of TPs and FNs. Those without the disease are the sum of TNs and FPs.
Figure 2 applies these concepts to the 3 performance tiers designed on the basis of performance criteria in Table 1. This figure plots PPV and NPV (green) as functions of prevalence and the corresponding false omission rates (black) for each tier. With a risk tolerance of 5%, the prevalence boundary for Tier 1 is 33.3% and for Tier 2, 50.6%. Tier 3 has no prevalence boundary because sensitivity is 100% and hence, there are no false negatives. That is, NPV = 1, and thus, RFO = 1 – NPV = 0.
Figure 3 illustrates the effects of repeated testing on 2 RAgTs with FDA EUA status (the top 2 listed in Table 2). The negative percent agreement (NPA) of 99.2% underlying curve A (purple) and NPA of 98.5% for curve B (blue) found in FDA EUA Information for Users (IFU) documents reflect high specificity that pushes PPV performance into upper Tier 2 and 3 levels where it peaks in low prevalence.
Theoretical analysis shows that repeated testing improves performance, such that the PPV curves for A* and B* appear almost perfect. However, performance based on repeated testing has not been proven in clinical trials. Additionally, the manufacturers include disclaimers for repeated testing in FDA EUA IFUs. Positive percent agreement (PPA) and NPA evaluation data supporting repeated testing were not included in FDA EUAs.
Figure 4 illustrates the NPV, false omission rates (RFO), and prevalence boundaries (PB) for the same 2 tests illustrated in Figure 3 and listed in the upper section of Table 2. Penalties for poor sensitivity, documented in FDA EUA IFUs as PPA of 83.5% for A and 84.6% for B, come into play. The PPAs (analogues of sensitivity) for both tests are substantially below Tier 1, that is, “subtier.” That causes test performance to fall off quickly and significantly as prevalence increases.
However, repeated testing pushes the prevalence boundary forward from 24.0% to 65.6% for A*, and from 25.2% to 68.3% for B*, nearly reaching levels of community immunity (herd immunity), which starts at about 70%. However, for the Delta variant, the prevalence boundary may have to reach as high as 85% or more (if attainable). Repeated test performance is indicated by the second box on the right in Figure 4. RAgTs achieving Tier 2 performance through recursive testing will have high impact in the ranges of prevalence where risk avoidance and management are needed the most.
From a clinical standpoint, testing once with low sensitivity RAgTs will generate high rates of false negatives as prevalence increases. With 1 test, the RFO becomes unacceptably risky, because missed diagnoses may lead to stealth spread of SARS-CoV-2. To maintain a RFO of 5% (horizontal red line toward the bottom of Figure 4), that is, to avoid missing more than 1 in 20 diagnoses of SARS-CoV-2 infection, repeated testing “jumps” performance over the Tier 1 and 2 prevalence boundaries of 33.3% and 50.6%, respectively, to make the RAgTs safer to use, but not as safe as Tier 3, for which NPV is 100% and RFO is zero.
In May 2021, the Infectious Disease Society of America (IDSA) issued guidelines11,12 for RAgTs as follows: (1) for symptomatic individuals suspected of having COVID-19, use standard nucleic acid amplification testing (NAAT); (2) for asymptomatic individuals with risk for exposure to SARS-CoV-2 infection, use a single standard NAAT; (3) for asymptomatic individuals with risk for exposure, use a single standard NAAT rather than 2 consecutive RAgTs; (4) in asymptomatic individuals with risk for exposure, the IDSA panel is neither for or against using a single RAgT over no testing; (5) in asymptomatic individuals with risk for exposure, the IDSA panel is neither for or against using a repeated RAgT over no testing.
These IDSA guidelines have merit. However, future editions must be based on detailed analysis of prevalence in relation to performance metrics, both FDA actions and clinical evaluations. The prevalence-based performance mappings in Figures 3 and 4 extend the IDSA guidelines to help create practical strategies for RAgTs. The IDSA panel should recommend minimum sensitivity and specificity metrics for RAgTs, such as Tier 2. It should facilitate clinical evidence for or against repeated RAgTs, provide a logic model for improving the quality of RAgTs, and create a roadmap with instructions for the FDA, manufacturers, and clinical investigators to follow.
False omission rates and prevalence boundaries must be taken into account, especially when the sensitivity of RAgTs falls below 90% (ie, subtier), because missed diagnoses increase exponentially as prevalence increases. Transformation of pretest to posttest probability of COVID-19 allows computation of the false omission rate, RFO, and determination of the prevalence boundary, PB, which creates significant limitation to the clinical use of an antigen test for decision-making.
