冠狀病毒檢測的操作分析

1、Contents HYPERLINK l _bookmark1 figures v HYPERLINK l _bookmark2 summary vii HYPERLINK l _bookmark3 Introduction 1 HYPERLINK l _bookmark3 The Operational Situation 1 HYPERLINK l _bookmark4 From Surface Positivity to Incidence Rate 2 HYPERLINK l _bookmark4 From a Point Estimate of the Incidence Rate

2、to a Range for the Incidence Rate 2 HYPERLINK l _bookmark4 From Incidence Rate to Risk Associated with Groups of Different Sizes 2 HYPERLINK l _bookmark5 A Worked Example 3 HYPERLINK l _bookmark6 Discussion 4 HYPERLINK l _bookmark7 Appendix Athe underlying Mathematical Models 5 HYPERLINK l _bookmark

3、8 Appendix BAccounting for Heterogeneity in test errors 7 HYPERLINK l _bookmark9 Bibliography 9 HYPERLINK l _bookmark0 Acknowledgments 11 HYPERLINK l _bookmark0 About the Authors 11Figures HYPERLINK l _bookmark5 figure 1. visual Representation of the Operational situation 3 HYPERLINK l _bookmark6 fi

4、gure 2. Risk versus group size 4SummaryEven though vaccines for coronavirus are increasingly available, it will be many months before sufficient herd immunity is achieved. Thus, testing remains a key tool for those managing health care and making policy decisions. Test errors, both false positive te

5、sts and false negative tests, mean that the surface positivity (the observed fraction of tests that are positive) does not accurately represent the incidence rate (the unobserved fraction of individuals infected with coronavirus). In this report, directed to individuals tasked with providing analyti

6、cal advice to policymakers, 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 incidence rate, and finally to the risk (defined as the probability of including one infected individual) associated

7、 with groups of different sizes. The method is summarized in four equations that can be implemented in a spreadsheet or using a handheld calculator. We discuss limitations of the method and provide an appendix describing the underlying mathematical models.IntroductionEven though coronavirus vaccines

8、 are increasingly available, it will be many months before sufficient herd immunity is achieved, so testing remains a key tool for those managing health care and making policy decisions.1 In this report, we provide a method for individuals tasked with providing scientific advice concerning the inter

9、pretation of coronavirus tests.2We focus on three questions. First, how does one go from surface positivity (the observed fraction of tests that are positive) to the incidence rate (the unobserved fraction of individuals infected with coronavirus), knowing that there are test errors?3Second, how doe

10、s one go from a point estimate for the incidence rate to a range of reasonably likely incidence rates? Third, how does one compute the risk (defined as the probability of including one infected individual) of coronavirus transmission in groups of different sizes, given the point estimate and range o

11、f values for the incidence rate?41 Allen et al., Roadmap to Pandemic Resilience; and Thompson, “Testing Remains One of Our Best Tools to Fight COVID-19.”2 Supporting analyses are described in Mangel and Brown,The Operational SituationWe assume that T coronavirus tests are admin- istered to a populat

12、ion and, for each individual tested, the result is either a positive or negative test (Figure 1). A fraction f of these individuals are infected with coronavirus and are antigen positive. However, these individuals have a probability pFN of a false negative result, where the test reports no infectio

13、n when the individual is in fact infected. We assume that the probability of a false negative test is known.5 The remaining individuals, a frac- tion 1 f of the sample, are not infected (i.e., are antigen negative) but have a probability pFP of a positive test result. We also assume that the prob- a

14、bility of a false positive test is known.6 It is likely that there is both medical and operational hetero- geneity in the values of pFP and pFN. When hetero- geneity can be characterized by distributions for the test, our method generalizes to account for it, as explained briefly in Appendix B here

15、and more fully in a forthcoming report.7Our goal is to begin with the test results, in this case P positive results out of T tests administered, giving surface positivity P/T, and obtain an estimate for the unobserved incidence rate of coronavirus. We denote this estimate by f and recognize that it

16、can never be known precisely.Given the unobserved incidence rate f, the probability of a false positive test p , and theOperational Analysis for COVID-19 Diagnostic Testing. InFPprobability of a false negative test pAppendix A here, we describe the assumptions underlying our analysis.3 Test errors i

