The first case of the coronavirus disease 2019 (COVID-19) was found in December of 2019 in Wuhan city, Hubei providence, P. R. China. On December 31, 2019, China informed the World Health Organization (WHO) of a case of novel viral pneumonia in Wuhan (Wong et al. 2020 ). Since the diagnosis of the first case, this virus has spread with astonishing speed and has caused many infections and a good number of deaths. The origin of the virus, however, remains unclear. An abrupt announcement was made on January 23, 2020 that the city of Wuhan was locked down to control the spread of the virus. Subsequently, almost all areas in China have begun to take serious public health measures to contain the virus (Xiao and Torok 2020).
On January 30, 2020, WHO declared COVID-19 as a public health emergency of international concern. By March 2, 2020, China had confirmed 80,174 infected cases and 2915 deaths. From February 27 to March 11, 2020 , the number of cases of COVID-19 outside China has increased 10-fold and the number of affected countries has increased to be 113. As COVID-19 has been found to be caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2); certain epidemiological and clinical characteristics of patients with COVID-19 have been reported. However, comprehensive knowledge of COVID-19 still remains incomplete. For instance, the risk factors for mortality and the clinical course of the illness have not been well understood (Zhou et al. 2020 ). The early presentation of COVID-19 infection is typically nonspecific. Some infected cases may be asymptomatic, while many infected individuals often show flu-like symptoms such as dry cough, sore throat, low-grade fever, or malaise in the first few days (Wong et al. 2020 ). The seeming-flu symptoms have created difficulties in differentiating COVID-19 from the common cold and seasonal influenza.
The mystery of the virus and the lack of effective treatment for COVID-19 have presented a striking threat to the public. In contrast to the rapid transmission of the COVID-19 pandemic, news on COVID-19 has traveled swiftly and broadly through internet, radio, newspaper, television, social media, and so on. The wave of fear has escalated in the public. Though it has been reported that people with medical complications are at a greater risk of suffering from COVID-19, the general public also has the fear of contracting the virus.
Clinicians have been under tremendous pressure to triage the high volume of patients for COVID-19 testing. To receive medical care as early as possible, an urgent question that has been asked by patients with flu-like symptoms is "Can I be tested for COVID-19?" More broadly, the public may be puzzled by the concern, "To achieve an effective containment of the virus, why does the government not take a proactive action to test everyone for COVID-19 to identify all infected individuals in a timely manner?" Due to the limited availability of trained personnel, testing kits, and PPE (personal protective equipment), it is impossible to test everyone with flu-like symptoms for COVID-19, let alone to test every individual. While these reasons can easily be perceived by the public, the underlying scientific reasonings do not seem to be considered. More importantly, due to the limited availability of resources, the protocol of screening patients for testing can be stringent to prevent a collapse in medical facilities. Based on the evolving global situation, the screening process is constantly being updated and differs from country to country.
Initially, the U.S. Centers for Disease Control and Prevention (CDC) recommended testing only people with respiratory symptoms such as fever, dry cough, shortness of breath, and those who had potentially been exposed to the virus. With the evidence for community transmissions, the CDC updated its recommendations on March 4, 2020 to allow anyone with respiratory symptoms to be tested as long as the request is approved by a doctor, though the CDC is encouraging physicians to minimize unnecessary testing by considering patients' exposure risks (Ferran 2020) .
From the medical perspective, testing for COVID-19 has crucial implications and importance. It is impossible to fight the virus blindly without knowing the target population.
The early diagnosis of infected patients is essential to manage the situation. Infected people must be isolated to control the virus spread; potentially infected individuals should be quarantined to minimize the possibility of infecting healthy people; and vulnerable people such as the elderly and patients with chronic health issues need to be secluded to prevent infection. Necessary medical attention must be focused on patients at high risk who require immediate medical intervention to prevent mortality. In a broader spectrum of learning and dealing with the virus, acquiring accurate data of infection and transmission is critical for researchers to unveil the correct profile of COVID-19 to implement more effective clinical management.