Figures 3 and 4 illustrate the theoretical merits of repeated testing. Repetition of testing takes advantage of the Bayesian transformation to improve post hoc knowledge of the presence or absence of COVID-19. In other words, it improves the yield of the test at the cost of duplicating time, effort, and reagents. Manufacturers who include disclaimers for repeated testing in FDA EUA IFU documents should provide clinical proof of the effectiveness and efficacy of the dual testing approach.
The theoretically derived hypothesis of improved performance when testing twice needs proof by conducting large multicenter clinical evaluations with diversified populations, including children. Such reports could not be found in the literature. No clinical research funds were allocated or recommended for RAgT clinical evaluations in the president's national strategy for COVID-19.13 Unfortunately, there is no way of singling out infectious patients of any age who have false-negative RAgT results without repeated or additional testing (eg, molecular diagnostics).
We do not know why the PPA and NPA of assays documented in FDA EUAs have not progressively improved during the past year, as inspection of recent FDA EUAs shows.9 However, the FDA has not required improvement. Liberal FDA authorization seems to have diminished competition to produce high performance tests. Uncertainty compounds the poor performance of low-sensitivity RAgTs over the range of prevalence.7,8 Please see Kost8 for the mathematics and visual logistics of uncertainty. Repeated testing can compensate for poor performance but can only partially alleviate uncertainty in test results.
One wonders what will become of subtier and Tier 1 tests in a competitive market following the end of the “EUA era,” if it actually ends. A Clinical Laboratory and Standards Institute (CLSI) white paper14 addressed this EUA life cycle issue but did not adequately analyze poor performers. CLSI emphasized quality control and like others,15 training, which underscores the importance of clear user instructions in RAgT kits. However, ultimately the FDA needs to tighten authorization criteria and unless improved substantially to Tier 2 or higher quality, retire EUAs for subtier and probably also marginal RAgTs in Tier 1.
There is unequivocal need for point-of-care testing of highly infectious diseases.5,16–23 Public health educators should teach point-of-care testing.24,25 Public health investment in RAgTs for COVID-19 is warranted to reduce human loss. In 2020 as a result of the pandemic, life expectancy in the United States dropped 1.87 years, 8.5 times the average decrease in peer countries, and deceases in life expectancy in Hispanic and non-Hispanic Black people were about 2 to 3 times greater than in the non-Hispanic White population.26
With ubiquitous access to testing comes responsibility on the part of academics, public health institutions, professional societies, governments, industry, and global organizations to promote high-quality testing, appropriate use, user training, and periodic public reporting of community prevalence. Prevalence is underestimated because of incomplete data and undiagnosed cases.27 The CDC estimates that by end-May 2021, a total of 120.2 million Americans had been infected.28 That would be 120.2 of 332.7 million, or 36.1% of the US population.
Viral infections that behave like COVID-19 eventually become endemic. Endemic means regularly found among a particular people or geospatially in a certain area, locality, or region. So far, the topic of endemic COVID-19 has received limited clinical and academic attention, probably because of insufficient testing as variants race ahead undetected. Annas et al29 showed the value of vaccination and isolation for forestalling endemic COVID-19 in a limited-resource setting. Lazizi et al30 pointed to safeguards that allow resumption of elective surgery, while Zaman et al31 highlighted the importance of diagnostic testing during endemic COVID-19.
Lum et al32 emphasized vigilance to detect endemic COVID-19 and ramp up for surges. McGuinness et al33 noted the importance of prevalence. Patterson et al34 recommended that “policymakers use lessons (learned) … to develop appropriate risk assessments and control plans for now-endemic COVID-19, and for future pandemics.” Clearly, optimal diagnostic strategies need to be integral to risk assessments and control plans. Strategy-based policy will be valuable in the race for survival, especially in limited-resource settings where Delta is changing the risk landscape.
People in the United States can purchase COVID-19 RAgTs online and in neighborhood stores, although supplies are temporarily depleted. Figure 5 portrays the accessibility, speed, convenience, and simple process steps of a RAgT. Instructions in English and Spanish are user-friendly. From purchasing the test kit, which was available on the shelf at a local pharmacy and then biking home, until observing the first negative result was 30 minutes. A second test is recommended 3 days (at least 36 hours) following an initial negative result. If the first result is positive, then immediate medical evaluation and isolation are in order.