17、nclude false negatives (infected individuals give a negative test) (Oran and Topol, “Prevalence of Asymptomatic SARS-CoV-2 Infection”; and Watson, Whiting, and Brush, “Interpreting a Covid-19 Test Result”) and false positives (noninfected individuals give a positive test) (He et al., “Diagnostic Per

18、formance between CT and Initial Real-Time RT-PCR”). False negative tests are unavoidable (Sethuraman, Jeremiah, and Ryo, “Interpreting Diagnostic Tests for SARS- CoV-2”); it may be possible to reduce the likelihood of false positive tests. What is important for our method is a recognition that these

19、 occur and can be at least approximately quantified.4 See Klein et al., Stepping Back to School, for the importance of this question in relation to reopening schools.FN, the expected positivity rate p+( f ) (the probability of a positive test) is composed of two terms: (1) the fraction of antigen-po

20、sitive individuals tested whose test results are accurate plus (2) the fraction of5 Watson, Whiting, and Brush (“Interpreting a Covid-19 Test Result”) report that in a clinical setting pFN ranges from about 2% to 30%.6 He et al. (“Diagnostic Performance between CT and Initial Real-Time RT-PCR,” Tabl

21、e 2) report pFP a bit less than 0.1.7 Mangel and Brown, Operational Analysis for COVID-19 Diagnostic Testing.antigen-negative individuals tested whose test results are inaccurate. That isVariation in P/T generates variation in f. Hence, the next step is to compute a range of possible valuesp+(f )= f

22、 (1 pFN )+ (1 f )pFP .(1)for f. A range having 95% probability of includingthe true but unknown value of the incidence rate is8It is clear from this equation that the surface positivity rate is not equal to the incidence rate unless there are no test errors. The analyticalsRange(f)= 3.92p+(f)(1 p+(f

23、) .T (1 pFN pFP )2(3)challenge is that we observe the surface positivity but want to know the incidence rate.From Surface Positivity to Incidence RateAn estimate of the incidence rate f from the test results (P positive tests out of a total of T tests) with test errors pFN and pFP isf = P/T pFP .(2)

24、1 pFN pFP As explained in Appendix A, on average, the valueMangel and Brown9 show via simulation testing that:Range(f) is symmetrically distributed around the true range, Range( f ), which is obtained by replacing f by f in Eq. 3.The mean error between the two is a fraction of a percent, so that Eq.

25、 3 is, on average, a very accurate characterization of the range.Thus, we are able to construct meaningful and reasonable lower and upper limits for the estimated infection rate as flower = f 0.5(Range(f) and fupper = f+ 0.5(Range(f). In these equations,of P will be p ( f )T. Replacing P/T in Eq. 2

26、by p ( f )the factor of 0.5 accounts for Range(f) being+symmetrically distributed around Range( f ).from Eq. 1 gives the result that f = f. That is, onaverage the estimate in Eq. 2 will capture the true value of the incidence rate. However, we do not live in a world of on average. Rather, each testi

27、ng event will give a single value of P/T that is randomly drawn from a distribution of possible values fromwhich we construct the estimate f. We would liketo know a reasonable range for the estimate of the incidence rate.From a Point Estimate of the Incidence Rate to a Range for the Incidence RateEq

28、. 2 provides a point estimate for the incidence rate given P/T and the test errors. Since P/T is a random variable, if one were to repeatedly sample the same population multiple times, the values of P/T would generally be different but around the expected positivity rate given in Eq. 1.From Incidenc

29、e Rate to Risk Associated with Groups of Different SizesHaving gone from the surface positivity P/T to a point estimate f for the incidence rate, and then to a range flower, fupper for the incidence rate, we illustrate how these values can be used to compute8 In Appendix A, we give an explanation of

30、 the determina- tion of range, which we interpret as a compatibility interval (McElreath, Statistical Rethinking, 54). It is a range of values for f that is compatible with the data and model and thus avoids the undesired implications of words such as confidence or cred- ible (Morey et al., “Fallacy