In response to the increasing need for testing for COVID-19, the U.S. Food and Drug Administration (FDA) announced on February 29, 2020, a new policy that made it easier for commercial and academic laboratories to develop their own tests and allowed other certified labs to test patient samples. Companies, hospitals and institutions are now racing to develop more tests to diagnose COVID-19. On March 10, 2020, Alex Azar, secretary of Health and Human Services, announced that 2.1 million testing kits were available and more than 1 million have shipped to certified labs for testing (Ferran 2020) .
With the urgency of identifying infected cases and increase in available test kits, screening criteria for testing COVID-19 have become less restrictive than at the initial stage. It now seems quite tempting to take aggressive action to adminster COVID-19 tests to as many patients as possible. However, an important yet overlooked issue is on the imperfectness of test procedures. It is imperative to enhance our understanding of testing for COVID-19 by factoring in the assessment of the uncertainty, randomness and imperfectness associated with the test procedures; otherwise, misleading and erroneous outcomes can be produced.
In this article, we examine the concerns of testing COVID-19 from the statistical standpoint. Our explorations are purely based on accounting for the uncertainty and randomness associated with medical test procedures. We will look at the uncertainty induced from the test procedures and assess the degree of the resulting false results. Our explorations are intended to shed light on the question "Should everyone be tested for COVID-19?", which would assist general people to assess situations with rational and evidence-based thinking.
Ultimately, as advocated by Sharkawy (2020) , the public should face the challenge of COVID-19 with educated reasoning and compassion for others. Everyone should seek truth and facts, as opposed to conjecture and speculation; we all must work together to battle COVID-19.
This article provides a dynamic framework to present the evolving features of COVID-19.
We examine test procedures in terms of their sensitivity and specificity, and quantify the degrees of false test results under various scenarios. Most importantly, we make sensible recommendations for physicians to balance the usage of limited test kits and the accuracy of the test outcome. We offer the assessment as to how likely we may miss identifying COVID-19 carriers based on consecutive negative results and how many times we should test a suspected COVID-19 patient to reduce the chance of errors. Our discussion provides the guidelines for discharging patients who are treated as COVID-19 carriers.
For generality, we use the term population to describe the group of subjects of our interest.
In the following discussion, population may represent the collection of all people in a country, a city, or a region; it may also refer to a cohort of individuals, a ward of patients, or a community of people. We first introduce abstract symbols to represent the quantities of our interest. To facilitate the dynamic feature of COVID-19, we include the dependence on time in the notations.
On day t with t = 1, 2, . . ., suppose our target population has N (t) people in total in which N h (t) people have no COVID-19 and N s (t) people have COVID-19. Let P (t) = N s (t)/N (t) be the prevalence on day t. Before patient zero (i.e., the first person who has COVID-19) appears, no one in the population has the virus. So we assume that when t = 0, N s (t) = 0, and when t = 1, 2, . . ., N s (t) ≥ 1. That is, P (t) = 0 for t = 0 and 0 < P (t) ≤ 1 for t = 1, 2, . . .. Due to the dynamic feature and the spread of the virus, the relative size of N h (t) and N s (t) varies with time t. Initially, N s (t) is negligible and N s (t) N h (t) for t in a certain interval, say [1, T 1 ], i.e., P (t) is near 0. As outbreaks occur and the pandemic is Consequently, the specificity p spe is also called the true negative rate. A good test is also expected to have p spe close to 1 to keep the number of misdiagnosed cases small.
Corresponding to the true positive rate and the true negative rate, the complement probabilities 1 − p sen = P (Y * = 0|Y = 1) and 1 − p spe = P (Y * = 1|Y = 0) are also useful to describe the test outcomes. These two measures are called the false negative rate and the false positive rate, respectively.
Because medical tests are not always accurate in testing diseased or non-diseased individuals, it is important to take into account the uncertainty and randomness when interpreting a test result. To make a sensible decision, it is necessary to understand how to evaluate the risk of receiving a false result when applying a test. Ultimately, our goal is to make an educational and evidence-based decision for the health care. Specifically, we are interested in evaluating two numbers of concern, # fp (t) = the number of false positive on day t if everyone in the population is tested, and # fn (t) = the number of false negative on day t if everyone in the population is tested.
