Fig 1.
Parallel and series testing protocols using two tests.
Positive (+) and negative (−) test outcomes are combined using the two Boolean functions AND () and OR (
). In parallel testing, both inputs are assessed simultaneously, while in series testing, the left input is examined before the right. Hence, if the initial test in a series protocol yields a negative result with aggregation through an AND gate, the assigned disease status will be negative, irrespective of the second input. In series testing with an OR gate, the assigned disease status will be positive if the first test is positive, regardless of the outcome of the second test.
Table 1.
Examples of parallel and series test protocols that have been used in COVID-19 seroprevalence studies.
Fig 2.
The ratio of the number of parallel tests to the number of series tests necessary to determine the aggregated output from n = 2 tests as a function of prevalence f.
Results in panels (A) and (B) are based on AND and OR aggregations of two tests, using Eqs (11) and (12), respectively. We consider three different combinations of true positive and true negative rates (solid black lines: TNR1 = 0.95 and TNR1 = 0.95; dashed red lines: TNR1 = 0.90 and TNR1 = 0.95; dash-dotted blue lines: TNR1 = 0.95 and TNR1 = 0.90). The critical values fc for which the ratios in panel (A) are larger than the ratios in panel (B) are given, respectively, by fc = 0.50, 0.47, 0.53. For f < fc greater savings are achieved by utilizing the AND-aggregated series tests, compared to the OR-aggregated series test.
Fig 3.
Positive predictive value (PPV) and negative predictive value (NPV) as a function of prevalence f.
The results that we show in panels (A,C) and (B,D) are based on AND and OR aggregations of n = 2 tests, using Eqs (14) and (15), respectively. We denote the sensitivities and specificities of the two tests i ∈ {1, 2} by TNRi and TNRi, respectively. We consider two different combinations of true positive and true negative rates (solid black lines: TNRi = 0.95 and TNRi = 0.95; dashed red lines: TNRi = 0.90 and TNRi = 0.90). As a reference, we also show results for single tests without further aggregation (dash-dotted blue line: TNR = 0.95 and TNR = 0.95; dash-dot-dotted orange line: TNR = 0.90 and TNR = 0.90). These curves are independent of the ordering (parallel or series) method used.
Fig 4.
Receiver operating characteristic (ROC) curves for various combinations of tests and aggregation functions.
(A) We consider n = 2 tests and two distinct aggregation functions (disks: AND aggregation; triangles: OR aggregation). (B) We consider n = 3 tests and the same aggregation functions as in panel (A) along with the majority function represented by inverted triangles. Markers in black, blue, and red represent combined tests where the underlying tests i ∈ {1, …, n} have sensitivities (TPRi) and specificities (TNRi) set to 0.8, 0.9, and 0.95, respectively. Dashed lines indicate the sensitivities and false positive rates (i.e., 1 − TNR) of the individual isolated tests. Under AND aggregation, both the sensitivities and false positive rates of the combined tests are smaller than those of the individual tests. The opposite holds for OR aggregation. When considering n = 3 tests, the majority function results in higher sensitivities and smaller false positive rates compared to the individual isolated tests. This function provides a tradeoff between the “all” and “any” characteristics of AND and OR aggregations. The results shown are independent of the ordering (parallel or series) method used. The error bars in both panels represent the bounds defined by the Boole–Fréchet inequalities (see Materials and methods), which apply irrespective of the dependence structure relating the individual tests.
Table 2.
Median sensitivities and specificities of three commonly used SARS-CoV-2 antigen tests that are based on studies involving symptomatic patients [1].
Numbers in parentheses denote 95% CIs.
Fig 5.
ROC curves associated with the aggregation of three antigen tests (Abbot, Innova, and Siemens).
The sensitivities and specificities of the n = 3 tests are listed in Table 2. (A) The ROC curve associated with the aggregation of the three antigen tests as derived from Eqs (33) and (35). We use Yi ∈ {0, 1} to denote the outcome of test i ∈ {1, 2, 3}. The dashed curve is a visual guide connecting the tests on the ROC curve. (B) A magnified view of the ROC curve without the trivial combined tests that classify all samples as either negative or positive. The error bars indicate the 95% CIs that we generated from 106 samples of beta distributions capturing the 95% CIs of the underlying individual sensitivities and specificities.
Fig 6.
Measured prevalence as a function of true prevalence f under the assumption that the measured, error-corrected prevalence
in Eq (37) can be identified with the true prevalence f.
The results shown in panels (A) and (B) are based on AND andOR aggregations of two tests i ∈ {1, 2}, respectively. We consider three different combinations of true positive and true negative rates (solid black lines:TNRi = 0.95 and TNRi = 0.95; dashed red lines: TNRi = 0.90 and TNRi = 0.95; dash-dotted blue lines: TNRi = 0.95 and TNRi = 0.90). Grey lines indicate measured prevalences associated with individual tests.
Table 3.
Measured and error-corrected prevalence in Norrbotten, Sweden (May 25—June 5, 2020) [56].
The error correction method we employed takes into account the two tests used in the seroprevalence study from Norrbotten: (i) the Abbott SARS-CoV-2 IgG kit and (ii) the Euroimmun Anti-SARS-CoV-2 ELISA (IgG). These tests have been combined using an AND function. We calculated the measured, error-corrected prevalence through Eq (39) and their corresponding 95% CIs by generating 106 samples from beta distributions capturing the measured prevalence as well as the underlying individual test sensitivities and specificities. Details of the study are listed in the Materials and methods section.
Table 4.
Main variables used in outbreak severity measures.
Population, fatality, hospitalization, and prevalence statistics are often reported for Na age intervals [ak−1, ak) (k ∈ {1, …, Na}) with . Here, a0 is the smallest age value in the data set and Δaℓ is the width of the ℓ-th age window. We assume that the population size N(ak) is constant in the considered time window. The closed interval [0, 1] contains 0, 1, and all numbers in between, and
denotes the set of non-negative integers.
Fig 7.
Probability density functions (PDFs) of dependence factors (A) (see Eq (46)) and (B)
(see Eq (47)).