Point-of-care testing is empowering individuals to take the steps necessary to care for themselves in the face of burgeoning populations, diminishing resources, infectious outbreaks, limited hospital access, packed emergency rooms, depleted oxygen supplies, and endemic Delta COVID-19. People purchasing test kits with their own funds for their own purposes should receive disclosure of performance metrics, ideally derived from large and diverse multicenter studies and presented as easily interpretable visual logistics.
In social settings, public gatherings, homes, schools, workplaces, factories, convalescent care, prisons, education, sports events, travel, airports, rural regions, and limited-resource settings abroad, RAgTs are easing the difficulty of transitioning risk avoidance to risk management. This is especially important in limited-resource settings where many people live hand-to-mouth and cannot afford extended lockdowns, expensive molecular diagnostics, long delays in results, and loss of employment, not to mention more than 1 month in quarantine if testing positive.35
CONCLUSIONS AND DIAGNOSTIC STRATEGIES
Weaknesses in COVID-19 RAgT performance, even for products introduced more than 1 year after the FDA first started granting COVID-19 EUAs, call for standardization, or at least a process for attaining consistency and improving sensitivity. PPA and NPA data originate from manufacturers, who typically have conducted limited evaluations. Well-populated multicenter studies with diverse populations are needed to establish performance in clinical practice.
Every step over a prevalence boundary magnifies chances of missing a diagnosis of SARS-CoV-2 infection (see Figure 4). Tier 3, with its 100% sensitivity, could eliminate false omissions and prevalence boundaries. However, Tier 3 appears out of reach for current RAgT technologies. Therefore, repeated testing offers a solution that improves performance substantially and advances prevalence boundaries to levels that are practical, even approaching herd immunity.
Repeated (recursive) antigen testing is supported by theoretical analysis (see Figures 3 and 4) but lacks clinical proof. Repeated testing after 36 hours allows time for viral incubation and an increase in viral load that potentially diminishes false negatives in infected persons. High Delta viral loads can be expected to improve RAgT performance. Clinical evidence of correlation with viral load would support the efficacy of repeated testing and help justify use of RAgTs.
If intrinsic (Tier 2 sensitivity) and extrinsic (repeated testing) enhancements are incorporated, higher performance will garner benefits. Communities could avail themselves of self-testing and home kits to detect COVID-19 quickly. Dual testing reagents, swabs for sample collection, a timed assay development interval, and a 37-hour repeated testing protocol are included in some commercial test kits (see Figure 5).
A major US airline is promoting RAgT kits that can be carried in luggage and then, before returning from international travel, used to credential the Internet-guided, self-swabbing traveler who obtains a single favorable negative test result valid for boarding the flight home.36 Two negative results spaced 36 hours apart would be more convincing of lack of SARS-CoV-2 infection. Sequential self-testing protocols will empower people to be responsible for their own health and for the safety of others.
Strategies for Endemic COVID-19
COVID-19 is propelling expansion of point-of-care strategies worldwide.6,19,35 The White House national strategy for COVID-19 recommended rapid point-of-care antigen testing without qualification in regard to prevalence level or clinical validation that would reveal performance in community settings.13
Table 3 presents multidimensional strategies for rapid antigen testing and importantly, also for improving its performance. This table is based on the synthesis of public health recommendations, peer-reviewed literature, external sources including industry, focus group surveys abroad, findings in this article, and the views of the author and academic colleagues.
Point-of-care testing represents a valuable pandemic response with the striking advantage of accessibility that allows people to confirm SARS-CoV-2 infection at the earliest possible moment in their homes, workplaces, or gatherings. From home to hospital, ready access raises public expectations for controlling transmission, combatting Delta COVID-19, and forestalling future pandemics. We should prepare now.37
The author thanks the creative students, research assistants, colleagues, and public in focus groups who inspired this article. The author is grateful to have received a Fulbright Scholar Award 2020–2022, which supports analysis of COVID-19 diagnostics, strategic point-of-care testing field research in ASEAN Member States, mainly Cambodia, the Philippines, Thailand, and Vietnam, and community and university lectures with the overall goal of improving standards of care in Southeast Asia. Figures and tables are provided courtesy and permission of Knowledge Optimization, Davis, California.
This work was supported in part by the Point-of-Care Testing Center for Teaching and Research (POCT•CTR) and by Dr Kost, its director.
Supplemental digital content is available for this article at https://meridian.allenpress.com/aplm in the January 2022 table of contents.
The author has no relevant financial interest in the products or companies described in this article.