It is easily seen that those numbers are determined by the size of the diseased subpopulation and the size of the non-diseased subpopulation as well as the sensitivity and the specificity of the test. That is,
To understand how medical tests with different sensitivities and specificities may perform, we consider the scenario where the population has possibly different prevalence for different days. We make recommendations by examining how the number of false negative # fn (t) and
the number of false positive # fp (t) are determined by the sensitivity and specificity of the test procedure as well as the prevalence of the disease.
To visualize the relationship between the false negative number # fn (t) and the sensitivity of the tests, we start with an example with N (t) = 10, 000 for a given day t and apply a sequence of tests with different sensitivities to everyone in the population. We show the results in Figure 1 for a range of prevalence. As expected, the false negative number drops as the sensitivity of the test becomes higher, and the drop rate is higher for the population with a bigger prevalence than that with a smaller prevalence. When the prevalence is very small, say, P (t) = 10 −4 , or equivalently, P (t) = 1/N (t) here (i.e., when patient zero just presents in the population), the false negative number is below 1 no matter what the sensitivity of a test is. In this case, testing everyone for COVID-19 would not virtually yield any false negative result.
In general, in the very beginning of the presence of patient zero (i.e., at time t = 1), the population has N s (1) to be 1 or nearly 1. If everyone is tested for COVID-19, then
regardless of the accuracy of test procedures. To track the origin of COVID-19 for a population, we recommend to test everyone in a group of presumptive patients who may potentially include patient zero; this is the only way to identify patient zero, yet it is unnecessary to worry about obtaining false negative results, no matter how inaccurate the test procedure could be.
In reality, it is often difficult to immediately identify the presence time of patient zero in a cohort based on the confirmation of infected cases. The discussion here offers a possible way to track patient zero retrospectively by examining the samples of suspected patients.
Checking the sample of every suspected patient by the reverse time order is needed to identify patient zero in the cohort.
Recommendation 1: To identify patient zero, it is recommended to test everyone in a presumptive group which may potentially include patient zero. 
We now visualize the relationship between the false positive number # fp (t) and the specificity of the test. Figure 2 shows the results for populations with a common size N (t) = 10, 000 but with different prevalence. The false positive number becomes smaller when the specificity of the test becomes higher. Interestingly, the decreasing rate appears fairly stable regardless of the prevalence value, though those drop rates are not identical. For a given test, the false positive number is bigger for a population with a smaller prevalence, and the difference between two populations with different prevalence tends to be negligible, especially for those tests with a high specificity. In a population with 10,000 people, the false positive number can be as high as 6,000 if a test of the specificity around 40% is applied to everyone in the population; the false positive number can be lowered to about 1,000 if everyone in the population is tested with a procedure having a high specificity (such as 95%). Given that even for the best scenario the false positive number is still around 1,000 for a population of size 10,000, we conclude that it is not sensible to test everyone in the population for a disease without discretion (even if it is affordable in terms of availability of resources and the cost). 
That is, three infected person would be missed, and 60,121 healthy people (i.e., near 10% of the population in Waterloo Region) would be misdiagnosed as infected with COVID-19. This clearly demonstrates the blunder of testing everyone in a sizable population for COVID-19 without discretion. Assume that this small cohort has the same prevalence as that of the Waterloo Region.
If all those patients were to be tested for COVID-19, we would then expect To this end, let Y * k represent the kth test result of applying the test to an individual, where k is a positive integer. This binary random variable has the same distribution as that of Y . We want to find the value of k so that the conditional probability
To find how the performance of the test comes into play, we express the conditional probability P (Y = 1|Y * 1 = 0, . . . , Y * k−1 = 0, Y * k = 1) using the sensitivity and specificity of the test as well as the prevalence. Assuming that the test is independently applied to an individual k times, Using the Bayesian theorem gives that
for k = 1, 2, . . ..
