From: bruck@math.usc.edu (Ronald Bruck) Newsgroups: sci.math.num-analysis Subject: Re: 90% acurate test done twice is 99% acurate Date: 18 Jul 1998 11:56:12 -0700 In article , Peter L. Montgomery wrote: >In article <35afafff.0@news1.igateway.com> "Bean" writes: >>I went to the doctor to get a test done for strep thoat. >>(but enough about my personal life) anyway the doctor >>said that the quick test for step is only 90% acurate and >>that he does two tests at the same time and then the test >>is 99% acurate. I assume he means that the test is positive >>90% if you have strep. Somehow this doesn't sit right with >>me, however I can't think of why it is not true. I'm thinking >>whatever caused the first test to be wrong would also cause >>the second test to be wrong. > >>If anyone could explain if this is true and why I would like >>to hear the explaination. Thank you. > > When the two tests are independent, the conclusion is accurate. >The first test is inaccurate 100% - 90% = 10% of the time. >The second test is inaccurate 10% of the time. >The assumption of independence means both will be inaccurate >(10%)*(10%) = 1% of the time. If both tests turn out negative, >AND IF THE TESTS ARE INDEPENDENT, you can be 99% confident >that you don't have strep throat. Uh, not quite. You can be 99% certain that it's not true that both tests were wrong; and that's undoubtedly what the doctor means. But if he claims (on receiving two negative tests) that there's a 99% chance the patient doesn't have strep, he's misunderstood conditional probability. In fact, he's considerably OVERSTATED the chances the patient has strep (provided strep is a rare disease). If the doctor wants to quote a probability that the patient has strep, GIVEN THAT both tests were negative, he MUST KNOW THE INCIDENCE OF STREP IN THE GENERAL POPULATION. That's because this is a Bayes problem. The "90% accurate" figure means that IF a person has strep, there's a 90% chance the test returns positive and a 10% chance it returns negative (i.e. 10% false negative). In the absence of any other indication, let's assume it also means a 10% incidence of false positive. The easiest way to visualize this is to draw a tree: label a point as root; then take two branches from the root, to "strep" and "not strep". Let the probabilities for those branches be p and 1-p. Now from each of the "strep" and "not strep" nodes, draw two branches to "1+" and "1-" nodes, indicating the results of the first test. The probabilities attached to these branches are 0.9 for strep->1+ and for not strep->neg, and 0.1 for strep->1- and for not strep->1+. From each of THESE four branches, draw branches to "2+" and "2-". Attach the appropriate probabilities, CHECKING BACK for whether you're on the "strep" branch or the "not strep" branch. Now suppose we want to find the probability that the patient has strep, GIVEN THAT both tests were negative. We are therefore restricting our sample space to the branches Root ------> strep -----> 1- ------> 2- Prob: p 0.1 0.1 and Root ------> not strep -----> 1- ------> 2- Prob: 1-p 0.9 0.9 which have probabilities 0.01*p and 0.81*(1-p), respectively. And the probability that the patient has strep, GIVEN that he has two negative tests--which are assumed to be independent of each other--is the first of these, divided by their sum: 0.01*p/(0.01*p + 0.81*(1-p)). Of course, this is the familiar Bayesian formula. Some sample values (assuming that strep throat is a relatively rare disease in the general population): p P(strep|both neg) ============================ 1% 0.000125 2% 0.000252 3% 0.000382 4% 0.000649 In other words, if the incidence of strep throat is just 1%, the chances are 1/80 of 1% that the patient actually has strep; the incidence of strep in the population must reach 45% before the probability that the patient has strep, given both tests are negative, reaches 1%. The question of whether the tests are really independent of each other would seem to me to require clinical trials. Anything else would be guesses; well, perhaps the lab is holding that missing 3.4cc of O.J. Simpson's blood, and looking to get rid of it... --Ron Bruck