{"id":58,"date":"2020-09-28T10:40:58","date_gmt":"2020-09-28T14:40:58","guid":{"rendered":"http:\/\/sites.miamioh.edu\/the-center-for-analytics-and-data-science\/?p=58"},"modified":"2020-09-28T10:40:58","modified_gmt":"2020-09-28T14:40:58","slug":"my-covid-19-test-is-positive-do-i-really-have-it","status":"publish","type":"post","link":"https:\/\/sites.miamioh.edu\/the-center-for-analytics-and-data-science\/2020\/09\/my-covid-19-test-is-positive-do-i-really-have-it\/","title":{"rendered":"My COVID-19 test is positive \u2026 do I really have it?"},"content":{"rendered":"\n

This blog post is reproduced from\u00a0<\/strong>ISI Statisticians React to the News<\/strong><\/em>\u00a0blog (<\/strong>https:\/\/blog.isi-web.org\/react<\/strong><\/a>). It was released on 25 August 2020 <\/strong><\/p>\n\n\n\n

Michael\nDeWine, governor of the state of Ohio in the United States, provided a visible\nexample of the  properties of screening tests.<\/p>\n\n\n\n

Gov. Mike DeWine of Ohio Tests Positive, Then Negative, for Coronavirus<\/em><\/a> by\nSarah Mervosh, The New York Times (6 August 2020 with 7 August 2020 update).<\/p>\n\n\n\n

After\ntesting positive for COVID-19 on a rapid antigen test, he missed an opportunity\nto meet with the US president who was visiting DeWine\u2019s state. After DeWine was\ntested again using a slower, more accurate (RT-PCR) test, he was negative for\nCOVID-19. A additional test administered a day later also was negative.<\/p>\n\n\n\n

Are\nthere benefits in having a rapid, less accurate test as well as having a\nslower, more accurate test? Let\u2019s consider what accuracy means in these tests\nand why you might be willing to tolerate different errors at different\ntimes. <\/p>\n\n\n\n

I won\u2019t\naddress how these tests are evaluating different biological endpoints. I\u2019ve\nbeen impressed at how national<\/a> and local<\/a> sources have\nworked to explain the differences between tests that look for particular\nprotein segments or for genetic material characteristic of the virus. Richard\nHarris (National Public Radio in the US) also provided a nice discussion<\/a> of\nreliability of COVID-19 tests that might be of interest. <\/p>\n\n\n\n

I want\nto talk about mistakes, errors in testing. No test is perfectly accurate.\nAccuracy is good but accuracy can be defined in different ways, particularly in\nways that reflect errors in decisions that are made. Two simple errors are\ncommonly used when describing screening tests \u2013 saying someone has a disease\nwhen, in truth, they don\u2019t (sorry Governor DeWine) or saying someone is disease\nfree when, in truth, they have the disease. Governor DeWine had 3 COVID-19\ntests \u2013 the first rapid test was positive, the second and third tests were\nnegative. Thus, we assume his true health status is disease free.<\/p>\n\n\n\n

These\nerrors are called false positive and false negative errors. (For those of you\nwho took introductory statistics class in a past life, these errors may have been\nlabeled differently: false positive error = Type I error and false negative\nerror = Type II error.)  Testing concepts include the complements of\nthese errors \u2013 sensitivity is the probability a test is positive for people\nwith the disease (1 \u2013 false negative error rate) and specificity is the\nprobability a test is negative for disease-free people (1 \u2013 false positive\nrate). If error rates are low, sensitivity and specificity are high.<\/p>\n\n\n\n

It is\nimportant to recognize these errors can only be made when testing distinct\ngroups of people. A false positive error only<\/em> can be made when\ntesting disease-free people. A false negative error only<\/em> can\nbe made when testing people with the disease. An additional challenge is that\nthe real questions people want to ask are \u201cDo I have the disease if I test\npositive?\u201d and \u201cAm I disease free if my test is negative?\u201d Notice these\nquestions involve the consideration of two other groups \u2013people who test<\/em> positive\nand people who test<\/em> negative!<\/p>\n\n\n\n

