Likelihood Ratios: What Actually Changes Your Mind

Two tests, the same accuracy, very different jobs

Imagine two tests validated on the same population, half of whom actually have the disease. Test One catches 95 percent of true cases but only correctly clears 60 percent of healthy people. Test Two catches just 60 percent of true cases but correctly clears 95 percent of healthy people. Despite looking almost like mirror images of each other, both tests land on the exact same overall accuracy, about 77.5 percent of all results correct.

If a validation study only reported that single accuracy number, the two tests would look identical. In practice, they are suited to opposite jobs. Overall accuracy blends sensitivity and specificity into one number and hides the shape of the tradeoff underneath it. The tool that actually reveals that shape is called a likelihood ratio.

What a likelihood ratio actually measures

A positive likelihood ratio, written LR+, is how many times more likely a positive result is in someone who truly has the disease compared to someone who does not. It is calculated as sensitivity divided by 1 minus specificity. A negative likelihood ratio, LR-, is how many times more likely a negative result is in someone who truly has the disease compared to someone who does not, calculated as 1 minus sensitivity divided by specificity.

Running the numbers on the two tests above resolves the mystery. Test One, high sensitivity and modest specificity, has an LR+ of about 2.4 but an LR- of about 0.08. Test Two, modest sensitivity and high specificity, has an LR+ of 12 but an LR- of about 0.42. Test One is the stronger tool for ruling a disease out when it comes back negative, sometimes remembered with the mnemonic SnNout, high sensitivity, a negative result rules the disease out. Test Two is the stronger tool for ruling a disease in when it comes back positive, remembered as SpPin, high specificity, a positive result rules the disease in.

From pretest probability to posttest probability

A likelihood ratio only becomes useful once it is combined with how likely the patient in front of you was to have the disease before the test was ever run, the pretest probability. Likelihood ratios multiply odds, not probabilities directly, so the pretest probability first has to be converted to odds, multiplied by the likelihood ratio, then converted back to a probability.

Take a patient with a pretest probability of 20 percent, odds of 1 to 4. A positive result on Test Two, LR+ of 12, multiplies those odds to 3 to 1, which converts back to a posttest probability of 75 percent. A negative result on Test One, LR- of about 0.08, shrinks those same starting odds to roughly 0.02, a posttest probability of about 2 percent. The same starting suspicion moves to a dramatically different place depending on which test was used and which way the result came back.

Rules of thumb for interpreting an LR

An LR+ above 10 or an LR- below 0.1 usually produces a large, often decisive shift in probability. An LR+ between 2 and 5, or an LR- between 0.2 and 0.5, produces a moderate shift, sometimes helpful, rarely decisive on its own. An LR close to 1 in either direction means the result barely changes what you believed before the test, no matter how official it looks on the page.

Overall accuracy tells you how often a test is right. A likelihood ratio tells you how much a single result should actually change your mind about the patient in front of you.

Key Takeaways

• Two tests can share the exact same overall accuracy while having very different sensitivity and specificity, and therefore very different clinical usefulness.

• LR+ measures how much more likely a positive result is in someone with the disease compared to someone without it; LR- does the same for a negative result.

• High sensitivity tests are strongest at ruling a disease out with a negative result, while high specificity tests are strongest at ruling a disease in with a positive result.

• Likelihood ratios combine with pretest probability, converted to odds, to produce a posttest probability, turning a raw result into an updated estimate for a specific patient.

• An LR above 10 or below 0.1 usually shifts probability substantially, while an LR near 1 barely changes anything regardless of how the result is reported.

Up Next:

Next, we will step away from diagnostic testing and into measuring treatment effects, starting with relative risk and odds ratio, two numbers that describe the same effect but can look very different depending on how the outcome is measured.