"Is India testing enough?" It seems to be the question on everybody's mind. With the opposition, media, and intellectuals alike hammering the government for not testing enough, let's look into the statistical mathematics of medical testing.
A medical test, like the one we use to detect COVID-19 virus (Swab tests, RT-PCR) should ideally give us a 'Yes or No' answer, but is that the case? No. In reality, no test is 100% accurate. Broadly speaking there are 4 different outcomes of a particular test concerning an individual.
TRUE POSITIVE- You are infected and the test says that too. (TP)
FALSE POSITIVE- You are not infected but the test says that you are. (FP)
TRUE NEGATIVE- You are not infected and the test says that too. (TN)
FALSE NEGATIVE- You are infected but the test says that you aren't. (FN)
TRUE POSITIVE- You are infected and the test says that too. (TP)
FALSE POSITIVE- You are not infected but the test says that you are. (FP)
TRUE NEGATIVE- You are not infected and the test says that too. (TN)
FALSE NEGATIVE- You are infected but the test says that you aren't. (FN)
Now the term 'accuracy' has a different connotation in medical testing, it generally implies how well the test is doing in relation to the sum of TP and TN as a percentage of the total tests. But this metric disregards the two other important outcomes FP and FN. To understand why FP and FN numbers are so important, let us look at the varying degree of risk associated with each of the four outcomes in the context of the COVID-19 disease.
TRUE POSITIVE(TP)
In this case, you have been accurately detected as a COVID-19 positive patient and now the ordeal begins. Depending on your underlying health condition, and severity of your symptoms, you will either be asked to self-isolate or be admitted into a hospital. The costs ascribed to this depend on the situation you find yourself in. But at least you have been assessed correctly.
TRUE NEGATIVE(TN)
In this case, you have been accurately detected as a COVID-19 negative patient I.e. you do not have COVID-19 and can go back home and keep following the Social Distancing Guidelines.
FALSE POSITIVE(FP)
In this case, you have been wrongly detected as a COVID-19 positive patient. As with TP, you'd have two scenarios depending on whether you are asked to self-isolate or be admitted. The only loss you'll have is the emotional turmoil for nothing. Though if you are admitted to a hospital, it'll greatly increase the burden on the healthcare system.
FALSE NEGATIVE(FN)
In this case, you have been wrongly detected as a COVID-19 negative patient I.e. even though you have the virus the test says otherwise. This mostly happens to asymptomatic patients. This is the worst situation as even though you have the virus, you'll probably go untreated and also become a huge threat to society in the form of an 'Asymptomatic Spreader'.
TRUE POSITIVE(TP)
In this case, you have been accurately detected as a COVID-19 positive patient and now the ordeal begins. Depending on your underlying health condition, and severity of your symptoms, you will either be asked to self-isolate or be admitted into a hospital. The costs ascribed to this depend on the situation you find yourself in. But at least you have been assessed correctly.
TRUE NEGATIVE(TN)
In this case, you have been accurately detected as a COVID-19 negative patient I.e. you do not have COVID-19 and can go back home and keep following the Social Distancing Guidelines.
FALSE POSITIVE(FP)
In this case, you have been wrongly detected as a COVID-19 positive patient. As with TP, you'd have two scenarios depending on whether you are asked to self-isolate or be admitted. The only loss you'll have is the emotional turmoil for nothing. Though if you are admitted to a hospital, it'll greatly increase the burden on the healthcare system.
FALSE NEGATIVE(FN)
In this case, you have been wrongly detected as a COVID-19 negative patient I.e. even though you have the virus the test says otherwise. This mostly happens to asymptomatic patients. This is the worst situation as even though you have the virus, you'll probably go untreated and also become a huge threat to society in the form of an 'Asymptomatic Spreader'.
Now that we have a rough idea of the qualms of medical testing, let's understand the application of Bayes' Theorem on COVID-19 testing. To understand Bayes' Theorem in-depth read "The Theory That Would Not Die: How Bayes' Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy"
The two fundamentals behind Bayes' Theorem are that first, no test is a 100% accurate or perfect (as explained above) and second is the concept of "prior probability" or "base rate" I.e. the probability of an event computed before the collection of new data based on intuition or belief. Let's look at an example to understand it better.
Let's assume that we have a highly accurate test with 99.99% accuracy (1 in 1000 chance of an FP or FN) with a prior probability rate of 1%. Now let's take 100,000 people, statistically, 1000 of them have the disease. This test will accurately denote 999 of them as TP and 1 as FN. Among the remaining 99000 people, 99 are denoted as 'Yes' as in have the disease and the rest 98,901 people are denoted as 'No' as in don't have the disease. So, of the people marked Yes, how many have the disease? Its 999/ (999+99) *100= 90.98% which amounts to about 10/11 people. So, using a highly accurate test when you test people 'randomly' you find that 1 in 10 people you find positive are not actually.
Now let's apply the principle of 'prior probability' I.e. in the context of COVID-19 those people who have travel history to affected countries, respiratory infections, or show symptoms. Let's assume the prior probability rate as 10% under the above conditions. Thus, 10,000 of the total 100,000 have the disease. The test denotes 9990 people as Yes and 10 people as No. Now of the remaining 90,000, 90 are denoted as Yes, and the other 89,910 as No. Now, the percentage of people who have the disease of those who were denoted as Yes is 9990/ (9990+90) *100= 99.10% way more accurate than the 90.98% from earlier albeit highly sensitive to the prior probability rate.
The data presented here is based on a highly accurate hypothetical test, in reality, the test for the COVID-19 virus could be, or most definitely is significantly less accurate. Hence, an effective way of testing for the virus would be to test when the prior probability is high I.e. individuals with travel history, those who were in contact with infected people, those with respiratory issues, those who show heavy symptoms, those who attended the Nizamuddin congregation, etc. Which is what the Indian model seems to be.
In no way does this article or data imply that testing shouldn't happen, only that individuals with lesser prior probability needn't be tested as vehemently as those with a substantially higher prior probability considering India is in lockdown and also the financial constraints of testing a population of 1.3 billion people
 Testing Bayes' Theorem){kind=link}
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