31、 of Placing Confidence in Confidence Intervals”). See Mangel and Brown, Operational Analysis for COVID-19 Diagnostic Testing, for a fuller discussion.9 Mangel and Brown, Operational Analysis for COVID-19 Diagnostic Testing.The surface positivity (fraction of positive tests) is observed, and from tha

32、t we want to infer the unobserved incidence rate f. See text for further details.Figure 1. Visual Representation of the Operational Situationthe risk associated with groups of different sizes, where we define risk as the probability that a groupA Worked ExampleTo illustrate the application of these

33、ideas, assumeof g individuals contains at least one infected individual.RWe denote this risk by (g, f) to emphasize that it depends on both the size of the group and the point estimate for the incidence rate. If we replace f bythat 1,000 tests, with errors pFN = 0.25 and pFP = 0.05, were administere

34、d and that the number of positive tests was 112,11 so that the surface positivity in the test pool was 0.112.Using Eq. 2, we obtain the point estimate for theflower or fupper, we obtain lower and upper boundsincidence rate = 0.0886 from Eq. 2 and the loweron the risk, respectively. We also emphasize

35、, andfwill clarify in the example below, that the level of acceptable risk is a policy question, not an analytical question. That is, analysis can provide the relationships among group size, infection rate, and risk but cannot delineate the thresholds at which risk should be considered acceptable.Th

36、e risk associated with a group of size g when the point estimate of incidence rate is f is10and upper limits for the range flower = 0.0606 and fupper = 0.1165 from Eq. 3 for the range and the equations given in the paragraph below Eq. 3.Using Eq. 4 three times with f, flower, and fupper, respectivel

37、y, we are able to make a plot of risk versus group size using the point, lower, and upper estimates for the incidence rate (Figure 2).R(g, f) =1 (1 f)g .10 In Appendix A we describe the underlying assumptions.(4) 11 We used a simulation with incidence rate f = 0.09 to generate the number of positive

38、 tests. R script for this simulation is available from the second author at HYPERLINK mailto:marcmangel marcmangelprotonmail. HYPERLINK mailto:marcmangel com. In an actual setting, the incidence rate remains forever unknown.As described above, the construction of Figure 2 is scientific advice. That

39、is, we developed analytical methods that produce a detailed description of the relationship between group size and risk (the probability of at least one coronavirus-infected person is in the group), but the choice of an acceptable level of risk is a policy decision.For example, a decision-maker migh

40、t consider a 50-50 risk acceptable. That person would then draw a horizontal line from 0.5 on the y axis to intersect the three curves, and from each intersection point draw a vertical line to the x axis to determine the group sizes associated with the 50-50 level of risk for the point, lower, and u

41、pper estimates of the infection rate. In such a case, groups of about511 (using fupper, f, and flower in Eq. 4) will beconsistent with the chosen level of risk. A risk-prone decision-maker might consider an 80% risk acceptable and thus conclude that groups of around 1225 are acceptable. A risk-avers

42、e decision-maker might consider a 20% risk acceptable, with groups of just a few individuals.1.00.8RiskDiscussionThe method embodied in Eqs. 14 is straight- forward, simple to use (it can be implemented in an Excel spreadsheet or using a handheld calculator), and has detailed justification.

43、12Although the discussion of group size and risk was written for policymakers, it also has implications for public health educators. For example, figures similar to Figure 2 can be used in a health education campaign to help individuals visualize what could happen if they are in contact with 5, 10,

44、30, 50, or 100 other individuals outside of their known circle. A visual tool such as a color-coded system will give individuals a way to assess the risk of their behaviors.Finally, we note two limitations. First, this method only identifies the incidence rate from knowledge of surface positivity an

45、d the test errors. In particular, it does not allow us to separate symptomatic and asymptomatic infections. Doing so requires a more complicated version of the methods reported here.Second, this method will not work all the time. For example, when the surface positivity P/T is less than the probabil

46、ity of a false positive test pFP, Eq. 2 gives a negative estimate for the incidence rate. This is clearly a meaningless result. Surface positivity is most likely to be small in the latter stages of the pandemic, when the incidence rate is falling to010203040500 and random fluctuations cause P/T to b