Understanding\nthe probabilities<\/h3>\n\n\n\n

Probability\ncalculations can be used to understand the probability of having a disease\ngiven a positive test result \u2014 if you know the false positive error rate, the\nfalse negative error rate and the percentage of the population with the\ndisease, along with testing status of a hypothetical population. The British\nMedical Journal (BMJ) provides a nice web calculator<\/a> for exploring the\nprobability that a randomly selected person from a population has the disease\nfor different test characteristics. In addition, the app interprets the\nprobabilities in terms of counts of individuals from a hypothetical population\nof 100 people classified into 4 groups based upon true disease status (disease,\nno disease) and screening test result (positive, negative). <\/p>\n\n\n\n

It is\nworth noting that these probabilities are rarely (if ever) known and can be\nvery hard to estimate \u2013 particularly when changing.  In real life,\nthere are serious challenges in estimating the numbers that we get fed into\ncalculators such as this \u2013 but that\u2019s beyond scope of this\npost.  Regardless, it is fun and educational to play around with the\ncalculator to understand how things work.<\/p>\n\n\n\n

These\nerror rates vary between different test types and even for tests of the same\ntype. One challenge that I had in writing this blog post was obtaining error\nrates for these different tests. Richard Harris (NPR)<\/a> reported\nthat PCR false positives from the PCR test were approximately 2%, with\nvariation attributable to the laboratory conducting the study and the\ntest. National Public Radio<\/a> reported\nthat one rapid COVID-19 test had a false negative error rate of approximately\n15% while better tests have false negative tests less than 3%.  One\ncomplicating factor is that error rates  appear to depend on when the test\nis given in the course of disease.<\/p>\n\n\n\n

Examples<\/h3>\n\n\n\n

The\nfollowing examples illustrate a comparison of tests with different accuracies\nin communities with different disease prevalence.<\/p>\n\n\n\n

Community\nwith low rate of infection<\/h4>\n\n\n\n

A\nrecent story about testing in my local paper reported 1.4% to 1.8% of donors to\nthe American Red Cross had COVID-19. Considering a hypothetical population with\n100 people, only 2 people in the population would have the disease and 98 would\nbe disease free. <\/p>\n\n\n\n

Rapid,\nless accurate test<\/strong>:  Suppose we have a rapid test with a 10% false positive\nerror rate (90% specificity), 15% false negative error rate (85% sensitivity)\nand 2% of people tested are truly positive. With these error rates, suppose\nboth of the people with the disease test positive and 10 of the 98 disease-free\npeople test positive. Based on this, a person with a positive test (2 + 10= 12)\nhas about a 16% (2\/12 x 100) chance of having the disease, absent any other information\nabout exposure. <\/p>\n\n\n\n
\n  \n <\/td>\n Disease\n <\/td>\n No Disease\n <\/td>\n Total\n <\/td>\n  \n <\/td><\/tr>
\n Test +\n <\/td>\n 2\n <\/td>\n 10 
\n (98 x .10)\n <\/td>		\n 12\n <\/td>\n 2\/12
\n (16%)\n <\/td><\/tr>
\n Test \u2013\n <\/td>\n 0 
\n (2 x .15)\n <\/td>		\n 88\n <\/td>\n 89\n <\/td>\n  \n <\/td><\/tr>
\n  Total\n <\/td>\n 2\n <\/td>\n 98\n <\/td>\n 100\n <\/td>\n  \n <\/td><\/tr><\/tbody><\/table>\n\n\n\n

For a hypothetical population of 100\npeople with 2% infected, a false positive rate of 0.10, and a false negative\nrate of 0.15, the chance of having the disease given a positive test is about\n16%.<\/em><\/p>\n\n\n\n

