Likelihood Ratios

Imagine that in trying to gather information about a particular patient you are seeing you come across an article that describes a diagnostic test that may help better confirm your suspected diagnosis. For example, let us assume that your patient is suffering from back pain, and you suspect that she might have a herniated intervertebral disc. You believe this because your work-up for that patient has found that there is pain in the low back which radiates down the leg and into the lateral side of the foot, that certain motions are quite painful, that this came on after a doing physical labor, and that it has worsened over the past three days. In addition, you have a number of positive diagnostic tests which have occurred, but they seem mildly contradictory and you feel that it may be due to the severe pain the patient is experiencing. What can you do to get a better handle on what is happening?

You can look at the likelihood ratio for one of the diagnostic tests you did. Why look at this? Well, the main reason is that doing so will help move you from your estimate of the likelihood the patient has a disc herniation (also called the pre-test probability) to a more accurate estimate (called the post-test probability of the target disorder). The likelihood ratio is a tool which moves us from pre-test probability to post-test probability.

Consider: the presence of a positive straight leg raise test is seen as indicative of a disc herniation. But it is not conclusive and there may be other possible explanations for its presence. But certainly, with the other information you have collected from your patient, a large likelihood ratio will likely move to you a more positive confirmation of disc herniation, and also may lead you to consider either diagnostic imaging or to a particular management protocol.

A likelihood ratio is the relative likelihood that a given test would be expected to be positive (or negative) in a patient with a disorder as opposed to one without (1). Likelihood ratios can be calculated easily from 2x2 contingency tables used to calculate sensitivity and specificity; a likelihood ratio is actually [Sensitivity/(1- Specificity)]. For the actual math involved, please see http://w3.palmer.edu/lawrence/RSCH841/Reliability%20and%20Validity.doc.

But here is the key to understanding a likelihood ratio. It indicates the extent to which a diagnostic test will increase (or decrease) the pretest probability of the target disorder. LRs of 1 mean that the pre and post-test probabilities are exactly the same; if greater than 1, it indicates an increase in the probability that the target disorder is present (and the greater this number, the greater the probability), while the converse is true for likelihood ratios of less than 1. With a likelihood ratio, you can use a nomogram (such as the one found at http://www.cebm.net/index.aspx?o=1161) to convert the pretest probability to a post-test probability.

In our case, let’s begin by assuming that the pre-test probability of the presence of herniated disc is 50%; this is based on our own clinical expertise plus any literature we may have read. From the paper we found, we see that the likelihood ratio for a straight leg raise is 12. Plugging this into the online nomogram shows us that the post-test probability of a disc herniation is now 94%. One could feel quite comfortable that this patient does indeed have a disc herniation, and can proceed accordingly.

You can find tables of likelihood ratios derived from the scientific literature, such as those seen in the Rational Clinical Examination series from JAMA. They are the most valuable test we have for the use of diagnostic tests.

References

1. Guyatt G, Rennie D, Meade MO, Cook DJ. Users’ Guides to the medical literature: a manual for evidence-based clinical practice, 2nd edition. New York, NY; McGraw Hill, 2008:426-430
2. Sackett D, Rennie D. The science of the art of the clinical examination. JAMA 1992;267:2650-2652