47、e veryGroup SizeThe methods embodied in Eqs. 14 allow one go to from the surface positivity to the risk of groups of different sizes, where risk is the probability that a group of size g includes at least one coronavirus-infected individual. These plots correspond to 1,000 tests, of which 112 are po

48、sitive, when the test errors are pFP = 0.05 and pFN = 0.25. See text for further details on how the plots can be used.Figure 2. Risk versus Group Sizesmall. At this latter stage of the pandemic, a very different method is needed.1312 Mangel and Brown, Operational Analysis for COVID-19 Diagnostic Tes

49、ting.13 See Mangel and Brown, Operational Analysis for COVID-19 Diagnostic Testing.Appendix A The Underlying Mathematical ModelsOur analysis is based on two classical probability distributions. Given the probability of a positive test in Eq. 1, the number of positive tests follows a binomial distrib

50、ution with parameters T and p+( f ).14The range computed in Eq. 3 is based on the Gaussian approximation to the binomial distribution,15 which is highly accurate as long as the expected number of positive tests is bounded away from either 0 or T. Since we are considering test numbers in the thousand

51、s, the expected number of positive tests will be close to 0 or T when (1) the incidence rate is close to 0 (in which case, as discussed above, a different method is needed) or (2) the incidence rate is close to 1 (in which case everyone is infected and the incidence rate is not relevant to the curre

52、nt pandemic). The factor 3.92 in front of the square root in Eq. 3 comes from the 95% compatibility interval for a normal distribution with mean 0 and standard deviation 1.The risk computed in Eq. 4 uses an assumption similar to the binomial distributionthat individuals in the group join independent

53、 of their infection status. Since 1 f is the probability that a single individual is not infected and probabilities are multiplied when they are independent, the probability that all individuals in a group of size g are uninfected is 1 f multiplied by itself g times.14 For example, Feller, Introduct

54、ion to Probability Theory and Its Applications, 146 and following.15 Feller, Introduction to Probability Theory and Its Applications, 174 and following.Appendix B Accounting for Heterogeneity in Test ErrorsEq. 2 requires single values of the test errors to estimate the incidence rate. If one knows t

55、he error probabil- ities for different kinds of tests and when those tests are done (so that a surface positivity is associated with single values for the test errors), Eq. 2 can be repeatedly applied.An alternative is that one has distributions for the test errors and does not know which test corre

56、sponds to a given surface positivity but only that the test errors come from the distributions. In such a case, we can apply the delta-method16 to generalize Eq. 2.We assume that the means, variances, and covariance of test errors are known and denote these by pFNand pFP (mean test errors), Vp , Vp(

57、variance in test errors), and Cov(pFN,pFP) (covariance between testerrors), respectively.FNFPIn a forthcoming report,17 we give the details of the derivation using the delta-method; here we report the result. For ease of notation, we letf (p, p)=P/T pFP.FNFP 1 pFN pFP This is the estimate in Eq. 2 i

58、n which we use the mean values of the test errors. The generalization of Eq. 2 is thenE (f)= f (p, p)+f (pFN , pFP )V+ f (pFN , pFP ) 1 V FNFP (1 pFN pFP )2pFN (1 pFN pFP )2pFP + 2f (pFN , pFP ) 1 Cov(p,p)(5)(1 pFN pFP )2FNFP ,wheredenotes the expectation over the distribution of the test errors.16

59、Hilborn and Mangel, Ecological Detective, 5859.17 Mangel and Brown, Operational Analysis for COVID-19 Diagnostic Testing.OperatiOnal analysis fOr COrOnavirus testing 9BibliographyAllen, Danielle, Sharon Block, Joshua Cohen, Peter Eckersley, M. Eifler, Lawrence Gostin, Darshan Goux, et al. Roadmap to

60、 Pandemic Resilience. Massive Scale Testing, Tracing, and Supported Isolation (TTSI) as the Path to Pandemic Resilience for a Free Society. Cambridge, MA: Edmond J. Safra Center for Ethics at Harvard University, April 20, 2020. HYPERLINK /files/center-for-ethics/files/roadmaptopandemicresilience_upd