Slower,\nmore accurate test<\/strong>: Now, suppose we have a more accurate test with a 2% false\npositive error rate (98% specificity) and 1% false negative error rate (99%\nsensitivity). With these error rates, both of the people with the disease test\npositive and 2 of the 98 disease-free people test positive. Based on this, a\nperson with a positive test (2 + 2= 4) has about a 50% (2\/4) chance of having\nthe disease. <\/p>\n\n\n\n
\n  \n <\/td>\n Disease\n <\/td>\n No Disease\n <\/td>\n Total\n <\/td>\n  \n <\/td><\/tr>
\n Test +\n <\/td>\n 2\n <\/td>\n 2
\n (98 x .02)\n <\/td>		\n 4\n <\/td>\n 2\/4
\n (50%)\n <\/td><\/tr>
\n Test \u2013\n <\/td>\n 0  
\n (2 x .01)\n <\/td>		\n 96\n <\/td>\n 96\n <\/td>\n  \n <\/td><\/tr>
\n  Total\n <\/td>\n 2\n <\/td>\n 98\n <\/td>\n 100\n <\/td>\n  \n <\/td><\/tr><\/tbody><\/table>\n\n\n\n

For a hypothetical population of 100\npeople with 2% infected, a false positive rate of 0.02, and a false negative\nrate of 0.01, the chance of having the disease given a positive test is about\n50%.<\/em><\/p>\n\n\n\n

Community\nwith a higher rate of infection<\/h4>\n\n\n\n

Now\nsuppose we test in a community where 20% have the disease. Here, 20 people in\nthe hypothetical population of 100 have the disease and 80 are disease free.\nThis 20% was based on a different news source suggesting that 20% was one of\nthe highest proportions of COVID-19 in a community in the US.<\/p>\n\n\n\n

Rapid,\nless accurate test<\/strong>: Consider what happens we use a rapid test with a 10%\nfalse positive error rate (90% specificity) and 15% false negative error rate\n(85% sensitivity) in this population. With the error rates described for this\ntest, 17 of the 20 people with disease test positive and 8 of the 80\ndisease-free people test positive. Based on this, a person with a positive test\n(17 + 8 = 25) has about a 68% (17\/25) chance of having the disease without any\nadditional information about exposure.<\/p>\n\n\n\n
\n  \n <\/td>\n Disease\n <\/td>\n No Disease\n <\/td>\n Total\n <\/td>\n  \n <\/td><\/tr>
\n Test +\n <\/td>\n 17\n <\/td>\n 8
\n (80 x .10)\n <\/td>		\n 25\n <\/td>\n 17\/25
\n (68%)\n <\/td><\/tr>
\n Test \u2013\n <\/td>\n 3 
\n (20 x .15) \n <\/td>		\n 72\n <\/td>\n 75\n <\/td>\n  \n <\/td><\/tr>
\n  Total\n <\/td>\n 20\n <\/td>\n 80\n <\/td>\n 100\n <\/td>\n  \n <\/td><\/tr><\/tbody><\/table>\n\n\n\n

For a hypothetical population of 100\npeople with 20% infected, a false positive rate of 0.10, and a false negative\nrate of 0.15, the chance of having the disease given a positive test is about\n68%.<\/em><\/p>\n\n\n\n

Slower,\nmore accurate test<\/strong>: Now suppose we apply a more accurate test with a 2% false\npositive error rate (98% specificity) and 1% false negative error rate (99%\nsensitivity) to the same population. In this case, all 20 people with the\ndisease test positive and 2 of the 80 disease-free people test positive. Based\non this, a person with a positive test (20 + 2 = 22) has about a 90% (20\/22)\nchance of having the disease. <\/p>\n\n\n\n
\n  \n <\/td>\n Disease\n <\/td>\n No Disease\n <\/td>\n Total\n <\/td>\n  \n <\/td><\/tr>
\n Test +\n <\/td>\n 20\n <\/td>\n 2
\n (80 x .02)\n <\/td>		\n 22\n <\/td>\n 20\/22
\n (90%)\n <\/td><\/tr>
\n Test \u2013\n <\/td>\n 0
\n (20 x .01) \n <\/td>		\n 78\n <\/td>\n 78\n <\/td>\n  \n <\/td><\/tr>
\n  Total\n <\/td>\n 20\n <\/td>\n 80\n <\/td>\n 100\n <\/td>\n  \n <\/td><\/tr><\/tbody><\/table>\n\n\n\n

