Today we talk a little bit about diagnostic interventions. That is to say, medical tests! What are they good for, what do they mean, and how do we interpret them? I'm going to restrict myself to the discussion of tests which have two outcomes: positive and negative. Many tests in fact have many outcomes, which may refine diagnoses into various categories. But for our case, we have this:
In this case, the "Reference Standard" can mean a direct observation, through, for example, autopsy. This is how we measure if diagnostic tests work in the first place. We compare the test outcome which we believe to be positive or negative to a reference standard, which is perfect, but which we would like to replace with a test. After all, if a simple blood test can reliably diagnose a tumor, that save a patient from having to be diagnosed with an invasive biopsy or with an autopsy later. I want you to mentally label the boxes a, b, c, and d, horizontally and then vertically. We'll use those designations later (so, false negative is 'c').
When determining if a test is reliable, we look at the same three basic concepts we looked at for the previous posts: Validity, Results, and Applicability. We test validity of a test by the blinded comparison against a known reference standard, and the reference standard must be used in all subjects of the comparison. We can't only apply the reference standard to, for example, the positive test results. And if the test were to involve multiple outcomes, it must be compared to all possible disease states.
Tests have a couple of basic properties: sensitivity, specificity, and the likelihood ratio. Much is often made of the so called "positive predictive value". Forget about it. It's useless. The sensitivity of the test is how frequently a positive disease state yields a positive test. Note the temporal direction of action here. If we know you have the disease, you will be correctly classified with a positive test at the rate of sensitivity. The specificity of a test is the opposite of this. If we know you do not have the disease, you will be correctly classified with a negative test at the rate of specificity. Sensitivity is equal to a/(a+c). Specificity is equal to d/(b+d).
So what we'd like to know is, if I have a positive (negative) test, what's the probability that I (don't) have the disease? To do that, I need to calculate the likelihood ratio of the test. The first thing we need to know is that there are multiple likelihood ratios(LHR) for each test. In fact, there's one for each outcome. An LHR > 1 means that that outcome increases my odds of having the disease. An LHR < 1 decreases those odds. So, in order to use LHRs, we need to be able to calculate odds from probabilities.
Luckily, this is dead simple. If you have a probability, you can calculate odds very easily. Odds = Probability/(1-Probability). Conversely, Probability = Odds/(Odds+1). Feel free to check the algebra yourself. I'll wait. Really. I'll just be here twiddling my thumbs and debating group heritability with Fonzie the winged lemur. Fine. I knew you weren't going to do it.
We use LHRs because the tests aren't perfect, and because individual patients have different pre-test probabilities from each other, and also from the population used to validate the test. So, for a 2x2 test like we are discussing here, the LHR+ = Sensitivity/(1-Specificity), and the LHR-=(1-Sensitivity)/Specificity. Again, when the LHR >1 that increases the odds that the subject has the disease. It may be easier in this case to not thing of tests as having positive and negative results, but simply "result A" and "result B". Because sometimes, the "negative" test result is the one which indicates presence of the disease.
So, to use an example from class, consider a white blood count test for appendicitis which has 76% sensitivity (that is, people with appendicitis will have a positive test 76% of the time), and 52% specificity (which is, people without appendicitis will have a negative test 52% of the time). Suppose I go in to the physician with a painful belly. Sawbones thinks I have about a 20% chance of having appendicitis based on my presentation. That's pretty unlikely, but worth refining with a test. So, let's do some arithmetic.
Let's figure out my a priori odds. So, 0.20/(1-0.2) = .25. Now, let's calculate the LHRs of this test. The LHR+ = 0.76/(1-0.52) = 1.58. The LHR- = (1-0.76)/.52 = 0.46. So let's say my test was positive. If we're naive, we might think that a positive test with correctly classifies the disease state with 76% sensitivity, then I should have a 76% chance of having appendicitis. But that's switching the order of things. 76% of people with appendicitis will have a positive test. It does not follow that 76% of people with a positive test have appendicitis. The test isn't specific enough for that. Many things which are not appendicitis also yield positive test results.
So, we take my a priori odds, 0.25, multiply by the LHC+ (I had a positive test), and get 6.32. These are my a postiori odds. To find out how this has changed my likelihood of having the disease, We convert those odds back t a probability, and get 0.28, or 28%. So, this positive test, which has pretty good sensitivity, has only increased my probability of actually having appendicitis by 8%. Similarly, if you repeat this with a negative test and find that my chances would go from 20% to 10%.
However, suppose this test were better. Suppose instead of 76% sensitivity and 52% specificity, we have a test which is 85% sensitive and 90% specific. Now, what does a positive test mean, assuming I have the same 20% a priori probability. Well, my a priori odds are the same. .25. But my LHR+ = 0.85/(1-0.9) = 8.5. So a positive test means I multiply my odds by 8.5. Which means my a postiori odds are 34. Which yield an a postiori probability of 68%. With a test this good, and a 20% chance going in, Sawbones is prepping the OR. Similarly, LHR- = (1-0.85)/.9 = .167. So if I had a negative test, my odds go to .67, and my chances of having appendicitis are 4.2%. Go on home, Capt. Stomach Ache.
So, this is how we use diagnostic tests to inform medical decisions. Quiz your physician next time you're in the emergency room. Tell 'em you read it on a blog. Do it. I dare you*.
*Dr24hours and Scientopia.org assume no responsibility for adverse outcomes from this foolish, absurd, and inadvisable course of action. Don't piss off your doctor in the emergency room. Or elsewhere. At least, not with any expectation of relief from me. Or Scientopia. Or Fonzie.