For any test with the specificity higher than the false negative rate, (2) shows that a larger value k suggests a smaller chance for an individual to be infected if the first positive test result appears at the kth test, no matter what the sensitivity of the test and the population prevalence are. However, for a smaller number of k, the sensitivity of the test and the population prevalence do matter for quantifying the probability.
In Figure 3 , we report the conditional probability P (Y = 1|Y * 1 = 0, . . . , Y * k−1 = 0, Y * k = 1) versus the sensitivity and specificity of the test for populations with different prevalence. In the top panel of Figure 3 , we display the results for k = 1. As long as the test has a high specificity, applying the test to an individual from the population with a high prevalence can be very informative. If the test result is positive, then it is highly likely that this individual is infected with the disease; this is true, even if the test does not have a high sensitivity.
However, for a population with a low prevalence (e.g., 1%), even if the test has both a high specificity and sensitivity, the probability of correctly confirming an infected case is very low if the test is applied only once. This finding further suggests that it is unwise to apply a test to everyone from the population with a low prevalence. It is advised that prior measures should be taken to identify a suspected subpopulation (or a group of presumptive cases) so that the resulting prevalence becomes high. Then applying the test to individuals in this subpopulation can increase accuracy to identify infected cases.
The bottom panel of Figure 3 shows the results for k = 3. For a test with a low sensitivity, there is a high chance that a patient is infected even when the first positive result appears at the third test, if we test individuals from the population with a high prevalence (such as 85%). However, if the population has a small prevalence (such as 1%), it is unlikely that the patient with the first positive result occurring at the third try is infected. The color shows the magnitude of the probability.
To further visualize how the cohort prevalence affects the probability of identifying an infected case, given consecutive negative results followed by a positive result, we examine the COVID-19 test described by Hutchison (2020) We graph the results in Figure 4 . There is a high chance that the tested person contracts COVID-19 if the result of applying the COVID-19 IgM/IgG Rapid Test is positive at the first test, unless the population prevalence is very small; the chance is higher for testing patients coming from a cohort with a higher prevalence. For the COVID-19 IgM/IgG Rapid Test, if the positive result occurs only at the 4th test, then the chance of the tested subject is infected is very slim unless the patient comes from a cohort with a high prevalence (such as 60% or higher); if the first positive result occurs at the 6th test, the chance of the tested subject is infected is almost 0 no matter which cohort this patient comes from. However, for the COVID-19 test described by Hutchison (2020), even the first five test results are negative, there is still a good chance that the tested subject is infected with COVID-19. To assess this perception, we evaluate the chance of incorrectly claiming a COVID-19
carrier to be cured after obtaining several consecutive negative results. We are interested in assessing the conditional probability P (Y = 1|Y * 1 = 0, . . . , Y * k−1 = 0, Y * k = 0) for k = 1, 2, . . .. Assuming that the test is independently applied to a patient k times, then using the Bayesian theorem, we obtain that
for k = 1, 2, . . ..
In Figure 5 , for populations with different prevalence, 3-dimensional graphs were made to show how the conditional probability for missing an infected case with k consecutive negative results may depend on the sensitivity and specificity of the test, where k = 1, 3. It is clearly seen that if the test has a low sensitivity, the chance of missing an infected case is high if the test is done only once for populations with a large prevalence, even if the specificity of the test is high; the larger the prevalence, the higher the chance of missing. For a test with a reasonably large sensitivity, the more we test, the smaller the chance of missing an infected case.