For a hypothetical population of 100\npeople with 20% infected, a false positive rate of 0.02, and a false negative\nrate of 0.01, the chance of having the disease given a positive test is about\n90%.<\/p>\n\n\n\n

Returning\nto the big question<\/h3>\n\n\n\n

So, if\nyou test positive for COVID-19, do you have it? If you live in a community with\nlittle disease and use a less accurate rapid test, then you may only have a 1\nin 6 chance (16%) of having the disease (absent any additional information\nabout exposure). If you have a more accurate test, then the same test result\nmay be associated with a 50-50 chance of having the disease. Here, you might\nwant to have a more accurate follow up test if you test positive on the rapid,\nless accurate test.  If you live in a community with more people who\nhave the disease, both tests suggest you are more likely than not to have the\ndisease. Recognize that these tests are being applied in situations with\nadditional information being available including whether people exhibit\nCOVID-19 symptoms and\/or live or work in communities with others who have\ntested positive.<\/p>\n\n\n\n

Final\nthoughts<\/h3>\n\n\n\n

You\nmight be interested in controlling different kinds of errors with different\ntests. If you are screening for COVID-19, you might want to minimize false\nnegative errors and accept potentially higher false positive error rates. A\nfalse positive error means a healthy disease-free person is quarantined and\nunnecessarily removed from exposing others. A false negative error means a\nperson with disease is free to mix in the population and infect others. So,\ndoes Governor DeWine have COVID-19? Ultimately, the probability that the\ngovernor is disease-free reflects the chance of being disease-free given one\npositive result on a less accurate test and two negative results from more\naccurate tests. The probability he is disease-free is very close to one, given\nno other information about exposure.<\/p>\n\n\n\n

About the Author<\/h2>\n\n\n\n
\"\"
Dr. A. John Bailer<\/a> is a University Distinguished Professor and Chair in the Department of Statistics at Miami University. <\/figcaption><\/figure><\/div>\n","protected":false},"excerpt":{"rendered":"

This blog post is reproduced from\u00a0ISI Statisticians React to the News\u00a0blog (https:\/\/blog.isi-web.org\/react). It was released on 25 August 2020 Michael DeWine, governor of the state of Ohio in the United States, provided a visible example of the  properties of screening tests. Gov. Mike DeWine of Ohio Tests Positive, Then Negative, for Coronavirus by Sarah Mervosh, The New […]<\/p>\n","protected":false},"author":3098,"featured_media":59,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_bbp_topic_count":0,"_bbp_reply_count":0,"_bbp_total_topic_count":0,"_bbp_total_reply_count":0,"_bbp_voice_count":0,"_bbp_anonymous_reply_count":0,"_bbp_topic_count_hidden":0,"_bbp_reply_count_hidden":0,"_bbp_forum_subforum_count":0,"_s2mail":"","footnotes":""},"categories":[2],"tags":[],"class_list":["post-58","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-faculty-research"],"_links":{"self":[{"href":"https:\/\/sites.miamioh.edu\/the-center-for-analytics-and-data-science\/wp-json\/wp\/v2\/posts\/58","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.miamioh.edu\/the-center-for-analytics-and-data-science\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.miamioh.edu\/the-center-for-analytics-and-data-science\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.miamioh.edu\/the-center-for-analytics-and-data-science\/wp-json\/wp\/v2\/users\/3098"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.miamioh.edu\/the-center-for-analytics-and-data-science\/wp-json\/wp\/v2\/comments?post=58"}],"version-history":[{"count":0,"href":"https:\/\/sites.miamioh.edu\/the-center-for-analytics-and-data-science\/wp-json\/wp\/v2\/posts\/58\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/sites.miamioh.edu\/the-center-for-analytics-and-data-science\/wp-json\/wp\/v2\/media\/59"}],"wp:attachment":[{"href":"https:\/\/sites.miamioh.edu\/the-center-for-analytics-and-data-science\/wp-json\/wp\/v2\/media?parent=58"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.miamioh.edu\/the-center-for-analytics-and-data-science\/wp-json\/wp\/v2\/categories?post=58"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.miamioh.edu\/the-center-for-analytics-and-data-science\/wp-json\/wp\/v2\/tags?post=58"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}