See a likelihood ratio in a study but don’t know how to apply it? Learn how to manipulate likelihood ratios just like an expert diagnostician.

We use diagnostic tests to change our understanding of the probability that a patient has a disease.  For instance, if you’re concerned your patient has heart failure, you might auscultate an S3. Then, your post test probability that the patient has a disease should go way up. But how do we actually quantify how much the probability goes up with a diagnostic test? We can do this with likelihood ratios. The post test odds of a patient having disease are equal to the pre test odds times the likelihood ratio.

You might be thinking, what are odds and how are they different from probability?  Probability is the chance an event will happen, expressed as a fraction or percentage of the total. So for instance, here there are four teal and six purple people. The probability of selecting teal is 4 out of 10 or 40%. Odds, on the other hand, is the ratio of an event happening versus not happening. So here, the odds of selecting teal are 4 over 6, or 0.67.  So let’s go back to our patient with heart failure. How does our post test probability change when we hear an S3? We can use this figure, called the Fagan Nomogram, to visualize converting in between odds and probability.

Start with your pre test probability, which you’ll find on the left side of the graph. Let’s just say that the pre test probability here was 50%. In the middle of the graph, you find the likelihood ratio. The S3 has a very strong likelihood ratio of 11 for the diagnosis of heart failure. To get the post test probability, you draw a line from the pre test probability through the likelihood ratio to get to the post test probability, which in this case is well over 90%.

What your pre test probability is really matters. So let’s say you had another patient for whom the pre test probability was only 1%. Even for that same strong likelihood ratio of 11, the post test probability would still only be 10%.  Some rules of thumb can be helpful to interpret likelihood ratios without using the nomogram.

In general, I think of a positive likelihood ratio of 1 to 2 as weak, not really moving my diagnostic needle much at all, 2 to 5 is okay, moving it somewhat, and greater than 5 is strong, substantially moving my diagnostic needle. Let’s recap. We can think of diagnostic tests as transforming pre test probability into post test.

Post test odds is equal to the likelihood ratio times the pre test odds.  Odds is the ratio of an event happening versus not happening. whereas probability is expressed as a fraction of the total. The Fagan Nomogram is a visual tool to convert between pre test and post test probability.  And a good benchmark is to consider a likelihood ratio of 1 to 2 as weak, 2 to 5 as moderate, and greater than 5 as strong.

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