Testing will remain a key tool for those managing health care and making health policy for the current coronavirus pandemic, and testing will probably be an important tool in future pandemics. Because of test errors (false negative tests in which an infected individual tests as uninfected and false positive tests in which an uninfected individual tests as infected), the observed fraction of positive tests out of a total of T tests, the surface positivity, is gen-erally different from the underlying incidence rate of the disease.

In a companion report,1 we describe a method for translating from the surface positivity to a point estimate for the incidence rate, then to an appropriate range of values for the inci-dence rate (the test range), and finally to the risk (the probability of including one infected individual) associated with groups of different sizes. Three key messages of that report are (1) surface positivity is not an accurate indicator of the incidence of coronavirus; (2) false negative tests lead to overestimation of the incidence rate, and false positive tests lead to underestimation of the incidence rate; and (3) the risk of groups of different sizes is not an either-or situation but can be graded according to the incidence rate, the size of the group, and the specified level of tolerance for risk.

The main purpose of this report is to provide supporting analysis for the recommendations for practice given in our companion report,2 summarized in Equations 1–4 in this report, and to provide an additional recommendation for practice. To do so, we model the process generating test data in which the true state of the world (incidence rate, probability of a false negative test, and probability of a false positive test) is assumed to be known. This allows us to compare analytical predictions with a known situation. We begin by showing how to compute the risk associated with groups of different sizes (defined to be the probability of including at least one infected individual) when one has an estimate for incidence rate.