More rigorously, we rewrite the conditional probability (3) as
for k = 1, 2, . . .. The probability (4) decreases as k increases if and only if
which is satisfied by a test with the sensitivity larger than the false positive rate (i.e., 1−p spe ) or the specificity higher than the false negative rate (i.e., 1 − p sen ). This condition must be met by any test in use, otherwise, there is no point of using a test that is even worse than a random guess. Hence, for any test in use, increasing the test number reduces the chance of mistakenly discharging infected patients. To see how doctors may implement this recommendation when they need to discharge inpatients with COVID-19-like symptoms, we compare the COVID-19 test described by Hutchison (2020) to the COVID-19 IgM/IgG Rapid Test. Figure 6 displays Without discretion, we would use the COVID-19 test described by Hutchison (2020) to test all passengers every day starting t = 17, then the daily numbers of false positive and negative results can be worked out by (1) . We visually display those numbers in the middle panel (blue dashed curve) and the right panel in Figure 7 . Clearly, when P (t) is very small, # f p (t) is close to N (t) and # f n (t) is near 0; when P (t) becomes larger, # f p (t) becomes
smaller but # f n (t) gets larger. It is sensible to focus on testing suspected patients to reach a balance between the use of limited medical resources and the result accuracy. By (2), the COVID-19 test should be repeated at least three times for a suspected patient, and all three negative results would ensure the chance of missing an infected case to be under 5%. 
In this article we take the statistical standpoint and examine several aspects of COVID-19
testing. We evaluate the uncertainty induced from the imperfectness of COVID-19 tests and make recommendations. In summary, we highlight the following points: Our discussion is carried out exchangeably at the population level and on the basis of an individual. When interpreting the results, caution should be taken for this difference.
In the initial stage, COVID-19 was transmitted in an unrecognized way. Limited testing capacity and strict testing criteria delayed the identification of COVID-19 carriers in many countries (Sullivan 2020) . Our discussion here is useful to help make prudent decisions from the administrative standpoint to optimize the usage of limited resources of test kits, healthcare workers, and medical facilities. However, from the perspective of an individual thinking "better safe than sorry", one might argue that having a false positive result is tolerable but a false negative is troublesome. From a physician's viewpoint, missing an infected case can be more detrimental than misdiagnosing a COVID-19-free patient as a carrier. Such an error would cause a spread of virus and delay medical care for the COVID-19 infected patients.
This article provides the assessment as to how likely we may miss identifying COVID-19 carriers based on consecutive negative results and how many times we should test a suspected COVID-19 patient to reduce the chance of errors. On equal footing, our discussion provides the guidelines for discharging inpatients who are treated for COVID-19.
There are several limitations of our discussion. In the discussion of the Diamond Princess
Cruise Data in Section 6, we ignore the fact that the population size N (t) and the prevalence P (t) are error-contaminated. In our discussion, the dynamic number of COVID-19
carriers N s (t) is taken as the confirmed cases for each day t. The true value of N s (t) is, however, highly likely to be larger than the reported number for day t since some infected passengers were asymptomatic and thus were not being tested on day t for t = 17, 18, . . . , 34.
While we consider a dynamic framework in Section 2 to characterize the change in the Another notable issue is that our discussion merely investigates testing errors from the statistical angle to quantify the randomness and uncertainty associated with the test inaccuracy. We have not explicitly accommodated in the development the characteristics of patients, such as age, severity of COVID-19-like symptoms, health conditions, and the medical history. This is caused by the lack of information on the sensitivity and specificity of available test kits estimated from different cohorts of patients with varying medical conditions. As a last recommendation, we suggest that rather than reporting an overall sensitivity and specificity for a developed test, developers of COVID-19 tests should evaluate a sequence of the sensitivities and specificities of the test applicable to different cohorts of patients. This will allow physicians and medical personnel to make more precise decisions to accommodate the personalized-features of the patients.
Recommendation 7: Instead of being assessed by an overall sensitivity and specificity, the performance of COVID-19 tests should be evaluated in a more refined measure by reporting their sensitivities and specificities obtained from the stratified population by the patient's medical conditions.
In the article, we compare two COVID-19 tests, the COVID-19 test described by Hutchi- conditions. It is useful to ponder the question: How do we use them effectively? We may apply a fast but less accurate test kit to do screening, and then apply a more accurate but time-consuming and costly test to do further checks. If repeated tests need to be done, we need to decide the number of the tests in order to not miss the infected patients. In addition, in our discussion of repeating the test for COVID-19, we have not looked into the issue of the gap time between two consecutive tests.