Paradoxically, Bayes rule says that the thing we want to know (i.e., how to interpret a positive test result), depends on how many people have the disease in general. Without going into the mathematics of it, what we require for doing this are three things: 1) the false negative rate 2) the false positive rate and 3) the prevalence of the disease in the population. Fortunately, we can use a mathematical trick termed ‘Bayes rule’ to reverse the conditional probabilities. Instead, what we want to know is p(disease | positive test) – i.e., the probability that an individual has the disease if they receive a positive test result. But in truth, this is not what we want to know. In the mathematical language of conditional probabilities, these are expressed as p(negative test | disease) or p(positive test | no disease), read as the probability (p) of a negative test result given that you have the disease or the probability of a positive test result given that you do not have the disease. Therefore, determination of the false negative and false positive rates requires testing of people who have been independently verified as having or not having the disease, respectively. These different kinds of errors are defined in terms of known states of the world (the “ground truth” of whether a person is infected or not). In this case, the cost is financial and emotional, including: 1) an unnecessary quarantine, with consequences for employment 2) a cost to friends and family who will also undergo contact tracing and testing, and likely quarantine and 3) a cost to mental health owing to fear, social isolation, and unwarranted shame if the local community blames the individual for perceived reckless behavior. The other kind of error is a false positive, which is a positive test even though the individual does not have the disease. This is considered the bad kind of error because a failure to detect would allow the cancer to grow, or, in the case of COVID-19, allow the individual to continue about their daily activities, potentially infecting others. The false negative rate is the probability that the test fails to detect the disease when the disease is present. Most doctors and labs are concerned with the false negative rate (which is one minus the ‘specificity’). However, screening for COVID-19 is not uniformly applied in truth, many people have never been tested, while others (e.g., college students) are tested every week, or even twice per week.īecause this level of screening is unprecedented, it is worth considering the two types of errors that can occur with a medical test. Hence, in the state of Massachusetts, everyone has been tested nearly twice on average. Among states, Massachusetts ranks third (after Rhode Island and Alaska), with 1.819 million tests per million. (retrieved from worldometers on January 20 th 2021). In other words, there has been nearly one test for every individual in the U.S. For instance, among countries with a population greater than 10 million, the United States ranks second in the world in its rate of testing, with 878,000 tests per million (the UK ranks first, with 968,000 per million). The numbers screened for COVID-19 are staggering. However, even HIV screening fails to come close to the level of screening that is currently deployed for COVID-19. Asymptomatic screening for infectious diseases is less common, with HIV screening a notable exception. This is common with cancer screening because the prognosis is better if cancers are identified before they cause symptoms. But in some cases, medical tests are used in the absence of symptoms, as a screening tool. In such cases, a doctor uses the test as a diagnostic tool, attempting to identify the root cause of the symptoms. Most medical tests are performed after a patient shows up in a doctor’s office or the emergency room and the test is chosen based on the patient’s symptoms. Considering this new guidance, and in light of the massive degree of testing currently taking place in the U.S., and Massachusetts in particular, I discuss what it means to receive a positive COVID test result. On January 20 th 2021, the WHO issued new guidance for the use of gold-standard PCR testing for detection of SARS-CoV-2 (COVID-19).
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